Algebraic and Combinatorial Methods in Operations Research

ALGEBRAIC AND COMBINATORIAL METHODS IN OPERATIONS RESEARCH

annals of discrete mathematics Generul Ediror Peter L. HAMMER, Rutgers University. New Brunswick, NJ, U.S.A. Advisory Editors C . BERGE, UniversitC de Paris M . A . HARRISON, UniversityofCalifornia, Berkeley, CA, U.S.A. V. KLEE, University of Washington, Seattle. WA, U.S.A. J . H.VAN LINT, California Instituteof Technology, Pasadena. CA, U.S.A. G . - t . ROTA, Massachusetts Institute of Technology, Cambridge, MA, U.S.A

NORTH-HOLLAND-AMSTERDAM

NEW YORK

OXFORD

NORTH-HOLLAND MATHEMATlCS STUDIES

95

Annals of Discrete Mathematics(19) General Editor: Peter L. Hammer Rutprs' University, New Bmnswick, MJ. U.SA

ALGEBRAIC AND COMBINATORIAL METHODS IN OPERATIONS RESEARCH Proceedings of the Workshop on Algebraic Structures in Operations Research Edited by:

R. E. BURKARD Technical University of Graz Graz Austria

R. A. CUNINGHAME-GREEN Universityof Birmingham Birmingham United Kingdom and

U. ZIMMERMANN University of Cologne Cologne Federal Republic of Germany

1984 NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD

Elsevier Science Publishers B.V. 1984 I

All righis reserved. No part of this publication may he reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 044487571 9

Publisher:

ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 1991 1000BZ Amsterdam The Netherlands Sole distribut0r.sfor the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY. INC. 52VanderhiltAvenue NewYork. N.Y. 10017 U.S.A.

Library of Congress Cataloging in Publication Data

Workshop on Algebraic Structures in Operations Research. Algebraic and combinstorial methods in operations research. (Annals of discrete mathemstics ; 19) (North-Holland mathematics studies ; 95) Bibliography: p. 1. Operations research. 2. Algebras. Linear. 3. Combinatorial analysis. I. BurJmrd, Rainer E. 11. Cuninghame-Green, Raymond A., 1933111. Zimmermsnn, U. (We), 1947N. Title. V. Series. VI. Series: North-Holland mathematics studies ; 95. T57.6.W67 1984 001.4'24'015125 84-13476

.

Ism 0-444-87571-9 PRINTED IN THE NETHERLANDS

V

FOREWORD

A recurring theme in operations research (O.R.) is that of optimization, and over the last 35 years the subjects of O.R. and mathematical programming have developed side by side and enriched one another.

Much traditional 0.R has been concerned with the behaviour of continuous real variables representing e.g. material stocks, time or money, and the corresponding Optimization theory is one in which real linear algebra, inequalities and the differential calculus have played important roles. However, many systems with which O.R. is concerned incorporate discrete structures for which the optimization questions are combinatorial rather than continuous: one thinks of sequencing, scheduling and flow-problems and of the great variety of questions which can be reformulated as path-finding circuit-finding or sub-graph-finding problems on an abstract graph. Correspondingly, we have witnessed a vigorous growth in the theory and practice of combinatorial optimization. A related, but perhaps less well-known, development has been in the application of ordered algebraic structures to optimization problems. This application is made relevant by the fact that many optimization questions depend essentially on the presence of two features: an algebraic language within which a system can be modelled and an algorithm articulated; and an ordering among the elements which enables a significance to be given to the concept of minimization or maximization.

By adopting this slightly abstract point of view, we can make useful reformulations: certain bottleneck problems become algebraic linear programs; certain machinescheduling problems reduce to finding eigenvectors and eigenvalues of a matrix over a semiring; certain path-finding problems reduce to the solution of linear equations over an ordered structure. Many optimization problems of the kind which have arisen in O.R. assume, under such reformulation, the appearance of problems of linear algebra over an ordered system of scalars. Hence we may look to the highly-developed classical theory of linear algebra over the real field to give us hints as to how we might approach these problems or, if appropriate adaptations of classical techniques cannot be found, we have a well-defined research program to elucidate the theory of linear algebra over such ordered structures, and to see how far the algorithms and duality principles, familiar to us from linear and combinatorial optimization over the real field, extend to more general structures.

vi

Foreword

These questions have stimulated a good deal of research over the last twenty-five years. From a few isolated publications by one or two researchers in the late 1960’s

the subject has matured into an identifiable branch of applicable mathematics, with an international following. We invited a number of those who have contributed to this development, to participate in the production of a publication featuring some of their more recent work. It is the result of their enthusiastic acceptance of this invk tation which we are now pleased to present as this volume in the series of Annals of Discrete Mathematics. R.E. Burkard R.A. Cuninghame-Green

U. Zimmermann

Acknowle&ement: I should like to add a personal note of gratitude to Tricia Carr, who made such a beautifuljob of preparing the manuscript. R.A. Cunjnghame-Green

CONTENTS Foreword

v

J. ARAOZ, Packing problems in semigroup programming

1

P. BRUCKER, A greedy algorithm for solving network flow problems in trees

23

P. BRUCKER, W. PAPENJOHANN, and U. ZIMMERMANN, A dual optimality criterion for algebraic linear programs

35

P. BUTKOVIC, On properties of solution sets of extremal linear programs

41

R.A. CUNINGHAME-GREEN, Using fields for semiring computations

55

R.A. CUNINGHAME-GREEN and W.F. BORAWITZ, Scheduling by non-commutative algebra

75

C k EBENEGGER, P.L. HAMMER, and D. DE WERRA, Pseudo-boolean functions and stability of graphs

83

R. EULER, Independence systems and perfect k-matroid-intersections

99

U. FAIGLE, Matroids on ordered sets and the greedy algorithm

115

A. FRANK and E. TARDOS, An algorithm for the unbounded matroid intersection polyhedron

129

A.M. FRIEZE, Algebraic Flows

135

M. GONDRAN, and M. MINOUX, Linear algebra in dioids: a survey of recent results

147

H.W. HAMACHER and S. TUFEKCI, Algebraic flows and time-cost tradeoff problems

165

H.W. HAMACHER, J.-C. PICARD, and M. QUEYRANNE, Ranking the cuts and cut-sets of a network

183

P. HANSEN, Shortest paths in signed graphs

201

B.L HULME, A.W. SHIVER,and P.J. SLATER,A boolean algebraic analysis of fae protection

215

B. MAHR,Iteration and summability in semirings

229

RH. MOHRING, and F.J. RADERMACHER, Substitution decomposition for discrete structures and connectionswith combinatorial optimization

25 7

K. ZIMMERMA", On max-separable optimization problems

357

U. ZIMMERMA", Minimizationof combined objective functions on integral submolar flows

363

ANNALS OF DISCRETE MATHEMATICS Vol. I :

Studies in Integer Programming edited by P. L. HAMMER, E. L. JOHNSON, B. H. KORTE and G. L. NEMHAUSER 1977 viii + 562 pages

Vol. 2 :

Algorithmic Aspects of Combinatorics edited by B. ALSPACH, P. HELL and D. J. MILLER 1978 out of print

Vol. 3:

Advances in Graph Theory edited by B. BOLLOBAS 1978 viii + 296 pages

Vol. 4:

Discrete Optimization, Part I edited by P. L. HAMMER, E.L. JOHNSON and B. KORTE 1979 xii + 300 pages

Vol. : Discrete Optimization, Part I1 edited by P. L. HAMMER, E.L. JOHNSON and B. KORTE 1979 vi + 454 pages

Vol. 6:

Combinatorial Mathematics, Optimal Designs and their Applications edited by J. SRIVASTAVA 1980 viii 392 pages

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VOl. 7: Topics on Steiner Systems edited by C. C. LINDNER andA. ROSA I980 x + 3jO pages VOl. 8 : Combinatorics 79, Part I edited by M. DEZA and I. G. ROSENBERG I 980 xxii + 3 I0 pages Vol. 9:

Combinatorics 79. Part I1 crlifcd by M. DEZA rind 1. G. ROSENBERG I980 viii + 3 I 0 pages

Vol. L 0: Linear and Cornbinatorial Optimization in Ordered Algebraic Structures edi/erlby U. ZIMMERMANN I98 I x + 380 pages

Vol. I I : Studies on Graphs and Discrete Programming edirrdby P. HANSEN I 98 I viii + 396 pages Vol. I.?:

Theory and Practice of Combinatorics rdiredbji A.ROSA,G.SABIDUSInridJ.TURGEON I982 x + 266 pages

Vol. 13: Graph Theory cditrd by B. BOLLOBAS 1987 viii + 204 pages

Vol. 14: Combinatorial and Geometric Structures and their Applications edired by A. BARLOTTI I982 viii + 292 pages Vol. 15: Algebraic and Geometric Combinatorics edifedby E. MENDELSOHN 1982 xiv 378 pages

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Vol. 16: Bonn Workshop on Combinatorial Optimization edifedby A. BACHEM, M. GROTSCHELL and B. KORTE 1982 x+312pages Vol. 17: Combinatorial Mathematics edited by C. BERGE, D. BRESSON, P. CAMIONand F. STERBOUL I983 x + 660 pages in preparation

Vol. 18: Combinatorics '81: in honour of Beniamino Segre edited by A. BARLOTTI,P. V. CECCERINI and G. TALLINI 1983 xii + 824

Annals of Discrete Mathematics 19 (1984) 1-22 0 Elsevier Science Publishers B.V. (North-Holland)

1

PACKING PROBLEMS I N SEMIGROUP PROGRAMMING

J. Ar&z I n s t i t u t f i r Okonometrie und O p e r a t i o n s Research U n i v e r s i t a t Bonn, W. Germany.

Departamento de Matema/tica,s y C i e n c i a de l a Computaction, U n i v e r s i d a d Sim6n B o l f v a r , Caracas, Venezuela.

We d e f i n e an o p t i m i z a t i o n problem i n t h e d i s c r e t e semimodule o v e r t h e n a t u r a l numbers g i v e n by an ordered c a n m u t a t i v e semig r o u p and show t h a t a c a n o n i c a l o r d e r induced i n any semig r o u p by t h e r i g h t - h a n d - s i d e e l e n e n t g i v e an ordered semig r o u p f o r which t h e o p t i m i z a t i o n problem i s e q u i v a l e n t t o t h e ( E q u a l i t y ) Semigroup Program, t h e r e f o r e t h e e x t e n s i o n i s cons i s t e n t . Packing Programs correspond t o p o s i t i v e l y o r d e r e d semigroups s a t i s f y i n g a s e l f - p o s i t i v e c o n d i t i o n , i n t h e s e cases t h e semimodule i s ordered. Packing Programs g i v e a g e n e r a l i z a t i o n o f I n t e g e r Packing Programs. We show t h a t t h e f a c e t s o f t h e convex h u l l o f s o l u t i o n s t o a Packing Program a r e s u p e r - a d d i t i v e s and we c h a r a c t e r i z e t h e p o l a r s and neopol a r s o f Master Packing Programs. 1.

INTRODUCTION

Semigroup programming has been developed by a l m o s t two decades because o f i t s r e l a t i o n t o i n t e g e r programming.

@qa b o u t g r o u p Johnson

B e g i n n i n g w i t h t h e work o f Gomory [17],

081,

problems and c o n t i n u e d by o t h e r s , see f o r example Burdet-

[lq , Gmory-Johnson

1201 , Jeroslow [ZZl,

[23],

Johnson [24],

i t was extended t o semigroups by Ara’oz and Edmonds [l], [4],

[5].

Wolsey [32],

Specially, a

course p u t t i n g t o g e t h e r those e a r l y works i s Johnson

[Zq.

Johnson [26]

Kelated t o t h i s t o p i c are

a l s o c o n s i d e r s g e n e r a l b i n a r y systems.

[lo],

Ara’oz-Johnson [9],

011.

I n t h i s paper we a r e concerned w i t h semigroup programs r e l a t e d t o p a c k i n g problems. E a r l y r e s u l t s were presented i n Arabz [l], [3]. I n S e c t i o n 2 we s t a t e w i t h o u t p r o o f t h e r e s u l t s i n p o l a r i t i e s needed.

I n Section

3 we d e f i n e ordered a b e l i a n semigroups and show t h a t t h e y cover c o n s i s t e n t l y a b e l i a n semigroup problems w i t h an o r d e r i n g o f t h e i r elements induced by t h e right-hand-side.

We a l s o prove t h e r e s u l t s we need f o r developing t h e p o l a r and

neopolar o f packing semigroup programs. packing semigroup programs. t i o n s o f t h i s work.

I n S e c t i o n 4 we c h a r a c t e r i z e neopolars o f

F i n a l l y i n S e c t i o n 5 we comment on p o s s i b l e c o n t i n u a -

2 2.

J. Alrioz POLAR FUJNDATIONS

The usefulness o f p o l a r i t y r e l a t i o n s i n polyhedral theory has been recognized because of h i s impact i n l i n e a r p r o g r a m i n g and d u a l i t y . work o f Farkas i f fundamental.

Cone p o l a r i t y and the

Polyhedral p o l a r i t y , as d e f i n e d by Minkowsky,

could be studied i n Rockafellar [28]

and Stoer and W i t z g a l l [29],

reverse p o l a r i t y

has been used i m p l i c i t l y by Gmory [19] and Fulkerson [ls], and s t u d i e d by o t h e r I t was developed and extended by authors l i k e Tind [30], [31] and Balas [13]. [l]. A study o f the p o l a r i t i e s g i v e n by general b i l i n e a r r e l a t i o n s can be found i n G r i f f i n [21], Ara/oz, Edmonds, G r i f f i n [ 6 ] , Edmonds and A r i a 2

[?I,

p],

Bachem-Grotschel [12].

2.1.

POLARITY DEFINED BY A RELATION

L e t X be a given s e t w i t h a symmetric r e l a t i o n R S X We denote (x,y)

Tn

polar

( r e s p e c t t o n) o f T i s t h e s e t T

(2.1.2)

If T

T"

(2.1.3)

(2.1.4)

Proof: Let T '

C

T ' t k n T n l TI'.

Then we have C j e a r l y since i f y G T "

i n p a r t i c u l a r x R y f o r a l l x e T E T ' , hence y E

x R y = y

always c o n t a i n s

T, because f o r any x

E

x ( a i s symnetric) and t h e r e f o r e x e (T")'.

=.

We always have Tm equal t o Tn.

We have (T')" = T",

E

T" we have

t Tn by (2.1.3),

hence T G T ' by (2,1.3),

using (2.1.2)

we o b t a i n Tn =. T I " = Tm

.

X, we say t h a t C i s n-closed i f C = C m .

(2.1.5)

Let C

(2.1.6)

m. C G

c_

then x Q y f o r a l l

T".

T and a l l y

sz

Therefore Tn = Tm

X i s n-closed i f and o n l y i f t h e r e e x i s t s T S X such t h a t

C = T'. -__ Proof:

n g i v e n by

= ( y E X: x Q y f o r a l l x d T I .

Let T and T ' be b o t h subsets o f X.

x e T',

X.

e n by x Q y.

For any s e t T 5 X the

(2.1.1)

x

I f C = Ts; then C" = Tm = Tn = C by (2.1.4). If C = C" l e t T = C". Hence C = C" = T'.

Packing problems in semigroup programming

2.2

3

a-POLARITY

When X i s a subset of Rn we consider t h e r e l a t i o n s x a y when xy 6 1 (polyhedral p o l a r i t y ) , x 8 y when xy + 1 (reverse p o l a r i t y ) ; group and covering semigroup programs a r e 8-closed b u t packing semigroup programs a r e a-closed. I n t h i s paper we will be concerned with a - p o l a r i t y . I t i s a l s o possible t o use anti-blocking polyhedra, Fulkerson n6], f o r t h e r e l a t i o n between anti-blocking and p o l a r i t i e s see Ardoz [2]. For any set P , of n-dimensional real vectors, t h e a-polar of P i s (2.2.1)

pa = {y E R":

xy

6

1 for all x

E

PI.

(2.2.2) In t h i s section we will consider P t o be a pointed f u l l dimensional polyhedron with vertex set V and extreme ray set R. (2.2.3

Lemma. Pa i s the set PVR 5 Iy e Rn: vy 6 1 f o r a l l v E V - {Ill and ry Q 0 f o r a l l r Moreover this system i s irredundant whenever 0 belongs t o P.

(2.2.4)

Theorem.

(2.2.5)

@.

P i s a-closed i f and only i f 0

E

E

R1.

P.

Pa i s f u l l dimensional and pointed.

The proofs a r e i n Ara'oz

Dl .

For a f u l l dimensional polyhedron t h e r e i s a unique irredundant defining system. Each inequality corresponds t o a f a c e t of the polyhedron. When P i s a-closed Lemma (2.2.3) means t h a t the f a c e t s corresponding t o i n e q u a l i t i e s of the type vx 1 a r e given by the non-zero v e r t i c e s o f Pa. (2.2.6) Let P be an a-closed polyhedron. We c a l l a polyhedron Q c_ Pa an a-neopolar of P whenever the non-zero v e r t i c e s of Pa a r e v e r t i c e s of Q. When t h e v e r t i c e s of Q a r e the non-zero v e r t i c e s of Pa we say t h a t Q i s a s t r i c t a-neopolar of P.

Hence f o r an a-closed polyhedron P , P will be the s e t of solutions t o I x : vx c< 1 f o r a l l v a vertex of Q and rx 6 0 f o r a l l r an extreme r a y of PI when Q i s an a-neopolar. The system i s irredundant when Q i s a s t r i c t . a - n e o p o l a r .

4

J. Arm2

3.

SEMIGROUPS

C(MMUTAT1VE SEMIGRCUPS

3.1.

(3.1.1)

A ( f i n i t e c u n m r t a t i v e ) semigroup i s t h e ordered p a i r (S,?)

non-empty f i n i t e s e t and

i s a b i n a r y o p e r a t i o n from S

+ i) f o r

a s s o c i a t i v i t y : ( a i e) C i = a i ( e comnutativity: a ie = e

The order o f t h e semigroup (S,;)

(3.1.3)

The

o

a l l a,e,i

ES.

a f o r a l l a,e E S.

(3.1.2)

s

where S i s a

S i n S which s a t i s f i e s :

x

i s t h e c a r d i n a l i t y o f S.

o f t h e semigroup i s an element s a t i s f y i n g a 4 u = a f o r a l l

a eS. C l e a r l y such element, i f t h e r e i s one, i s unique.

When (S,;)

i d e n t i t y we can always a d j o i n one element a k S d e f i n i n g a and

(I

= IJ and the p a i r ( S U [ a } ,

t IJ

d o e s n ' t have an a = a for a l l a t S

$) w i l l be a semigroup w i t h i d e n t i t y a.

Therefore we w i l l denote by u the i d e n t i t y o f (S,;)

when i t has one o r t h i s new

element added i n the way explained above.

(3.1.4.)

The i n v e r s e o f an element a i n S i s another element e i n S, when t h e r e

i s one, such t h a t a

e =

U.

C l e a r l y a can have a t most one inverse.

When a l l

-

the elements have inoerses t h e semigroup i s c a l l e d a group. (3.1.5) e

For a,e E S we d e f i n e a

4 x = a , i.e. a

(3.1.6)

1,

e E {x E S

For any non-negative

1,

e t o be t h e s e t o f s o l u t i o n s x t o t h e equation

e i x = a). n t e g e r k and any a i n S we d e f i n e

k.a by

the

recursion:

i

(J

k . a =

when k = 0 (k-1)

.a

a when k

>

0,

w i t h t h i s o p e r a t i o n t h e semigroup d e f i n e s a semimodule over t h e n a t u r a l numbers. (3.1.7)

1

.

a, 2

Since we are considering o n l y f i n i t e semigroups the sequence 0

.

a,

...

.

a,

has o n l y f i n i t e l y many d i f f e r e n t elements f o r any a i n S, the

order of a, w r i t t e n o ( a ) , i s t h e minimum i n t e g e r which repeats one element i n t h e sequence, i . e .

Packing problems in semigroup programming o ( a ) = m i n { k : t h e r e i s j < k such t h a t j

.a

.

k

=

5

a}, t h a t

s o(a) i s the order

kbO

o f t h e semigroup generated by a (see ( 3 . 1 . 1 4 ) ) .

k

-

j from j

1

>I

j, m

.a

+O.

t o (k

-

1)

.a

and we have 1

The elements r e p e a t w i t h p e r i o d

.

a = (1 t (k

The l o o p of a i s t h e s e t { e e S: e = k

belongs t o i t s l o o p we c a l l a a l o o p element.

.

i the i t e r a t i o n o f the operation k s i o n a1, a2, ..., a o f elements f r o m S , We denote b y

c

{l, ...,k }

j

.

a for all When a

k

G,

. a.

t h a t i s f o r any succes-

... ?. a k

denotes a1 ?. a2

aJ

j)m)

I t i s easy t o see t h a t a i s a l o o p

element i f and o n l y i f t h e r e i s a k > 0 such t h a t a = a

(3.1.8)

-

a, k >I o ( a ) } .

L e t A be c o n t a i n e d i n S and t = ( t ( a ) ; a c A ) be a non-negative i n t e g e r A v e c t o r . We c a l l t t h e i n c i d e n c e v e c t o r o f f t ( a ) a; t h e r e f o r e N i s t h e s e t aeA o f i n c i d e n c e v e c t o r s . We s a i d t h a t t r e p r e s e n t s e when i t ( a ) . a = e. a 4 (3.1.9)

.

The r e p r e s e n t a t i o n f u n c t i o n 8 : NA

+

S i s d e f i n e d as e ( t ) =

t(a) aeA

. a;

denote by g a t h e v e c t o r s a ( e ) equal t o 1 i f e = a and zero o t h e r w i s e . a e(6 ) = a. S i n c e (S,;)

(3.1 . l o )

i

.a

t(a)

t

acA

i

we

Clearly

i s a s s o c i a t i v e and c o m u t a t i v e .

t'(a)

.a

equals t o

aeA

c

(t(a) t t'(a))

.

a,

a 4

that i s O(t) t e ( t ' ) = O(t t t o ) .

e. L e t t, t '

(3.1.11)

e(t

-

.

t " t N A be such t h a t t ' 6 t and e ( t ' ) = e ( t " ) .

Then

t ' t t") = e l t l .

Proof: S i n c e t e(t (3.1.10),

-

-

t' E

N A we have

t' t t") = e ( t

-

t ' ) t e(t") = e(t

t h e h y p o t h e s i s and (3.1.10)

-

t ' ) t e ( t ' ) = e ( t ) by

again.

T h i s Lemma i s u s e f u l because i t a l l o w s us t o r e p l a c e a p o r t i o n of a v e c t o r t by another r e p r e s e n t i n g t h e same element, i n p a r t i c u l a r i f ta > 0 we can s u b t r a c t A a f r a n t and add t o t h e v e c t o r t any r e p r e s e n t a t i o n o f a w i t h o u t c h a r g i n g t h e

semigroup element represented. (3.1.12) then ( R , ? )

I f a subset R o f S i s c l o s e d under $, t h a t i s a i s a semigroup c a l l e d a s u b s m i g r o u p o f ( S , ; ) .

4e

R f o r a l l a,e E R,

6

J. Araoz

m. L e t

(3.1.13) subset o f S.

be a semigroup w i t h i d e n t i t y a and A be a non-empty where H = e(N A ), i s a subsemigroup o f (S,;)

(S,;)

Then t h e p a i r (H,:),

Moreover i f (A,$)

and o i s represented by e ( 0 ) .

U

H = A

i s a subsemigroup o f ( S , ; )

then

{a}.

The p r o o f i s t r i v i a l from t h e d e f i n i t i o n s o f 8 and k

. a.

be a semigroup and A a subset o f S. Then t h e semigroup A That i t i s a semigroup f o l l o w s from generated by A i s t h e semigroup (e(N ),+). lerrma (3.1.13). A i s a generating s e t f o r (S,t) when t h e semigroup generated by Let (S,+)

(3.1.14)

A i s (S,t), and a minimal generating s e t i s a basis. A c y c l i c semigroup i s one having a basis of c a r d i n a l i t y one. A s i n k element a s a t i s f i e s a

(3.1.15)

e = a f o r a l l elements e o f S.

we c o u l d add always a s i n k element, b u t we c o u l d loose s t r u c t u r e e.g.

Clearly

i n a group.

Hence we w i l l e l i m i n a t e a s i n k element when i t i s n o t generated by the r e s t o f t h e

A s i n k element i s always a l o o p element and i t i s unique.

elements. (3.1.16)

Lemma.

e s e ' imply a Proof:

Let

e

5

3

be an equivalence r e l a t i o n i n S s a t i s f y i n g a

e'

a'

.

be equivalence classes. L e t { a } , { e l , {i}

We have-{a} $ ( { e l 1 {il) = {a1

4 { e T il =

4 { e l ) 7 {il= { a e l ( i l { a ] ;{ e l = { a 1 e l = t e } i { a ] . ({a}

3.2

a ' and

Then t h e q u o t i e n t S/= i s a semigroup.

= {a

e

{a % e

il

il. Furthermore

ORDERED SEMIGROUPS

For a general d e s c r i p t i o n o f ordered semigroups and t h e i r p r o p e r t i e s see Zimnermann p3],

t h i s book a l s o c o n t a i n s o t h e r a p p l i c a t i o n s o f ordered semigroups

t o Operations Research. We w i l l be concerned w i t h p a r t i a l o r d e r r e l a t i o n s mainly, hence we r e s t r i c t t h e

d e f i n i t i o n o f ordered semigroups t o p a r t i a l l y ordered f i n i t e commutative semigroups. (3.2.1)

An ordered semigroup (S,%,4)

a p a r t i a l order r e l a t i o n

(3.2.2)

a

6

6

i s a f i n i t e comnutative semigroup ( S , ? ) w i t h

between i t s elements, such t h a t

e i m p l i e s a ?. i s e

;i

f o r a l l a,e,i

E

S.

Packing problems in semigroup programming

This i s equivalent to: (3.2.3)

a \< a ' and e 6 e ' i m p l i e s a a t e c a

Lemma.

(3.2.4)

+

e' 6 a'

Let

+

G

e < a'

e ' since

e'.

be a preorder r e l a t i o n ( t r a n s i t i v e and r e f l e x i v e ) which

6

Then t h e q u o t i e n t S / E , where a :e whenever a 6 e and e 6 a,

s a t i s f i e s (3.2.2).

i s an ordered semigroup w i t h r e s p e c t t o t h e p a r t i a l o r d e r associated t o 6. we o n l y need t o prove t h a t a E a ' and e :e ' i m p l i e s

Proof: by Lemma (3.1.16) that a

ie

ie'.

:a '

But a 4 a ' and e

Q

obtain a '

a

el

&

e e a'

e' implies t h a t a e.

Hence a

e

5

e ' by (3.2.3).

S i m i l a r l y we

ie ' .

a'

We c a l l t h i s q u o t i e n t a reduced semigroup and a semigroup i s reduced whenever i t i s isomorphic t o t h e q u o t i e n t . (3.2.5)

Since t - s

e(t)

(3.2.6)

=

= e(t-s)

+ e(s)

L e t :be an equivalence e.

+

and e ( r ) d e ( s ) .

e :a '

t

>/

e(t-s)

t

e(r) = e(t-str)

E q u a l i t y holds i f e ( r ) = e ( s ) (Lenuna (3.1.11)).

and (3.1.10).

e l imply a

6 t

NA we have

= e(t-sts)

using (3.2.2)

e

E

e NA be such t h a t s

E q u a l i t y holds when e ( r ) = e ( s ) .

Then e ( t - s t r ) 6 e ( t ) . Proof:

L e t r,s,t

S u b s t i t u t i o n Lemma.

r e l a t i o n i n S s a t i s f y i n g a : a ' and

e ' and whenever a s e i m p l y a '

4

e'.

Then t h e q u o t i e n t

S / z i s an ordered semigroup.

Proof:

By Lemma (3.1.16) S / z i s a semigroup, hence we o n l y need t o prove t h a t i t

i s ordered. {a}

3.3

{ e l , {i} be equivalence classes and l e t { a } s { e l .

Let {a},

{il= {a

4 il 6

{e

We have

i l = {el 3 {il.

SOLUTION VECTORS

L e t A be a subset o f S, and l e t ( S , i , < ) be an ordered semigroup. F i x (3.3.1) some element b E S, b # u and c a l l b t h e right-hand-side. t t N A i s a solution v e c t o r i f e ( t ) s b.

Denote by F(A,b) t h e s e t o f s o l u t i o n vectors and by C(A,b)

t h e convex h u l l o f F(A,b).

Our i n t e r e s t i s t o determine p r o p e r t i e s o f C(A,b).

8

J. Arboz The i n t e r v a l s e t o f a w i t h r e s p e c t t o b i s

(3.3.2)

IS(a,b)

a

(3.3.3)

E

0x

= t x E S: a

,< b } .

S i s i n f e a s i b l e when IS(a,b) =

0, we

denote by

m

the set o f infeas-

C l e a r l y , i f a e A, a i n f e a s i b l e i m p l i e s t ( a ) = 0 i n any s o l u t i o n

i b l e elements. v e c t o r t. (3.3.4) S.

Proof: a

%.

L e t (S,+,c)

Then we have a

Since e

+

Let a

4 (e ix)

4

e 6

e

m,

t.

-

be an ordered semigroup, a e and i f a

Then e x i s t s x

s b; i . e . ( e $ x )

ix

b we have a

S such t h a t (a

E

bm.

i x s b by (3.2.2).

x I e

0.

and e any element o f

m

e then e e m .

IS(a,b) hence a

E

e q u i v a l e n t IS(e,b) c IS(a,b) = (3.3.5)

6

Therefore e

ie)

x 6 b, t h a t i s

s e and x

Let a

That i s x

d

-

f o r a l l { a 1 e S/- and

The p r o o f i s imnediate frmi Lenmas (3.2.6)

m

a t S i s t e r m i n a l when IS(a,b)

(3.3.75

The b-complementor o f a, denoted by 2 , s a t i s f i e s a

(3.3.8)

I n o t b e r words,

ie

b

01

m.

Then S/-

i s a l o o p element.

and (3.3.4).

(3.3.6)

a.

IS(a,b)

e w .

L e t a : e i f and o n l y i f e i t h e r a = e o r a,e t

Corollary.

i s an ordered semigroup w i t h t a l

s b then e s

t

IS(e,b).

E

= Col

a and e s

z

i c o u l d be undefined, b u t when

i

= b and i f a

ie

i f o r a l l e e IS(a,b).

i t i s d e f i n e d i t i s unique s i n c e

6

i s the

maximum of IS(a,b). (3.3.9)

An ordered semigroup i s b-complementary i f every f e a s i b l e element has a

b-complementor. b-complementors keep many p r o p e r t i e s o f b-a i n a group and have shown t o be a u s e f u l s u b s t i t u t e when d e a l i n g w i t h semigroup p r o g r a m i n g . (3.3.10)

S i m p l i f i c a t i o n s and r e d u c t i o n s :

The arguments g i v e n here w i l l l e a d t o

assumptions on (S,?), w i t h o u t l o s s o f g e n e r a l i t y .

T h i s assumptions a r e i m p o r t a n t

i n order t o s i m p l i f y the proofs. (3.3.11) element

-

Infinite

m.

We can c o l l a p s e a l l i n f e a s i b l e elements i n t o t h e s i n g l e

u s i n g C o r o l l a r y (3.3.5).,is

semigroup has

m

then a s i n k element and we say t h a t t h e

i f i t cannot be e l i m i n a t e d (see (3.1.15)),

t h a t is, there e x i s t

Packing problems in semigroup programming

a,e #

such t h a t a

m

most one

m,

e =

and a
0. Since 0

Hence n t o

6 a 4 to, we have:

n ( a ) = ~6~ = max {rrsl, by lemma (4.3.1).

KN*

e(s)

e(sa)

Hence, because e ( s a ) = a, t h e lemma i s proved. (4.3.4)

Theorem.

I n a packing problem, l e t

IT

be a proper v e c t o r .

Then

T

satis-

f i e s the following conditions. (4.3.5)

I f a E A and a i s t e r m i n a l then n ( a ) = 1.

I n particular b e A implies

a ( b ) = 1. (4.3.6)

Monotonic: f o r a l l a, a 1

(4.3.7)

Superadditivity:

I f a, a',

Complementarity:

I f a,

A, if a 6 a' then * ( a ) d n ( a 1 ) . a2

E

A and a2 = a

a 1 then n ( a )

+

n ( a 1 ) \
, e > u by (3.5.2), then we have e ( r + 6 ) = e ( r ) P sa) = u i m p l i e s t - ha = 0. Therefore n(a) = n t = 1. Since b i s t e r m i n a l

any r

>I

-

e(t

-

Hence e ( t ) = e ( t

= 1.

E IS(a,b)

= {a].

0 and any e e S

by Lemma (3.5.4),

n ( b ) = 1.

u s i n g (4.3.3) we have nsa = n ( a ) s n(a 1 ). 2 A 2 O f (4.3.7): By (4.3.3) we have T(a ) = max I n t l over t E: N s a t i s f y i n g e ( t ) s a 1 ) = a ia1 2 Therefore n ( a 2 ) > ~ ( + 6tia1~) = " ( a ) t n ( a1 because 8(sa + = a .

O f (4.3.6):

Since a s a',

Of (4.3.8):

Since a

Thus " ( a )

+

.(a)

a

= n(6'

o n l y need t o prove n ( a )

1

s a t i s f y i n g n t = 1 and t ( a ) 3 1.

-

4.4

&

F(A,b).

+ s a ) c 1 because n t s 1 + ~ ( i3 )1. However, by

b-complementor o f a we have e ( t nt

+

= b we have d a

Hence e ( t ) = O ( t

-

a,

sa) s

" ( a ) o r e q u i v a l e n t n ( a ) + n(i) 3

n t

f o r a l l t c F(A,b). (4.3.2),

- aa)

Hence we

t h e r e e x i s t s t EF(A,b) a

by Lemma (4.3.3)

6

b.

2 i s the

Since

e ( a ) 0.

x.

0).

If we s e t

for v = k

V

else

V

= 0 f o r a l l v c P ( i ) and T h i s i s t r u e because lv

t o o because

... 5

Uj(1) 6 U k ( l ) . < Uk(2) < Let

>

for v = j

'1

we g e t a f e a s i b l e s o l u t i o n

31

Uk

( 2 ) be t h e problem d e r i v e d f r o m ( 2 ) by r e p l a c i n g fj(t)

= f.(o)

J

... *

*

Uj(1)

*

*

Uj(2)

* ... *

u.

J

by Tj(t) =

f-(0)

J

u

j

keeping f v ( t ) f o r v # j unchanged and r e d u c i n g t h e bounds as i n s t e p s 14., and 17. o f t h e a l g o r i t h m .

15.,

(2) has a f e a s i b l e s o l u t i o n which can be seen as

(2)

Denote by 1 ' b! t h e bounds o f i' 1 v e r t e x h . Then we c o n s i d e r two cases.

follows.

and assume t h a t 1;1 > b;

f o r some

Case 1: h i s a predecessor o f i which i m p l i e s

0

+ 1;

>

T h i s i s a c o n t r a d i c t i o n t o t h e f a c t t h a t b;

b.,l

>/

0.

Case 2: h i s n o t a predecessor o f i. Then h and i have a f i r s t common successor

s.

Then 1;

have lh> bh

b;

>

-

i m p l i e s bs = bh and bh = bh

1 (see s t e p 17.).

= lh.With ( 8 ) and li

1 because 1;

bh 6 bh-1

-

+

1.

i

1 +1 s 1 h i s


e.

ui(j)

can be represen-

33

Solving network flow problems in trees

A l l we have t o do i s t o i n s e r t between statements 8. and 9. o f t h e a l g o r i t h m t h e statement lr+ br and t o r e p l a c e t h e word " m i n i m a l " i n statement 11 by t h e word "maximal" ( B r u c k e r (1982)). The c o m p l e x i t y o f t h i s m o d i f i e d a l g o r i t h m i s O(nb) with b = b

.

r By a d i f f e r e n t approach an O(n'1og

1 ) - r e s p . O(n210g b ) - a l g o r i t h m can be d e r i v e d

f o r t h e s e problems ( B r u c k e r (1982)).

Thus t h e y a r e p o l y n o m i a l l y s o l v a b l e .

T h i s work i s m o t i v a t e d by a p r o d u c t i o n - s a l e s p l a n n i n g model which l e a d s t o t h e f o l l o w i n g m a x i m i z a t i o n problem (Tamir (1980)).

n Maximize

C

fi(xi)

i=l (11) subject t o j

z

i=l

xi

xi 6 b . f o r j = 1, J

...,n

a 0 i n t e g e r f o r i = I, ...,n

i n which a l l fi a r e monotone nondecreasing and concave f u n c t i o n s .

(11) may be

d e r i v e d f r o m (10) by s p e c i a l i z i n g t h e t r e e s t r u c t u r e and t h e o b j e c t i v e f u n c t i o n .

A n a t u r a l q u e s t i o n i s : how c o u l d t h e s e r e s u l t s be extended t o more g e n e r a l n e t work s t r u c t u r e s ?

I t can be shown t h a t c i r c u l a t i o n problems w i t h n o n l i n e a r

a l g e b r a i c o b j e c t i v e f u n c t i o n s as d e f i n e d i n t h i s paper can be s o l v e d u s i n g an o u t - o f - k i l t e r method f o r t h e l i n e a r a l g e b r a i c c i r c u l a t i o n problem developed by B r u c k e r and Papenjohann (1 982).

However t h e c o m p l e x i t y i n c r e a s e s c o n s i d e r a b l y .

T h e r e f o r e a g e n e r a l i z a t i o n o f t h e concepts discussed i n t h i s paper t o a b r o a d e r c l a s s o f network s t r u c t u r e i s d e s i r a b l e .

ACKNOWLEDGRENT

I thank t h e anonymous r e f e r e e s f o r t h e i r h e l p f u l suggestions.

REFERENCES [I]

Brucker, P., Network f l o w s i n t r e e s and knapsack problems w i t h n e s t e d cons t r a i n t s , t o appear i n : Proceeding o f t h e Workshop on " G r a p h t h e o r e t i c conc e p t s i n computer s c i e n c e " , 1982.

[2l

Brucker, P. and Papenjohann, W., An o u t - o f - k i l t e r method f o r t h e a l g e b r a i c c i r c u l a t i o n problem, Osnabrucker S c h r i f t e n z u r Mathematik, U n i v e r s i t a t Osnabrtick, Reihe P r e p r i n t s , H e f t 48 ( 1982).

[31

Burkard, R.E., Hahn, W . and Zimmermann, U., An a l g e b r a i c approach t o assignment problems, Mathematical P r o g r a m i n g 12 (1977) 318-327.

34

[4;

P Brucker Burkard, R.E. and Zimnermann, U., Weakly admissible t r a n s f o r m a t i o n s f o r s o l v i n g a l g e b r a i c assignment and t r a n s p o r t a t i o n problems, Mathematical Programming Studies 1 2 (1980) 1-18.

_ -

-5,

Tamir, A . , E f f i c i e n t a l g o r i t h m f o r a s e l e c t i o n problem w i t h nested cons t r a i n t s and i t s a p p l i c a t i o n t o a production-sales p l a n n i n g model, SIAM Journal on Control and O p t i m i z a t i o n 18 (1 980) 282-287.

Annals of Discrete Mathematics 19 (1984) 35-40 0 Elsevier Science Publishers B.V. (North-Holland)

35

A DUAL OPTIMALITY CRTTERION FOR ALGEBRAIC LINEAR PROGRAMS P. Brucker W. Papenjohann Fachbereich Mathemati k U n i v e r s i t a t Osnabruck A l b r e c h t s t r . 28 D - 4500 Osnabriuck

U. Zimermann Mathmatisches I n s t i t u t U n i v e r s i t a t zu Koln Weyertal 86-90 D 5000 Kaln 41

-

We g e n e r a l i z e a dual o p t i m a l f f y c r i t e r i o n f o r a l g e b r a i c l i n e a r programs from [4] and discuss a p p l i c a t i o n s t o primal dual methods.

ALGEBRAIC LINEAR PROGRAMS We b e g i n w i t h a s h o r t d e s c r i p t i o n o f t h e a l g e b r a i c s t r u c t u r e .

For a d e t a i l e d

[s].

discussion we r e f e r t o

We consider l i n e a r l y ordered semigroups (H,*,&) a s s o c i a t i v e composition

*

w i t h i n t e r n a l comnutative and

and l i n e a r ( t o t a l ) order r e l a t i o n s. We assume t h a t H Thus H i s a l i n e a r l y ordered monoid.

contains a n e u t r a l element e.

If t h e

a d d i t i o n a l axiom a < b i s s a t i s f i e d f o r a l l a,b

+

3 c

E

a*b = a*c

E

H: a*c

= b

(divisor rule)

H then H i s c a l l e d a d-monoid. +

( 1 .I)

If

b = c o r a*b = a

(1.2)

H then H i s c a l l e d weakly c a n c e l l a t i v e . Throughout t h i s note H denotes a weakly c a n c e l l a t i v e d-monoid. We d e f i n e H,:= {a e H I a 2 e l . Due t o

f o r a l l a,b,c

E

Furthermore c 3 e.

(1.2) c i n (1.1) i s uniquely defined. and b

-

Let b

-

a:=c i f a

c

b

a:=e i f b = a.

L e t (R,*,.)

be a subring o f t h e f i e l d o f r e a l numbers and assume Z G R .

s i d e r an e x t e r n a l c o m p o s i t i o n n : R% ,H

+

We con-

H satisfying

aa( m a ) = (aB)aa

(a+B)na =

(am)*(m a )

an(a*b) = (aoa)*(anb) Ona = e h a = a for a l l

a,B

s R,

and f o r a l l a,b

a semimodule over R.,

If

E

H.

Then (R+;u;H,+, t. a)

a

k = b

@

t i f and o n l y i f a

a @ t = b

@

t implies a

3

@ k

@ k = b

@

x

o

c

(5)

b) = a

(6)

= b,

@ k. Then k o x @ a = t

Lemma 2: L e t 5 ( a ) be s a t i s f i e d and k , t E E, k > t. i f and o n l y i f k

c and b

Then

b)

@

(a

o

>/

a = b f o r a l l a,b,x

E

O X

@ b

E.

We v e r i f y o n l y Lemna 2. Assume x # 0 otherwise the a s s e r t i o n i s t r i v i a l . t

o

x

@ b

Supposing b = k

-

x

k

@

o

x

@ a ak

a >I k

x > t

0

b

@

o

0

t

x > t

Together w i t h ( 2 ) i t y i e l d s

x, hence t

o

o

x

@ b

= b.

x we g e t o

x = b, QED.

SYSTEMS OF EXTREMAL EQUATIONS We denote En = E

x

... x

v -

E.

Elements o f En w i l l be c a l l e d vectors.

n

We can extend t h e operations

@,

o

and the r e l a t i o n

4

r i c e s and vectors over E (denoting the product by z ) .

i n a n a t u r a l way t o matThe symbol X T means t h e

t r a n s p o s i t i o n o f the vector X. The f o l l o w i n g p r o p e r t i e s o f these operations w i l l be used l a t e r (A,B,D r i c e s and X,Y,Z

a r e mat-

column vectors o f the a p p r o p r i a t e type):

A ,c B i m p l i e s A

@

andAzDc As a c o r o l l a r y t h e r e holds:

D

Q

B

B G D .

@

D

(7) (8)

P. ButkoviE

44

Y

z

c

xT

implies

and A z Y d A

xT is z


1 a1 , ..., a M , p l y .. . , BN

1; nl

.. ., nN,

deg (0) =

with




0 w i t h Ip as i n (3.2)

Then (7(z),b) i s a c l a s s i c a l ( l i n e a r l y ) ordered f i e l d .

Moreover

i f $,$ e 0 then deg(9 t $ ) = (deg $ ) @ (deg $ ) .

I(z) equivalent i f deg u = deg v and l e t P now be the p o s i t i v e cone o f Call u,v 2(z). For each u E P l e t be the i n t e r s e c t i o n w i t h P o f the equivalence class o f u w i t h respect t o deg i n Z(z).

Let ii be the c o l l e c t i o n o f a l l such .

R.A. Cuningliame-Green

58 For each u,

V t

P, we now d e f i n e

< u + v>

=

t

x = 1 =

and i f v # 0

(3.8)



I t i s e a s i l y v e r i f i e d t h a t these operations a r e well-defined.

Define a l s o

deg = deg u.

(3.9)

Then since deg = n, deg i s s u r j e c t i v e and c l e a r l y i n j e c t i v e over u . (3.4),

(3.5),

So from

(3.7) deg c o n s t i t u t e s the isomorphism

(I,0 Q , I/ )

= (u,+,x,/).

I

(3.10)

Thus the d i v i s i o n semiring i s isomorphic t o a c e r t a i n algebra o f classes o f e l e ments o f Z ( z ) .

The d e t a i l e d v e r i f i c a t i o n s u n d e r l y i n g t h e isomorphism (3.10) may

be found i n B o r a w i t z ’ s t h e s i s [ll]. But c l e a r l y we may c a r r y o u t t h e c l a s s algebra by using general elements o f t h e classes combined according t o the algebra o f I ( z ) , thereby i n e f f e c t r e p l a c i n g semiring computations by s t r a i g h t f o r w a r d c l a s s i c a l algebra.

As a simple example, suppose we r e q u i r e d t o f i n d x E

(2,@,@ )

t o satisfy

6 @ 9 ~ ( ~ ) @ 3 Q x= 4 0 x 0 1 where x ( ~ denotes ) x@x

.

(3.11)

(The p r o p e r t i e s o f maxpolynomials such as t h e expressions on e i t h e r s i d e o f (3.11) have been discussed i n 1142.

By sketching these maxpolynomials, i t i s r e a d i l y

seen t h a t (3.11) has t h e unique s o l u t i o n x = - 2 ) . To solve (3.11) v i a isomorphism (3.10), =

z>.

,2

(3.16)

The problem over Z ( z ) corresponding t o (3.15) i s Solve <X t z> = <X t 9,. However t h e r e l a t i o n X

t

z = X

(3.17)

t z2 has no s o l u t i o n .

I n t h e f o l l o w i n g section, we s h a l l extend t h e formalism t o i n c o r p o r a t e t h e desi r e d improvements, and then apply i t t o a range o f problems.

4.

THE EXTENDED 2-METHOD

...,

n ) i s an a l g e b r a i c expression formed by combining nll' 9 n e Z a n d t h e v a r i a b l e s 5 = (xl, x ) u s i n g t h e operations 9 P @ , @ ,//, then l e t G(;) = G(X1 ,X ;znl ,znq) E Z(z)(Xl X ) be formed P P n by combining t h e constants z nl z e Z(z) and t h e v a r i a b l e s 5 = (X1 ,...,XP)

If g ( 2 ) = g ( x l,...,xp; t h e constants nl,

...

,...

...,

,...

,...,

,...,

r e s p e c t i v e l y u s i n g t h e operations t,x,/ manner.

r e s p e c t i v e l y i n a p r e c i s e l y analogous

Then we say t h a t g(5) and G(5) a r e corresponding expressions.

Suppose now t h a t we have a s e t o f simultaneous a l g e b r a i c r e l a t i o n s over Z ( z ) :

G j ( X ) = Hj(X) ( j = l,...,r).

(4.1

1

To r e s t r i c t s o l u t i o n s t o (4.1) t o those i n which a l l v a r i a b l e s take values i n t h e cone P, i t s u f f i c e s t o consider (4.1)

together w i t h n o n - n e g a t i v i t y c o n s t r a i n t s

on a l l v a r i a b l e s : Gj(Z) = Hj(X)

x

3

( j = 1,

...,r )

(4.2)

g.

For typographical convenience we w r i t e a i n (4.2)

and henceforth.

T h i r t y y e a r s o f mathematical programming have g i v e n us a m u l t i t u d e o f techniques f o r h a n d l i n g problems i n v o l v i n g n o n - n e g a t i v i t y c o n s t r a i n t s ; by adapting these from t h e r e a l f i e l d t o t h e ordered f i e l d 2 ( z ) , we hence d e r i v e a technique, i n c o r p o r a t i n g improvement I1 , f o r s o l v i n g relation-systems such as (4.2). Next suppose t h a t we a r e g i v e n a s e t o f simultaneous r e l a t i o n s over we must f i n d a l l s o l u t i o n s :

2 for

which

R.A. CuninghamcCreen

60

gj(x) = hj(x)

...,r ) .

( j = 1,

(4.3)

With a view t o improvement 12, we s h a l l need t h e f o l l o w i n g

Lemna 1 For u, v E P , = i f and o n l y i f u Proof -

= vn where n e .

(4.4)

Immediate.

L e t us c a l l a v a r i a b l e , which ranges a r b i t r a r i l y over t h e c l a s s , a

unit

parameter. Now, by t h e isomorphism (3.10),

t o s o l v e (4.3) i t s u f f i c e s t o s o l v e



= J-. ( j = 1,

x Using Lemma 1, (4.5)

>,

...,r )

(4.5)

0

i s equivalent t o n J- GJ. ( X ) = HJ. ( X- )

( j = l,...,r)

(4.6)

z. >, g. Where

nl,

...,T

r a r e u n i t parameters.

This i s an e s s e n t i a l l y s i m i l a r problem t o (4.2),

b u t w i t h parametric c o e f f i c i e n t s .

By considering how t h e s o l u t i o n v a r i e s as t h e u n i t parameters v a r y over , we s o l u t i o n s t o (4.3) v i a the isomorphism (3.10).

may generate

The technique extends t o i n e q u a l i t i e s also.

Suppose we have t o solve, over

2,

t h e simultaneous r e l a t i o n s (j= 1

g j ( x ) a hJ. ( x ) These are e q u i v a l e n t t o f i n d i n g y,

,. .. ,yr

,...,r ) .

(4.7

e 2 such t h a t

gj(x) = h j ( x ) O y j

( j = l,...,r)

(4.8

and hence, by t h e f o r e g o i n g discussion, we may e q u i v a l e n t l y s o l v e n.G.(X) = H.(X) J J J -

+ Y.J

(j = l,...,r);

5

( j = l,...,r)

(4.9)

over Z ( z ) , s u b j e c t t o n . e

J

a

0;

Yj

2

0

( j

But t h e Y . now p l a y t h e r o l e of s l a c k v a r i a b l e s , and (4.9), J to n.G.(X) :: H . ( X ) J J J -

x.

0.

( j = l,...,r)

1,

...,r ) .

(4.10)

(4.10) a r e e q u i v a l e n t (4.11)

Using fields for semiring computations

61

Where ~ ~ , . . . , na~r e u n i t parameters. I n p a r t i c u l a r , we have the one-way i m p l i c a t i o n t h a t i f t h e v a r i a b l e s

5

>,

0

take

values s a t i s f y i n g Gj(;)

a Hj(:,)

g j ( x ) >c hj(x)

then where t h e components o f

5

( j = lY...,r)

(4.12)

( j = l,....r),

(4.13)

a r e obtained from those o f

T. by

t h e operator deg,

Comparing (4.6) w i t h (4.11) we see t h a t equations and i n e q u a l i t i e s transform i n e s s e n t i a l l y the same way when l i f t e d from

2

t o Z((z).

F i n a l l y , suppose we have an o p t i m i s a t i o n prDblem over

2:

max f ( 5 ) Subject t o

gj(x) = hj(x) gj(x)

( j = lY...,r)

(j = r+lY...,s).

hj(z)

&

(4.14)

Consider over 2 ( z ) t h e o p t i m i s a t i o n problem using corresponding expressions: max F(X) Subject t o

(j = 1,

n-G.(X) = H . ( Z )

J J

J

n-G.(_X) 3 J J

...,r )

(4.15)

( j = r+l,...,s)

Hj(X)

X & $

where

,. ..,rS

II~

are u n i t parameters.

We s h a l l c a l l t h e o p t i m i s a t i o n problem (4.15) a r e g u l a r problem i f i t has the f o l l o w i n g two c h a r a c t e r i s t i c s : 1.

For each choice

;of t h e u n i t parameters, F(X) i n (4.15) a t t a i n s a con-

s t r a i n e d maximum

i , depending

on

P

2.

:.

f T i s independent o f

The value o f deg

..

p.

For such a problem, l e t $(?) be obtained by applying t h e operator deg t o t h e components of a maximising s o l u t i o n o f (4.15).

f(r(?))= deg

Then

A

F(X(?)) = f (say) i s independentof E

Moreover, i f xo i s any f e a s i b l e s o l u t i o n o f the c o n s t r a i n t s o f (4.14), obtained by applying the o p e r a t o r

T

By (4.14) and t h e isomorphism (3.10),

o f (2.2) t o t h e components o f

^x0

(4.16) l e t Xo be

ro.

s a t i s f y i e s t h e c o n s t r a i n t s o f (4.15) f o r

, s u i t a b l e u n i t parameters F ~ whence:

F(_xo) 6 Fr "0

(4.17)

62

R. A. CuninghameGreen

and hence by the i m p l i c a t i o n from (4.12) t o (4.13): f(,xo) 4 f. Hence, as the u n i t parameters t o (4.14).

TI

(4.18)

vary over < 1 > , the

i ( ~produce ) otpimal

F u r t h e r s o l u t i o n s may be d e r i v e d by s e n s i t i v i t y a n a l y s i s .

solutions We

i l l u s t r a t e a l l the f o r e g o i n g ideas i n the ensuing examples.

5.

ILLUSTRATION: LINEAR EQUATIONS OVER (1,@ ,0 ,// )

Given an (m x n) m a t r i x A and an m-vector b t h e problem

A@: to find

5

2

(5.')

=

s a t i s f y i n g (5.1) a r i s e s i n e.g.

c e r t a i n minimax-earliness machine-

scheduling problems, and has been e x t e n s i v e l y s t u d i e d [13].

As a consequence o f

the presence o f r e s i d u a t i o n , (5.1) ( i f s o l u b l e ) always has a p r i n c i p a l s o l u t i o n ji =

A*@'b_

(5.2)

where A* i s t h e negated transposed o f A , a n d @ ' i n d i c a t e s t h a t t h e algebra i s c a r r i e d o u t i n t h e dual s t r u c t u r e

( r , @ ' , Q , /where /)

?@q

f o r m,n e

m @ ' n = min(m,n) f o r example t h e equation

c

4

-42

4

x2

2,

(5.3)

=

(5.4)

I

LX3 has t h e p r i n c i p a l s o l u t i o n

L e t us now address t h e problem over Z ( z ) corresponding t o (5.4). e t e r s r 1 , TI^, f i n d XI.

X2,

X3 such t h a t

For u n i t param-

Using fields for semiring computations

63

To s a t i s f y a s e t of l i n e a r equations and inequality c o n s t r a i n t s , we may use

Phase 1 of t h e Simplex Method having introduced a r t i f i c i a l variables U1, Up. Maximising Q = - ( U 1 + U2) gives an optimal tableau, w i t h a l l e n t r i e s having the common denominator ( z 3 - l ) , as follows

"1

x3

o

-z3+1

Q

O

x,

71,-iT2Z

x2

IT~Z'+-IT,Z~

-I

The problem i s c l e a r l y regular.

-z2

z-'+

-z~+I

z'+

-Z

-z5+24

Moreover

( 93-1

71 24-71

deg X 2 = deg

deg X3

(5.7)

=

z2 = 1

=

deg ( 0 )

(5.8)

-m

I t may be confirmed t h a t

i s indeed a solution t o (5.4). If we wish t o find t h e principal s o l u t i o n , we may use s e n s i t i v i t y analysis. T h u s , in ( 5 . 7 ) , the lack of a positive entry i n t h e X3 column shows t h a t the basic variables a r e already a t maximum values f o r given u n i t parameters. Moreover, since X3 has zero c o e f f i c i e n t in the optimal objective function expression, i t may be increased a s f a r as will just send one basic variable t o zero, i . e . as f a r as

-x

z =--- 1 2

-1

71 -71

z2-1

Since deg

P,

=

- 2 , we have now completed the principal solution.

(5.10)

R.A. Cuntnghame-Green

64

Clearly, the d e t a i l s o f the parametric s e n s i t i v i t y a n a l y s i s w i l l vary between examples. Our aim i s t o g i v e a f i r s t , simple i l l u s t r a t i o n ; (5.2) remains the e a s i e s t s o l u t i o n t o (5.1).

6. ILLUSTRATION: A DIMENSIONALITY THEOREM L e t En = (En, @ ) be the semimodule o f n-tuples over

...,jm e En,

.,uyxl,

Y

(2, @

,...,xm e 9 such t h a t

i f there a r e x1

@ ,//)

.

= ~ 1 @ ~ 1 @ ~ ~ ~ O x m @ V m

For

(6.1

1

we say t h a t u i s l i n e a r l y dependent on yl, ...'vm. Theorem.

w i t h m > n then g i s l i n e a r l y

I f u i s l i n e a r l y dependent on Vl,..,vm

dependent on some subset {yi

1

,...,v-i,1 c_ {vl,.. .,vm}

w i t h k s n.

R e l a t i o n (6.1) i s soluble o n l y i f r e l a t i o n s (6.2) a r e soluble: m g = 1 Xj"Vj

Proof.

j=l

X . a 0 ( j = l,...,m)

J

where

n

i s a u n i t parameter.

But (6.2)

says t h a t U l i e s i n the convex cone generated by T~'~,...,T~,, so by l i e s i n the convex cone generated by a t most n o f the Caratheodory's theorem nV.

-J

and t h e r e s u l t f o l l o w s by the isomorphism (3.10).

This theorem, which i s a k i n d o f dimensionality statement f o r En, may be looked a t i n various l i g h t s . equation

A s e t o f x . t o s a t i s f y (6.1) c o n s t i t u t e s a s o l u t i o n t o the J

B @ 5 = u_

(6.3)

where B i s t h e ( n x m) m a t r i x

B

=

[q ,. -.,vJ

.

(6.4)

I f we solve (6.3) by the use o f Phase 1 o f t h e Simplex method as o u t l i n e d i n

Section 5 then i n the optimal tableau a t most n o f the x . w i l l receive non-zero J values since the tableau has o n l y n basic rows. Hence the theorem. To look a t i t another way, l e t us r e c a l l t h e method o f s o l v i n g equations such as (5.4) as explained i n [13], Chapter 15. We s u b t r a c t through each row o f the m a t r i x by t h e RHS value, and mark the maximum element i n each column o f the

Using fields for semiring computations

65

resulting matrix

I f element (r,s)

i s r i n g e d then by s e t t i n g v a r i a b l e xs t o minus t h a t value t h e

r t h row i s s a t i s f i e d .

Hence t o s a t i s f y a l l rows we need o n l y assign values t o n

o f t h e v a r i a b l e s and t h e others can be taken t o be

7.

ILLUSTRATION: EXTREMAL L.P.

Formal l i n e a r programs over

DUALITY

(2, @ ,@ , J ) have come t o be known as extremal

l i n e a r programs and f o r them a formal the f a m i l i a r elementary L.P. L.P.

--.

duality.

dual problem may

be d e f i n e d by analogy t o

The r e l a t i o n s h i p between an extremal

and i t s dual has been w e l l - s t u d i e d [15],

i n i t i a l l y by Hoffman [8].

By use

o f t h e z-method, however, we may b r i n g the study o f t h i s t o p i c under t h e aegis o f o r d i n a r y l i n e a r programming. Consider for example t h e f o l l o w i n g extremal L.P. max

m in

3 0 x 10 x 2

subject t o

and i t s dual.

x1 @ 2 Q x 2 6 5

5Q~y1@8@~2

y 1 0 2@y2 a 3

subject t o

(7.1)

2 Q ~ y 1 @ 3 8 Y 2a 0

2 @ ~ 1 @ 3 @ ~\< 28

We now formulate over Z(z) the f o l l o w i n g p a i r o f symnetric-dual L.P.s:

+ e2x2

max

P E elz3x1

subject t o

X1 t z2X2 6 n1z5

z2x1 t z 3 x 2 X1’

where el,

e 2 , nl,

xp

5

m in

nlz5Y1 + 7r2z8y2

subject t o

Y1+z2Y2

a2z8

>I

z2Y1 t z3Y2 Y,,

1 0

e1z3 >I

(7.2)

e2

Y2 >I 0

n2 a r e u n i t parameters.

A f t e r i n t r o d u c i n g slack v a r i a b l e s U1, U2 and V1,

V2 i n the primal and dual

problem r e s p e c t i v e l y , we can apply the Simplex algorithm, regarding t h e u n i t parameters t e m p o r a r i l y as f i xed. tableau.

We d e r i v e the f o l l o w i n g optimal primal-dual

R.A. Cuninghame-Green

66

P

a1n128

-a 123

-e 1z5+e 2

x1

n1z5

-1

-22

u2

n2z8-n1z7

z2

24-23

(7.3)

I t i s c l e a r t h a t t h i s t a b l e a u remains f e a s i b l e and d u a l - f e a s i b l e however t h e

u n i t parameters a r e chosen i n .

For a l l such choices o f t h e u n i t parameters,

deg Pmax = 8, so t h e problem i s r e g u l a r .

P a r t i c u l a r s o l u t i o n s t o LP's (7. 2)

may be read o u t as x1 = 5

x2 =

Yl = 3

--

y2 =

(7.4)

--

Applying s e n s i t i v i t y a n a l y s i s t o (7.3) we see t h a t X2 c o u l d be increased from zero t o nz3, where

B

e ~ 1 i>s chosen s m a l l e r than

f e a s i b i l i t y o r t h e v a l u e o f

.

nl,

without a f f e c t i n g primal

However, i f X2 i s chosen w i t h deg X2 > 3, b o t h

primal f e a s i b i l i t y and t h e value o f

w i l l be a f f e c t e d . p r i m a l s o l u t i o n i n (7.2)

Thus t h e general

is x1 = 5

and

x2 ,< 3.

(7.5)

S i m i l a r l y a p p l y i n g s e n s i t i v i t y a n a l y s i s t o t h e dual s o l u t i o n i n (7.3) we f i n d t h e is

general dual s o l u t i o n i n (7.2)

y1 = 3

(7.6)

and y2 s 0.

Obviously, t h e d e t a i l s o f t h e s e n s i t i v i t y a n a l y s i s w i l l v a r y i n d i f f e r e n t numerical examples, b u t f o r any p a i r o f dual extremal L.P's such as (7.1), t h e primal and t h e dual L.P. over

both

o f t h e corresponding symmetric-dual p a i r (7.2)

Z(z) w i l l have f e a s i b l e s o l u t i o n s .

F o r t h e p r i m a l L.P. always has t h e

o r i g i n a s a f e a s i b l e p o i n t , and a f e a s i b l e s o l u t i o n t o t h e dual L.P.

i s always

o b t a i n a b l e by a s s i g n i n g values o f s u f f i c i e n t l y h i g h degree t o a l l v a r i a b l e s . Hence f o r any g i v e n choice o f t h e u n i t parameters

J,

g. these two L.P.'s

always

have optimal f e a s i b l e s o l u t i o n s , w i t h equal o p t i m a l o b j e c t i v e f u n c t i o n values Pmax ( T Y ? )

(say).

Now, f o r t h e two extremal L.P.'s

over

f o l l o w i n g weak d u a l i t y p r i n c i p l e [15]:

2,

i t i s very s t r a i g h t f o r w a r d t o prove t h e

Using fields for semiring computations

67

No f e a s i b l e v a l u e o f t h e p r i m a l o b j e c t i v e f u n c t i o n can exceed any f e a s i b l e v a l u e o f t h e dual o b j e c t i v e function. I t f o l l o w s t h a t t h e v a l u e o f deg Pmax(~,g) i s independent o f t h e c h o i c e o f values

f o r t h e u n i t parameters, e l s e we c o u l d f i n d two s e t s of u n i t parameters (C,CI),

(!',ti) such t h a t o v e r 2 ( z ) t h e dual o b j e c t i v e f u n c t i o n c o u l d a t t a i n t h e v a l u e Pmax (2,g) and t h e p r i m a l o b j e c t i v e f u n c t i o n t h e v a l u e Pmax

(y',!')

>

deg Pmax

(s,$),

(TI,!.')

w i t h deg Pmax

which would v i o l a t e t h e weak d u a l i t y p r i n c i p l e o v e r

Thus we have used c o n v e n t i o n a l L.P.

2.

t h e o r y o v e r t h e o r d e r e d f i e l d Z(z) as a con-

v e n i e n t d e m o n s t r a t i o n o f t h e s t r o n g d u a l i t y p r i n c i p l e f o r a p a i r o f d u a l extremal o v e r 2 D5]:

L.P.'s

Both extremal L.P.'s

have an o p t i m a l f e a s i b l e s o l u t i o n , w i t h e q u a l i t y o f o p t i m a l

o b j e c t i v e f u n c t i o n value.

8.

ILLUSTRATION: EXTREMAL QUADRATIC PROGRAMMING

O p t i m i s a t i o n problems o v e r (2, @ , 8 ) i n v o l v i n g q u a d r a t i c o b j e c t i v e f u n c t i o n s do n o t seem t o have f i g u r e d p r o m i n e n t l y i n t h e l i t e r a t u r e . As an example o f such a problem, c o n s i d e r m in

2 @x ( ~@ ) 2

subject t o

8x Qy

@3

@ 2 @ x @y

(8.1)

x @ 3 @ y = 1.

Over I ( z ) we c o n s i d e r m in

Z'X2

subject t o

X

t

+

Z2XY

z3Y =

+

+

z3Y2 t z2x

Y

ITZ

X>,O,Y>,O

where

'TI

i s a u n i t parameter.

N o t i n g t h a t t h e Hessian m a t r i x o f t h e minimand, i.e.

i s p o s i t i v e - d e f i n i t e , we know t h a t t h e f o l l o w i n g Kuhn-Tucker C o n d i t i o n s (8.4), (8.5),

(8.6) g i v e s u f f i c i e n t c o n d i t i o n s f o r a minimum:

2z2x

+

z 2 ~t

x +

z2Y t

22 t h

2 z 3 ~+ 1 z3Y

-

ITZ

+

-

p = 0

-

2 3 ~

q =

o

= o

(8.4)

R.A. Cuninghame-Green

68

x >,o,

Y

>,o

(8.5)

p>,o,q>,o

px = qY = 0

(8.6)

Here (8.4) are the s t a t i o n a r i t y conditions; (8.5) the non-negativity conditions on the variables X , Y and the Lagrange multipliers p,q; and (8.6) the usual complementari t y conditions.

a i s the Lagrange multiplier f o r the equality constraint. These conditions a r e s a t i s f i e d by X = O ;

-2

Y =nz

= z2tn-2nZ-2-z-3;

=

o

(8.7)

1 = -2nz-'-z-3

whence x = --;y = -2; giving an otpimal objective function value of -1. Sensitivity analysis shows t h a t x may r i s e i n value t o 1 before i t violates the equation constraint in (8.1), b u t may only r i s e to value t o -3 before i t begins t o increase the object function value. Hence the general optimal solution i s :

x

d

-3, y = -2.

In general, a minimising extremal quadratic programing problem with linear equation and/or inequality constraints, such as (8.1), may be solved via a corresponding problem such as (8.2) over Z ( z ) , provided the l a t t e r problem has p o s i t i v e definite Hessian. For then i t can be shown by standard algebraic arguments t h a t the Kuhn-Tucker conditions are s u f f i c i e n t f o r a solution, and these conditions may be solved by L.P. derived processes such a s Wolfe's method, or principal pivoting. But,if we consider e.g. mi n

x ( 2 ) 0 3 @ x @Y 0 2 QY(2)

02 @ x O y

subject t o x @ 3 @ y = 1 ,

(8.8)

then i t i s easily seen t h a t the otpimal solution i s again x \< -3, y = -2. However, the corresponding problem over 2(z) now has non-positive-definite Hessian and leads us into the realms of non-convex quadratic programming.

9

ILLUSTRATION: THEORY OF POSITIVE MATRICES

Matrices over ( I , @ .@ ) correspond t o matrices over Y ( z ) with non-negative entries. The theory of such matrices i s , of course, highly developed and we can

Using fields for semiring computations

69

f i n d i n t e r e s t i n g correspondences between t h i s theory and t h a t o f matrices over

(2, 0 , Q ), by using the isomorphism (3.10). For example a square p o s i t i v e m a t r i x always has a Perron root, i.e. p o s i t i v e eigenvalue w i t h associated p o s i t i v e eigenvector.

a greatest

Hence a square m a t r i x

over (2, @ ,@)) always has a f i n i t e l y soluble eigenvector-eigenvalue problem ( i n f a c t t h e eigenvalue i s unique [6]). Again, from the theory o f i t e r a t i v e schemes r e l a t e d t o maximal-path-finding problems over

(2, @ ,6 )

i t i s known that, i f a m a t r i x a has no p o s i t i v e cycles,

then there e x i s t s a m a t r i x y s a t i s f y i n g (9.1 1

a@v@i = Y where i i s the i d e n t i t y matrix. To deduce t h i s r e s u l t using the isomorphism (3.10) we f i r s t remark t h a t i t i s s t r a i g h t f o r w a r d t o show t h a t i f a has no

p o s i t i v e cycles then the m a t r i x I - A , where A i s t h e corresponding m a t r i x over

I ( z ) and I i s the i d e n t i t y matrix, has p o s i t i v e p r i n c i p a l minors. known theorem f o r p o s i t i v e matrices and i s p o s i t i v e .

Thus

[lo]

=

Then by a w e l l -

r (say) e x i s t s

r i s p o s i t i v e and s a t i s f i e s Ar

Hence the corresponding m a t r i x

10.

we know t h a t (I-A)-'

Y

+I

=

r.

(9.2)

e x i s t s and s a t i s f i e s (9.1).

ILLUSTRATION: POWER SERIES OVER (IR, @ ,@ )

I n n 7 ] , we studied the properties o f power-series bo @ bl 0 x 0 b2@ x(') @ over (IR,

0 ,@ )

showing t h a t they converged f o r x

...

< P

(10.1)

and diverged f o r x >

p

where p = lim ~

-'

i n f br -r

(10.2)

Via the isomorphism (3.10), P corresponds t o the usual radius o f con.vergence -1 /r l i m i n f larl (10.3) rbr Of power-series w i t h c o e f f i c i e n t s ar = 2 The theory may be extended t o power-series o f a square m a t r i x a over (R, @ bo @ bl @ a @ which [17]

converges (resp. diverges) i f h(a)

the (necessarily unique) eigenvalue o f a.

... < p

,@ ) : (10.4)

(resp. A(a)

> P)

where x(a) i s

I n the l i g h t o f the present theory,

R.A. Cuninghame-Green

70

t h i s r e s u l t corresponds t o the convergence (resp. divergence) o f a m a t r i x powers e r i e s w i t h p o s i t i v e c o e f f i c i e n t s , f o r a p o s i t i v e m a t r i x whose Perron r o o t l i e s i n s i d e (resp. outside) the c i r c l e o f convergence o f the s c a l a r power-series.

11.

THE METHOD I N GENERAL

If ( G , @ ) i s a t o t a l l y ordered group and .I i s an i n t e g r a l domain, we may by standard algebraic methods c o n s t r u c t the group-ring Io(G) = I 1 (say). Then I l is again an i n t e g r a l domain, t o t a l l y ordered i f . I

is.

(We may thus continue the

... .

sequence Io , ,I ) l

(i,@,

Arguing exactly as i n Section 3 and i n [ll],we may show t h a t @ ), i.e. ( 6 , max, @ ) i s isomorphic t o t h e algebra o f equivalence classes o f t h e p o s i t i v e

6

cone o f t h e q u o t i e n t f i e l d F1 o f 11,where

= G

UI--l.

For the case when I. and G a r e Y, the z-method i s simply a convenient and i n t u i t i v e method o f c a r r y i n g o u t t h e c o n s t r u c t i o n o f F1, which i s i n t h i s case z(Z). For general p1 there i s a mapping d:F1

+

E

analogous t o the mapping deg, i . e .

d(X) e G f o r X # 0 d(0) =

-

(11 .l)

d(ab) = d(a) @ d ( b ) d(a + b)

d(a) @ d ( b )

The mapping d i n e f f e c t defines a non-Archimedean v a l u a t i o n

on F l .

It i s a

c l a s s i c a l r e s u l t f o r f i e l d s F~ so constructed t h a t they have a topological completion i n f i e l d s of formal Laurent series, which f o r t h e z-method a r e j u s t c l a s s i c a l Laurent s e r i e s w i t h i n t e g r a l c o e f f i c i e n t s and a t most a f i n i t e number o f terms of p o s i t i v e power. t o the valuation. [l]

Such Laurent s e r i e s are always convergent r e l a t i v e

The Giffler-Wongseelashote method consists e s s e n t i a l l y o f a d i r e c t construction o f generalised Laurent s e r i e s as well-ordered sequences o f elements o f a given z r b i t r a r y t o t a l l y ordered group G, as discussed i n [lZ] when .I

,

[15].

For t h e case

and G are 2, the two methods a r e t h e r e f o r e equivalent.

F i n a l l y , i f we c a r r y o u t the c o n s t r u c t i o n described above, f o r the dual s t r u c t u r e (G, @',Q ) = (G,min, 0 ) we o b t a i n the same q u o t i e n t f i e l d , and t h e mechanics o f

Using fields for semiring computations t h e isomorphism, v a l u a t i o n and c o m p l e t i o n a r e analogous.

71 We make use o f such a

dual c o n s t r u c t i o n i n t h e n e x t s e c t i o n .

12.

ILLUSTRATION: (SEMI) GROUP MINIMISATION PROBLEM

Suppose we a r e g i v e n c e r t a i n

m.

J

E

costs (j = 1

,...,n )

(0 < ml

f o r c e r t a i n elements o f an a b e l i a n semigroup ($,

$

g. E J

(j = 1

$,

We w i s h t o f i n d t h e cheapest element o f


q ) o r i f a j E N(ar) qr 9 1J u)

resp. (E,Yi),

be an independence system and (E,yl)

k t h a t ;1= .n3..Then 1=k

i = 1 ,... ,k.

,...,( E,Jk)

be

we c a l l (E,^3) p e r f e c t r e l a t i v e t o (E,gl),.

(E,jk), i f f o r a l l c E N " any m u l t i p l i c a t i o n of m i n - p r o p e r t y r e l a t i v e t o (Ec13,) ,. .. ,(EC,gk 1.

(EJ)

b y c, (EcJc),

.. ,

has t h e max-

C

Obviously, f o r any X G E t h e r e s t r i c t i o n (XJX) (EJ),

given b y Y X : = { I € 3 : I

m a t r o i d s , b u t r e s t r i c t e d t o X.

C o r o l l a r y 2.5

G

Moreover, b y P r o p o s i t i o n 2.2 we i m m e d i a t e l y o b t a i n

Any m u l t i p l i c a t i o n (Ec,JC)

o f an independence system (E,J),

i s p e r f e c t r e l a t i v e t o t h e m a t r o i d s (E,jl) c o r r e s p o n d i n g m a t r o i d s (Ec,J1)

o f such an independence system

i s p e r f e c t r e l a t i v e t o t h e same s e t o f

XI,

,. . . ,(

E

,...,( E , j k ) ,

which

i s perfect relative t o the

J ). kc

We remark a t t h i s p o i n t t h a t an independence system (E,J)

need n o t b e p e r f e c t k r e l a t i v e t o any s e t of m a t r o i d s (E,Il) ,...,( E,&) such t h a t 7 = .n1., a l t h o u g h 1=1 1 i t i s p e r f e c t r e l a t i v e t o some s p e c i f i c such s e t . However, i f (EJ) i s p e r f e c t

103

Independence systems and perfect k-matroid-intersections r e l a t i v e t o (EJ1) (E,jk),

...,

,..., (E,gk),

(E,rkJ,such

gi.

that’J=.r)

t h e m a x i n i n - p r o p e r t y n o t o n l y fo:=IE,’I1) multiplications.

,...,

t p n t h i s i s t h e case f o r any s u p e r s e t (EJl)

Besides, i t i s r e a l l y necessary t o c l a i m i t s e l f , b u t a l s o f o r a l l o f i t s proper

To see t h i s c o n s i d e r f o r i n s t a n c e those m a t r o i d s (E,7A), which

c x 6 r ( A ) , i . e . whose systems o f c i r c u i t s j u s t eEA e c o n s i s t o f a l l subsets o f A h a v i n g c a r d i n a l i t y r ( A ) + l , and r e l a t i v e t o which (E,2)

a r e induced by t h e i n e q u a l i t i e s has always t h e max-min-property.

One s t a r t i n g p o i n t f o r i n t r o d u c i n g t h i s concept o f p e r f e c t i o n has been t h e c l a s s of s t a b l e set-independence systems ( E J )

o f p e r f e c t graphs, i . e .

those graphs,

f o r which minimum number o f c l i q u e s i n r ( X ) = G[X] needed t o cover t h e s e t X

for all

cE,

C l e a r l y , (E,T) can be r e p r e s e n t e d as t h e i n t e r s e c t i o n o f t h o s e m a t r o i d s , whose system o f c i r c u i t s a r e g i v e n by t h e edges o f a maximal c l i q u e . {K1,

...,K k l

has t h e max-min-property r e l a t i v e t o (E,jl) ,... ..., k. By a lemma o f Berge

n o t d i f f i c u l t t o see t h a t (EJ) (EJk), (cf.

[l])

Moreover, i f

i s t h e s e t o f a l l maximal c l i q u e s i n a p e r f e c t graoh G, t h e n i t i s

where

(E,Yi)

,

i s induced by Ki f o r i = 1,

t h e m u l t i p l i c a t i o n o f (E,’JI) by c €LNn i s a g a i n t h e s t a b l e - s e t indepen4

dence system o f a p e r f e c t graph Gc, whose maximal c l i q u e s correspond, up t o r e p e t i t i o n s , t o t h e m a t r o i d s (Ec,’Ji),

i = 1 ,... ,kc.

Hence, (Ec,Zc) has t h e max-

,. . . ,(E ,2 ) and, t h e r e f o r e , (E,2) i s p e r f e c t

m i n - p r o p e r t y r e l a t i v e t o (Ec,fl)

kc i n t h e sense o f D e f i n i t i o n 2.4.

r e l a t i v e t o (E,Y1), ...,( EJk)

A POLYHEDRAL DESCRIPTION I n t h i s s e c t i o n we w i l l prove t h a t a p o l y h e d r a l d e s c r i p t i o n o f an independence system (EJ),

which i s p e r f e c t r e l a t i v e t o (E,jl)

c

xe 6 r l ( A )

for all

,...,( E , j k ) , AS

i s g i v e n by

E,

eEA

c xe

4 rk(A)

(3.1)

f o r a l l A G E,

eEA xe

3

0

for all e

E

where ri i s t h e r a n k - f u n c t i o n o f t h e m a t r o i d (E,Ti),

E,

i = 1,

... ,k.

r e f e r t o a theorem, which has a l r e a d y been used b y Edmonds [4],

For t h i s we

Chvdtal [2]

others:

Theorem 3.2

L e t S be a f i n i t e s e t o f s o l u t i o n s x

E

W”

o f t h e system o f

and

R. Euler

104

i nequal i t i e s

xe aiexe

I

B

f o r a l l e c E,

0

for all i

\c b .

1

eF E

(3.3)

I.

E

Then the s e t of a l l s o l u t i o n s of ( 3 . 3 ) i s t h e convex hull of S i f and only i f , n f o r every vector c r Z we h a v e max =

ICX :

min

i

x

-

iLI

E

Si

W ic1,

b. : li:O i i

x

~

~

aW ~ ek.E?.~

a

c

~

icI

n Now l e t c = ( c l ,..., c n ) E 72 . W.1.o.g. we can d e l e t e a l l 5 0 - c o e f f i c i e n t s of c and go over t o t h e corresponding r e s t r i c t i o n (E;2E,). I n a d d i t i o n , we can d e l e t e the f r e e elanentsof ( E , j ) as well as t h e i r c o e f f i c i e n t s in c , s i n c e e s t a b l i s h i n g t h e max-min-equality in Theorem 3.2 f o r such a r e s t r i c t i o n (E"JE,,) of (E,Y) will immediately lead t o the r e l a t e d one f o r (E,X). I t i s now our aim t o apply Theorem 3.2 t o the system of i n e q u a l i t i e s ( 3 . 1 ) and v i a the max-min-equality given t h e r e show (3.1) t o represent a polyhedral desc r i p t i o n of (E,?I), i . e . the v e r t i c e s of t h a t polyhedron correspond exactly t o the independent s e t s . For t h i s we multiply ( E J ) by c and obtain ( E c J C ) . Let t h e r e s u l t i n g k c matroids be indexed such t h a t any of the following blocks of k of than, ( Ec,jl

1,. . . * ( E c , 3 k )

. .. ;( Ec*Ikc-k+l correspond to

(E,7,),. . . , ( E , & ) ,

3 .

; ( Ec,lk+l ) 9 .

'.*(E

..

3

(Ec,Y2k) ;

kc

i .e. ( E c J 1 ,(Ec$xk+l1 ,. . . ,(

Ec'flkc-k+,)

arise

from ( E J 1 ) by s u b s t i t u t i n g E by a transversal T i of A , where { T . ) . is a 1 1 = 1 , ... fixed sequence of a1 1 t r a n s v e r s a l s of A , ( Ec,y2),( Ec,gk+2) ,. . . ,( E c ,lkc-k+2) in exactly the same way, and so on.

a r i s e from ( E J 2 )

Claim 3.4

Proof

Let I

L

z c and

i z 1.1,. . . , n i .

Now we s t a t e the following

Then

By assumption, t h e r e e x i s t s a c i r c u i t C of (Ec,JC) i n IU : e l . B u t then, by d e f i n i t i o n o f (Ec,&), the set ( C \ i e ? ) U i e ' ! i s a l s o a c i r c u i t of (Ec,&)

105

Iirdependence systems and perfect k-matroid-in tersections for a l l e'

( I e i } U E i ) , which proves t h e c l a i m .

E

Claim 3.4 says t h e f o l l o w i n g :

(Bn E)

If B i s a base of Ec, IBI = r c ( E c ) , t h e n

E

3

and, i n p a r t i c u l a r ,

161 = cxgnE, where xBnE i s t h e i n c i d e n c e v e c t o r o f BnE.

What we s t i l l need i s a

s o l u t i o n y o f t h e system o f i n e q u a l i t i e s 1

YA k

i = 1 ,..., k,

(3.5)

i

z

c

for all A GE,

3

yA >, c j

f o r a l l j = 1,

..., n

i=l e.EAcE J such t h a t

k 161 =

C

L e t (Ei,

i = l,...,kc)

c

c

yb r i ( A ) .

i=lAGE

be a p a r t i t i o n o f E c s a t i s f y i n g rc(Ec) =

kC

c

ri(EF).

i=1

,..., (Ec,Ijtk),

To any b l o c k o f k m a t r o i d s (Ec.Ijtl)

j = O,k,Zk

,..., kc-k,

there

o f A , which has been s u b s t i t u t e d f o r E i n (j/k)+l and a l l these t r a n s v e r s a l s a r e d i s t i n c t . i n a d d i t i o n , t o any element

e x i s t s a unique transversal T (EJ),

eEEc t h e r e i s a t l e a s t one b l o c k o f k r n a t r o i d s , i n whose i n t e r s e c t i o n e i s n o t a f r e e element. Now c o n s i d e r t h e s e t s

blockwise.

Then, o b v i o u s l y , e

E

Ec i s c o n t a i n e d i n one o f t h e s e t s , say E g .

3.

'f

such t h a t e i s n o t f r e e i n fl belongs t o a b l o c k o f k s e t s EC ,..., Eilk J+1 i=l I f , however, t h i s i s n o t t h e case, i . e . e i s ., we l e a v e e i n t h a t s e t E i .

t h i s E: J+1

f r e e i n t h a t i n t e r s e c t i o n , we can t a k e i t o u t f r o m E:

and p u t i t i n a s e t E:,

such t h a t e i s n o t f r e e i n t h e corresponding i n t e r s e c t i o n . T h i s i s p o s s i b l e f o r kC C a l l e E Ec w i t h o u t i n c r e a s i n g t h e v a l u e Now, l e t (E:, i = l,...,kc) ri(Ei). be a l r e a d y m o d i f i e d a c c o r d i n g l y .

j=l

Then i n any o f t h e s e t s E F t h e r e can be a t most

one element f r o m { e i } U E; f o r i = 1,. ..,n.

Now we r e p l a c e i n any o f t h e s e t s

Ei t h e elements e ' f r o m E; b y t h e corresponding ei, r e s u l t i n g f a m i l y (At,

t = l,...,kc)

i = 1, ..., n, and c o n s i d e r t h e

( o f n o t n e c e s s a r i l y d i s t i n c t subsets o f E ) , k -1. t = l , k t l , ...,p k t l ) , where p = To any

"4

i n p a r t i c u l a r t h e s u b f a m i l y (At, 1 A s E we a s s i g n t h e v a l u e yA, which i s equal t o t h e number o f times A i s o c c u r r i n g i n t h a t subfamily.

Clearly, rl(At)

=

rt(Ei)

for t

=

1 ,k+l,.

.. ,pk+l,

106

R. Eulcr

and i n t h i s manner we o b t a i n a number o f d u a l v a r i a b l e s f o r t h e system o f inequalities

z

f o r a l l A 6 E.

x e \c r l ( A )

eEA Next, we c o n s i d e r t h e i n d i c e s 2,k+2, ...,p k+2 and f i n d a v e c t o r yf\, and so on, ,Ek u n t i l we have y

E

WL

, which i s

f e a s i b l e f o r (3.5).

Since

k

we have, t o g e t h e r w i t h Theorem 3.2:

Theorem 3.6

L e t (E,’3) be an independence systen,which i s p e r f e c t r e l a t i v e t o t h e

m a t r o i d s (E,S1)

,... ,(E,’jk).

Then a p o l y h e d r a l d e s c r i p t i o n o f (EJ),

convex h u l l of t h e i n c i d e n c e v e c t o r s o f a l l members o f f ,

i.e.

the

i s g i v e n by ( 3 . 1 ) .

I t f o l l o w s , t h a t a l s o t h e f o l l o w i n g l i n e a r system c o n s t i t u t e s a p o l y h e d r a l

d e s c r i p t i o n of such an (E,T):

x

e

3 0

for all e

E

E.

TOTAL DUAL INTEGRALITY D e f i n i t i o n 4.1

L e t A r e s p . b be a r a t i o n a l - v a l u e d mxn-matrix r e s p . m-vector.

Then we say t h a t t h e l i n e a r system Axgb has t h e p r o p e r t y o f t o t a l d u a l i n t e g r a l i t y (TDI), i f , f o r any i n t e g e r o b j e c t i v e f u n c t i o n c such t h a t max{cx: Axsbl e x i s t s , t h e r e i s an i n t e g e r optimum dual s o l u t i o n . T h i s p r o p e r t y has been i n v e s t i g a t e d w i t h i n t h e c o n t e x t o f 2 - m a t r o i d - i n t e r s e c t i o n s by G i l e s

[8l

( s e e a l s o [16]),

o f submodular f u n c t i o n s on g r a p h s b y Edmonds and

G i l e s [7l and o f i n t e g e r p o l y h e d r a by G i l e s and P u l l e y b l a n k [ll]. I n p a r t i c u l a r , S c h r i j v e r [15]

has shown t h a t any r a t i o n a l p o l y h e d r o n i s t h e s o l u t i o n s e t o f a

unique m i n i m a l i n t e g e r l i n e a r system h a v i n g t h e T D I - p r o p e r t y . The p r o o f o f Theorem 3.6 shows t h a t , i f (E,’2) i s p e r f e c t r e l a t i v e t o m a t r o i d s

( E J , ) , . .. , ( E , & ) , t h e n t h e l i n e a r system ( 3 . 1 ) has t h e T D I - p r o p e r t y . Moreover, as i n t h e case o f 2 - m a t r o i d - i n t e r s e c t i o n s (see 0 6 ] ) , a l s o t h e l i n e a r system (3.7) has t h e T D I - p r o p e r t y .

107

Independence systems and perfect k-ma troid-in tersections

L e t us now deduce a converse r e l t i t i o n :

Theorem 4 . 2

I f t h e l i n e a r system ( 3 . 1 ) has t h e T D I - p r o p e r t y , t h e n (E,Y)

proof BY T D I , we have f o r e v e r y v e c t o r c

is

,... , ( E , l k ) .

p e r f e c t r e l a t i v e t o t h e g i v e n m a t r o i d s (E,j,)

LNn

E

max {cx : x i s t h e i n c i d e n c e v e c t o r o f an independent s e t } k = min

I c

c yAi ri(A)

k :

z

z

i >, c j yA

f o r j = 1,

i yA 3 0

f o r a l l A&,

...,n,

i=l e .EASE

i=l AGE

J

and t h e r e i s always an i n t e g e r optimum s o l u t i o n y.

i = 1,

...,k l

F o r t h i s s o l u t i o n y we can

always achieve k

i c y = c . i=l e.EAC-E A J J

c

f o r j = l ,

i f we have ,, > ' I i n ( 4 . 3 ) f o r some j A o f E c o n t a i n i n g e . as w e l l as an i n d e x i J by t h e t r a n s f o r m a t i o n since,

E

E

...,n,

(4.3)

11 ,...,nl, we can c o n s i d e r a s u b s e t such t h a t y; > 0. Then

{l,...,k}

we can decrease t h e sum i n (4.3) by 1 w i t h o u t i n c r e a s i n g t h e optimum v a l u e k c z yAi ri(A), s i n c e r a n k - f u n c t i o n s a r e monotone. By r e p e t i t i o n o f such a i=lAGE t r a n s f o r m a t i o n ( 4 . 3 ) can b e achieved. Now we m u l t i p l y (E,J)

by c t o o b t a i n (Ec,&)

and f r o m o u r optimum d u a l s o l u t i o n y

we w i l l c o n s t r u c t a p a r t i t i o n (ET, i = l , . . . , k c )

o f Ec such t h a t

k-

C

z ri(E:).

rc(Ec) =

i=1 L e t AsE, i

{ l ,...,k } be such t h a t yb >, 1.

To any e . E A we choose a d i s t i n c t J i element e l f r o m { e . ) U E l and r e p l a c e A by t h e s e t { e ' : e . E A}. I f yA 3 2, J J J J J we r e p e a t t h i s replacement b y a s e t { e " : e . E A, e " # e l } and so on, u n t i l we i j~ J J g e t yA p a i r w i s e d i s j o i n t subsets o f Ec. Now we choose t h e n e x t A: i ' such t h a t i' y A , b l and proceed s i m i l a r l y t o o b t a i n ybyb, p a i r w i s e d i s j o i n t subsets o f Ec and E

.

I

so on, u n t i l we g e t a p a r t i t i o n (ET, i = 1,. ..,kc)

o f Ec.

Note t h a t a l l t h e s e

s e t s ET c o n s t i t u t e p a r t i a l t r a n s v e r s a l s Ti o f t h e f a m i l y A ( s e e s e c t i o n 2) and,

R. Eiilrr

108

therefore, we can enlarge these Tls t o t r a n s v e r s a l s of A and assign a unique 1

T h i s matroid

matroid (Ec,2j.) t o Ei.

a r i s e s from one of

(€,Il) ,...,( E,&)

by

s u b s t i t u t i n g t i e enlarged transversal f o r E and, t h e r e f o r e , a f t e r a l l these

c

Hence, r '(E ) = C

C

1 r . (E.). Jj 1

C

We can show the corresponding r e l a t i o n f o r X 5 E by choosing a vector c '

E

Zn

E X , c! 0 f o r e . i X, as well a s f o r an a r b i t r a r y J J J 3 Ec, c already given, by choosing a n appropriate vector c ' and then constructing an optimal p a r t i t i o n of X . This completes the proof.

such t h a t c! = 1 f o r e . X

C_

FURTHER EXAMPLES

Example 5 . 1 Matroids (E,g) have t h e max-min-property r e l a t i v e t o themselves. I t remains t o show t h a t they keep t h i s property a f t e r every proper m u l t i p l i c a t i o n by a vector c

g:

CN'.

Let us consider t h e family A = (ie,-U

E; ,..., {e,iL) E;).

Since multiplying ( E J ) by c does n o t depend on the order of t h e c i , we may assume t h a t c1 ;c 2 p . . . ,c holds. Now we p a r t i t i o n E ( s e e Figure 5 . 2 ) in a s e t of n C ( p a r t i a l ) t r a n s v e r s a l s T i , i = 1 , . - . , c l , o f A , such t h a t T1 = E, T 2 = { e1l ,..., e nl ) ,

.... .

I f , f o r instance, c1 a c 2

5

1 , the l a s t transversal Tc

{ e1c , - l : .

just c o n s i s t s of 1

Figure 5.2

hideperidcwce

SIs t e m

109

arid perfect k-matroid-iritersections

To any o f these ( p a r t i a l ) t r a n s v e r s a l s Ti we a s s i g n t h a t u n i q u e m a t r o i d ( E c , r . ) , Ji

which we o b t a i n from s u b s t i t u t i n g TiU T i f o r E i n t h e g i v e n m a t r o i d (E,J),

,. . . , e l T . l + l i .

where T i c o n s i s t s o f t h e s e t Ien,en-l m u l t i p l i c a t i o n o f (E,Y) f o r a1 1 i = 1 ,.

. . ,cl,

where Ei = E\T!.

__ Proof

C l e a r l y , r . (T.) = r ( E . ) 1 Ji 1

Moreover, we have t h e f o l l o w i n g

1

P r o p o s i t i o n 5.3 where r

By t h e d e f i n i t i o n o f a

by c t h i s i s always p d s s i b l e .

rc(Ec) =

I: r . (Ti),

i=l Ji

denotes t h e r a n k - f u n c t i o n o f (Ec,Tc).

C

I t i s s u f f i c i e n t t o f i n d a member

I

of

2c such

t h a t (11 =

t h i s we a p p l y a s l i g h t v a r i a n t o f t h e Greedy-Algorithm (see [5]) Step 1 )

1:=0=:J,

i:=1

c1 I: r . (Ti). i=l Ji

For

t o (Ec,IC):

Step 2)

-+

I := IU( [ei }UE; )

1

Yes :

Step 2)

J

I f i = n, Stop.

One e a s i l y v e r i f i e s t h a t I 1 f ) Ti\ C. -1

]I! = c

Step 3 )

U ieil ~ 2 ? No :

Step 3 )

+

J :=Jut ei }

+

Step 3 )

Otherwise, i:=i+l and =

f

Step 2).

r . ( T . ) f o r a l l i = 1, Ji 1

... ,cl

and t h u s

r . (T.) J .

i=1

1

1

By an analogous c o n s t r u c t i o n t h e max-min-property

can be shown f o r a l l X relative t o itself.

s Ec,

r e l a t i v e t o (Ec,Il)

so t h a t t h e m a t r o i d (EJ)

i s shown t o be p e r f e c t

We p o i n t t o t h e f a c t , t h a t t h i s r e s u l t i s c l o s e l y r e l a t e d t o

Edmonds' work on m a t r o i d s and t h e Greedy-Algorithm ( c f . constitutes

,. . . ,(Ec,gkc)

PI),which,

obviously,

a n e f f i c i e n t procedure f o r computing t h e rank o f Ec3(Ec,2,)

a m u l t i p l i c a t i o n o f t h e m a t r o i d (E,Y).

being

Such a procedure even e x i s t s f o r t h e

problem of d e t e r m i n i n g a maximum w e i g h t independent s e t i n (Ec,Tc),

given a weight-

f u n c t i o n c ' over Ec.

Example 5 . 4

L e t two m a t r o i d s (E.2,).

corresponding 2 - m a t r o i d - i n t e r s e c t i o n

(EJ2) (EJ).

be g i v e n and l e t u s c o n s i d e r t h e We do n o t know o f a d i r e c t proof

( i n t h e sense o f D e f i n i t i o n 2.4) f o r showing t h a t (E,J) (E,Jl)

and ( E , j 2 ) .

However, i t i s w e l l known ( s e e [6],

i s perfect relative to [8]) that the l i n e a r

110

R. Euler

system

z xe
)

i s perfect relative

(EsT2).

t o (E,?),

Example 5.6

Let G = (V,E)

be a f i n i t e , undirected, loopless graph having vertex

s e t V and edge s e t E . For S c V l e t scs)denote the s e t of edges of G having exactly one end i n S and u(s)t h a t s e t of edges having both ends i n S; moreover, l e t Q:= {S c V : I S \ >, 3, IS/ odd1 and qs:= 1/2( IS\-1) f o r a l l S E Q. Edmonds [4] showed t h a t a polyhedral d e s c r i p t i o n o f t h e matching independence system (EJ) of G , i . e . t h e convex hull o f incidence vectors of a l l those subsets of E, no two elements of which a r e incident t o a common vertex, i s given by for all e

E

E,

Xe"l

for all v

E

V,

xe

for all S

E

Q.

xe

z

b

0

(5.7)

eE6(v)

z eEY(

\c

qs

S)

proved t h a t the system (5.7) a l s o has the TDI-property so t h a t again by Theorem 4.2 (E,T) i s shown t o be perfect r e l a t i v e t o t h e matroids ( E , & ) , v E V , and ( E , I S ) , S E Q , a s induced by the i n e q u a l i t i e s z x e < 1 resp. z xe c qs.

Cunningham and Marsh [3]

ess(v)

ecy( S )

Exmple 5.8 We would l i k e t o present now an independence system (E,C), which i s not perfect r e l a t i v e t o any s e t of matroids. Consider t h e s t a b l e - s e t independence system of the graph G = (E,C), E = t l , ...,6 1 , a s shown i n Figure 5.9.

I t i s well known ( s e e f o r instance [14])

5

z x . + 2x6

4 2 i=l defines a f a c e t of t h e convex hull of incidence vectors o f s t a b l e s e t s in G .

that the inequality

'

Clearly, by Theorem 3 . 6 , (E,C) cannot be p e r f e c t r e l a t i v e t o a s e t of matroids. k To show t h i s d i r e c t l y , suppose i t i s . Then '=I = n 3. f o r a s e t of k matroids i=l (E.2,) ,..., ( E , I k ) . I n p a r t i c u l a r , r ( E ) = ,I r i ( E i ) f o r sane p a r t i t i o n 1=1

(Ei, i

=

1 , ...,k ) of E.

Since r(E) = 2 , r i ( E )

3

2 f o r i = 1 ,..-,k .

Suppose t h e r e

Independence systems and perfect k-matroid-intersections

111

1

Figure 5.9 i s an i n d e x j such t h a t r . ( E . ) = 1.

J

J

Then GEEj]

can o n l y be a s i n g l e v e r t e x , a

s i n g l e edge o r a t r i a n g l e i n G, s i n c e r ( X ) = 2 f o r any 4-element s u b s e t X of E . However, r(E\E .) = 2 f o r any such s e t E and so r ( E ) = 2 cannot be achieved. J j Hence, r . ( E - ) = 0 o r 2 f o r a l l i = 1, k and, t h e r e f o r e , r ( E ) = ri(E) = 2 f o r 1 1

...,

some i n d e x i

E

...

{l, , k l .

Now we m u l t i p l y (E,I)

by c = (l,l,l,l,l,Z)

and

observe t h a t t h e r e i s no c l i q u e o f s i z e 4 i n t h e corresponding graph Gc = (EC$,) (see F i g u r e 5.10). 1

F i g u r e 5.10 So a g a i n 2 = r ( E c ) = ri(Ec)

m u l t i p l i c a t i o n o f (E,q)

f o r some index i

ri(Ec)

2,

{l,...,k,

...,Z k } .

But a f t e r t h e

by c t h e rank o f any o f t h e m a t r o i d s (E,;Ji),

i n c r e a s e s b y 1 and s i n c e (Ec,Ii), (Ec,jk),

E

3 for a l l i

E

i = 1, ...,k,

...,2k a r e " c o p i e s " o f (Ec,I,) ,..., 11, ..., k ,..., 2k1, a c o n t r a d i c t i o n t o t h e p e r f e c t i = k+l,

ness assumed f o r ( E , j ) .

CONCLUSIONS AND OPEN PROBLEMS I n t h i s paper we have i n t r o d u c e d t h e concept o f independence systems, which a r e

R. Euler

112

p e r f e c t r e l a t i v e t o a s e t o f matroids.

I n p a r t i c u l a r , a polyhedral d e s c r i p t i o n

has been presented, an i n t e r r e l a t i o n t o t o t a l d u a l i n t e g r a l i t y has been establ i s h e d and i t c o u l d be shown t h a t t h i s c l a s s i s c l o s e d under m u l t i p l i c a t i o n . I n c o n n e c t i o n w i t h independence systems (EJ)

one i s o f t e n i n t e r e s t e d i n s o l v i n g

a problem o f t h e f o r m Maximize g i v e n a weight f u n c t i o n c

I ce s u b j e c t t o I e: I

(6.1)

EJ,

..IR E ' . The q u e s t i o n a r i s e s , i f , f o r t h e case o f an ~

i n d e p e n d e m system, w h i c h i s p e r f e c t r e l a t i v e t o ( E J l ) a l g o r i t h m f o r t h e s o l u t i o n o f (6.1) e x i s t s .

,. . . ,(E,Xk),

a polynomial

More s p e c i f i c a l l y , c a n e l l i p s o i d

methods b e used, as i n t h e case o f p e r f e c t graphs ( s e e [12]),

t o s o l v e (6.1)

polynmially? As a l r e a d y p o i n t e d o u t f o r m u l t i p l i c a t i o n s o f a m a t r o i d , p o l y n o m i a l a l g o r i t h m s f o r s o l v i n g ( 6 . 1 ) o v e r s p e c i f i c c l a s s e s o f independence systems ( E , j ) may be used t o s o l v e t h e r e l a t e d problems o v e r m u l t i p l i c a t i o n s o f (E,Y).

Moreover, i f

I C ! i s p o l y n o m i a l i n / E l , i t seems t o be p o s s i b l e t o i n v e r t t h e o D e r a t i o n o f (E,2) by an a p p r o p r i a t e c h e c k i n g o f t h e c i r c u i t s and, t h e r e b y , t o

multiplying

r e d u c e ( 6 . 1 ) o v e r (E,?)

t o a s i m i l a r problem o v e r ( E ' , Y ) ,

which i s n o t a proper

m u l t i p l i c a t i o n o f any o t h e r independence system, i n a p o l y n o m i a l number o f steps. We conclude w i t h a l i s t o f o t h e r open q u e s t i o n s w i t h i n t h i s framework:

-

Are t h e r e p o l y n o m i a l a l g o r i t h m s o f p u r e l y c o m b i n a t o r i a l a n a t u r e t o d e t e r m i n e a maximum ( w e i g h t ) independent s e t r e s p . a minimum p a r t i t i o n of E i n t o independ e n t subsets f o r a g i v e n p e r f e c t ( r e l a t i v e t o (E,gl),.

. . ,(E,Yk))

independence

system;

-

a r e t h e f a c e t s of t h e p o l y t o p e d e s c r i b e d by (3.1) g i v e n b y t h o s e subsets o f E, which a r e c l o s e d and i n s e p a r a b l e r e l a t i v e t o t h e r a n k - f u n c t i o n r o f

-

(E,2);

how do concepts and r e s u l t s on p e r f e c t graphs such as odd c y c l e s and a n t i c y c l e s , g e n e r a l i z a t i o n s of which c o u l d g i v e more i n s i g h t i n t o t h e f a c e t t i a l s t r u c t u r e o f independence system p o l y h e d r a , t h e c h a r a c t e r i z a t i o n o f p e r f e c t graphs as g i v e n b y Lov6sz L13:

e t c . c a r r y o v e r t o such p e r f e c t independence

systems; and l a s t , b u t n o t l e a s t

- how do o t h e r w e l l known c l a s s e s o f independence systems such as those, which a r i s e from d e g r e e - c o n s t r a i n e d subgraphs (see [4]) and

clq) f i t

i n t o t h i s framework,

o r m a t c h i n g - f o r e s t s ( s e e [9]

and a r e t h e r e i n t e r e s t i n g examples beyond

those p r e s e n t e d h e r e and those, which a r e known from t h e t h e o r y o f p e r f e c t graphs?

Independence systems and perfect k-ma froid-intersections

113

ACKNOWLEDGEMENT

I am most g r a t e f u l t o P r o f e s s o r Claude Benzaken f o r v e r y v a l u a b l e s u g g e s t i o n s .

REFERENCES

[l] Berge, C . ,

Graphes e t Hypergraphes (Dunod, P a r i s , 1973).

[2]

Chvdtal, V., On c e r t a i n p o l y t o p e s a s s o c i a t e d w i t h graphs, J . C o m b i n a t o r i a l Theory B 18 (1975) 138-154.

[3]

Cunningham, W.H. and Marsh, A.B., A p r i m a l a l g o r i t h m f o r optimum matching, Math. P r o g r a m i n g Study 8 (1978) 50-72.

[4]

Edmonds, J . , Maximum matching and a p o l y h e d r o n w i t h 0 , l - v e r t i c e s , Nat. Bur. Stand. Sect. B 69 (1965) 125-130.

[5]

Edmonds, J . , M a t r o i d s and t h e greedy a l g o r i t h m , Math. Programming 1 (1971) 127-136.

[6]

Edmonds, J . , M a t r o i d i n t e r s e c t i o n , Annals o f D i s c r e t e Math. 4 (1979) 39-49.

[7]

Edmonds, J . , and G i l e s , R., A min-max r e l a t i o n f o r submodular f u n c t i o n s on graphs, Annals o f D i s c r e t e Math. 1 (1977) 185-204.

[8]

G i l e s , R., Submodular f u n c t i o n s , graphs and i n t e g e r polyhedra, Thesis, Univ. o f Waterloo, 1975.

[9]

G i l e s , R.,

Optimum matching f o r e s t s I, Math. Programming 22 (1982) 1-11.

PO]

G i l e s , R.,

Optimum matching f o r e s t s 11, Math. Programming 22 (1982) 12-38.

el]

G i l e s , R., and P u l l e y b l a n k , W . , T o t a l dual i n t e g r a l i t y and i n t e g e r polyhedra, L i n e a r Algebra and i t s A p p l i c a t i o n s 25 (1979) 191-196.

e2]

G r o t s c h e l , M., Lovasz, L., and S c h r i j v e r , A., Polynomial A l g o r i t h m s f o r P e r f e c t Graphs, Report No. 81178-0R, I n s t i t u t f u r Okonometrie und Operations Research d e r U n i v e r s i t a t z u Bonn (1981).

e3]

Lovdsz, L., A c h a r a c t e r i z a t i o n o f p e r f e c t graphs, J . C o m b i n a t o r i a l Theory B 13 (1972) 95-98.

e4]

S b i h i , N., Etude des s t a b l e s dans l e s graphes sans @ t o i l e s , Thesis, Univ. S c i e n t i f i q u e e t Medicale de Grenoble ( 1 9 7 8 ) .

c5]

S c h r i j v e r , A., On t o t a l d u a l i n t e g r a l i t y , L i n e a r Algebra and i t s A p p l i c a t i o n s 38 (1981) 27-32.
a and

E

=

~

a b, and s i n c e b i s antisymmetric, t h i s i m p l i e s a =

E

E.

Note t h a t axiom ( A 3 ) may be unnecessary f o r t h e study o f c e r t a i n classes o f problems.

This axian i s nevertheless i m p o r t a n t t o prove uniqueness o f s o l u t i o n s

f o r epuations i n dio'ids. The s t r u c t u r e s

(R+U {+-I,

max, min),

(RU I + - } ,

min, + ) , ([O,l],

max, x ) ,

Linear algebra in dioids @?,

+,

x ) f o r i n s t a n c e , a r e dio’ids.

Gondran

3

-

149

F o r a d e t a i l e d r e v i e w o f such examples, c f .

Minoux 1979 Ch. 3 and Zimmerman 1981.

SOLVING LINEAR EQUATIONS AND SYSTEHS I N DIOIDS

L e t (S, @ t h e form

, @)

be a

dio’id and

suppose we have t o s o l v e a l i n e a r e q u a t i o n o f x = a

0

x @ b

(1)

(where a,b E S a r e g i v e n ) . We n o t e t h a t ( 1 ) reduces t o f i n d i n g a f i x e d p o i n t f o r t h e ( l i n e a r ) mapping: rdx : f ( x ) = a @ x @ b. A p p l y i n g t o ( 1 ) t h e s u c c e s s i v e a p p r o x i m a t i o n scheme: xo =

(2)

E

Xk+l = a

@

@

’k

(3)

l e a d s by i n d u c t i o n t o : x ~ =+ ( e~ @ a @ a’

w h e r e v k : ak = a @ a @

...

@

...

@ ak) @ b

@ a (k times).

Convergence o f ( 2 ) - ( 3 ) i s t h u s s t r o n g l y r e l a t e d t o t h e convergence o f : a(k) = e

0

a

0

For m o s t examples o f p r a c t i c a l imoortance, t o p o l o g i c a l p r o p e r t i e s on dio’ids: sufficient.

a*

0

... 0

ak .

i t i s n o t necessary t o i n t r o d u c e

f i n i t e convergence ( d i s c r e t e t o p o l o g y ) w i l l be

Thus we a r e l e d t o :

Definition 1 = a(p)

An element a e S i s c a l l e d p - r e g u l a r i f :

We say t h a t an element a E S i s r e g u l a r i f t h e r e e x i s t s a p such t h a t a i s pr e g u 1a r . The d e f i n i t i o n above a l s o i m p l i e s :

= a(’)

(Ur

>

0).

A s p e c i a l case f o r p - r e g u l a r elements a r e p - n i l p o t e n t elements i . e . such t h a t ap = E . Assuming p - n i l o p t e n c y i s s t r o n g e r t h a n p - r e g u l a r i t y b u t sometimes necessary when o t h e r p r o p e r t i e s (such as c o m m u t a t i v i t y o f @ ) a r e l a c k i n g .

T h i s i s t h e case,

f o r i n s t a n c e , f o r t h e g e n e r a l i z e d p a t h a l g e b r a s s t u d i e d b y Minoux 1976.

M. Gondran and M. Minoux

150

Property 1 For each p - r e g u l a r e l a n e n t a r S, t h e r e e x i s t s a * € S c a l l e d quasi-inverse o f a and such t h a t : a* = l i m a(‘) k-w

= ,(PI

= a(P+l)

...

Property 2 Let a e S be p - r e g u l a r w i t h quasi-inverse a*, and consider t h e f o l l o w i n g l i n e a r equation : x = a @

(4)

b

x @

then :

(i)t h e successive approximation scheme ( 2 ) - ( 3 ) converges i n a t most p t 1 steps t o : a* @ b s o l u t i o n o f ( 4 ) ( i i ) moreover, a* @ b i s t h e unique minimum s o l u t i o n o f ( 4 ) .

As a consequence o f Property 2 a* i s recognized as t h e (unique) minimum s o l u t i o n o f b o t h equations : x = a

6

x @ e and : x = x @ a

0

e.

Example 1 Suppose t h a t ( S ,

0 , 0 )i s

a d i o i d i n which @ and @ a r e idempotent.

Then

any element i s 1 - r e g u l a r since: a(’)=,@

a

0

0

a 2 = e e a

a = e

0

a=a(’).

This applies, i n p a r t i c u l a r , t o a l l d i o i d s which a r e d i s t r i b u t i v e l a t t i c e s .

Example 2 I n t h e case o f t h e d i o i d

(R,

Min, + ) (where

E

=

i s 0-regular and t h e quasi-inverse i s : a* = e. n o t r e g u l a r s i n c e : a ( k ) = Min {O, a, Za,

...

+-

and e = 0) any element a a 0

However, i f a

ka) = ka and a ( k )


perm ( A ) ( V j ) .

The converse o f the above r e s u l t can be proved i n t h e f o l l o w i n g case:

Theorem 3 (Gondran and Minoux 1978) We assume t h a t , Va, b

r S a @ b

= a

or b and

@ i s invertible.

Linear algebra in dioids

Then, i f A r M n ( S ) i s such t h a t al(A)

157

= a2(A) t h e columns ( a n d t h e rows) o f A a r e

1 i n e a r l y dependent. I n t h e case o f d i d i d s which a r e d i s t r i b u t i v e l a t t i c e s ,

nl(A)

however, t h e c o n d i t i o n

i s g e n e r a l l y n o t s u f f i c i e n t f o r g e t t i n g a dependence r e l a t i o n on

= A2(A)

t h e columns o f A.

1; :; lj

Consider, f o r i n s t a n c e , t h e m a t r i x :

(IR'U

i n t h e dio'id

{+=I, Max, M i n )

We check t h a t : A (A) = w ( U ) = M i n {11,14,101

=

10

a2(A) = wA(u2) = M i n {12,15,10)

=

10.

1

Thus, nl(A)

A 1

= a2(A) = perm (A) = 10, b u t no dependance r e l a t i o n between t h e

columns ( o r the rows) o f A can b e found w i t h A ~ , A ~ , x z~ 10.

I n view o f t h i s c o u n t e r example, t h e c o n d i t i o n s f o r a m a t r i x t o b e s i n g u l a r (columns l i n e a r l y dependent) i n l a t t i c e - d i o i d s have t o b e g i v e n a s l i g h t l y d i f f e r e n t form. We f i r s t d e f i n e t h e s k e l e t o n o f a m a t r i x A, as t h e 0-1 m a t r i x S(A) = (sij) sij

= 1 i f aij

where:

>

perm (A)

s . . = 0 otherwise

75 we say t h a t a 0-1 n x n m a t r i x B = ( b . .) i s p a r t l y - b a l a n c e d i f and o n l y i f t h e 1J

s e t o f i t s columns J may be p a r t i t i o n e d i n t o J, and J 2 such t h a t :

( b a l a n c e d m a t r i c e s have been s t u d i e d b y Berge 1972).

Then we have:

Theorem 3 ' (Minoux 1982) Assume t h a t a @ b = a o r b and a @ b = i n f (a,b).

ThenA i s s i n g u l a r i f and

o n l y i f S(A) i s p a r t l y balanced. Theorem 3' above a l l o w s t o g i v e a new c h a r a c t e r i z a t i o n o f balanced m a t r i c e s of Berge (1972) i n terms o f l i n e a r dependence and independence i n t h e d i o ' i d ( { O , l l ,

M. Gondran and M. Minoux

158 Max, Min)

.

L e t A be a 0-1 m a t r i x , I and J the s e t o f i t s rows and columns. For K c J and K L c I , we denote by AL t h e submatrix d e f i n e d by t h e subset o f columns ( r e s p o f rows) K (resp. : L ) . For any K c J ,

& ( K ) denotes t h e subset o f rows i E I i n

which t h e r e e x i s t a t l e a s t two t e r n s equal t o 1 and belonging t o columns i n

K.

Then we can s t a t e :

Corol l a r y 1 The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t and c h a r a c t e r i z e a balanced m a t r i x A: (i)

Every subset K c J can be p a r t i t i o n n e d i n t o K, and K2 i n such a way t h a t :

(ii)

4Kc

J, L

= 6 ( K ) t h e submatrix

K AL has i t s columns dependent i n t h e dioi'd

( t O , l I , Max, Min) (iii) For any 0-1 v e c t o r b t h e l i n e a r program: z = Max c x subject t o Ax < b x,o has an optimal i n t e g e r s o l u t i o n f o r a l l c . Condition ( i ) corresponds t o t h e d e f i n i t i o n , and c o n d i t i o n ( i i i ) has been given by Berge (1972).

Condition ( i i ) i s t h e new c h a r a c t e r i z a t i o n i n terms o f l i n e a r

dependence and independence i n dio'ids, which can be recognized as a formal analogue t o a well-known c h a r a c t e r i z a t i o n of unimodular m a t r i c e s due t o GhouilaHouri (1962).

6 A

EIGENVP.LUE PROBLEMS AND APPLICATIONS S i s c a l l e d an eigenvalue o f AcMn(S) i f and o n l y i f t h e r e e x i s t s V

Sn

(eigenvector) such t h a t : A@v = x k ' v . There i s a s t r o n g r e l a t i o n between eigenvalues and l i n e a r dependence i n general d i d i d s as shown by:

Theorem 4 (Gondran and Minoux 1978) i s eigenvalue o f Ae Mn(S) i f and o n l y i f t h e 2n x 2n m a t r i x

Linear algebra in dioids

' e

E

159

' e

E

has i t s columns dependent. By considering h as a v a r i a b l e , each term o f t h e bideterminant o f A ( h ) may be

considered as a polynomial i n A. Pl(A) then

P(A) =

]:l[

Thus i f we note

= Al(A(h))

and P2(h) = A2(A(h))

w i l l be c a l l e d t h e c h a r a c t e r i s t i c b i p o l y n m i a l o f A.

A f i r s t important property o f t h e c h a r a c t e r i s t i c bipolynomial i s t h e generalizat i o n t o dio'ids

o f the Caley-Hamilton theorem which reads:

Theorem 5 (Stranbing 1983, Gondran 1983) The m a t r i x A i t s e l f solves the c h a r a c t e r i s t i c equation P1(A) = P2(A). On t h e o t h e r hand, combining Theorem 2 and Theorem 3 leads t o t h e f o l l o w i n g g e n e r a l i z a t i o n o f a c l a s s i c a l r e s u l t i n o r d i n a r y l i n e a r algebra:

Theorem 6 L e t @ be such t h a t a @ b = a o r b and

6

invertible.

Then, h i s eigenvalue

o f A EMn(S) i f and o n l y i f h i s s o l u t i o n o f the c h a r a c t e r i s t i c equation: P1(X) = P2(X). Other c o n d i t i o n s f o r the existence o f eigenvalues and eigenvectors have been given by Gondran and Minoux (1977) such as:

Theorem 7 If a @ b = a (i)

or b,

i f @ i s idempotent and commutative, then:

A* (quasi i n v e r s e o f A) e x i s t s ;

M. Gondrun and M. Minoux

160 (ii)

every >

t

5 i s an eigenvalue of A;

( i i i ) J ( h ) ( t h e s e t of a l l eigenvectors f o r h ) i s t h e moduloid generated by those 6 1 @ ui vectors of the form where ui = (

[ATi

'A(')

denotes t h e i t h column o f A*,

and C i i i s the set of a l l c i r c u i t s origina-

ting a t i in t h e graph associated w i t h A ) . As an i l l u s t r a t i o n , we g i v e below a number o f examples where a p p l i c a t i o n of t h e

above r e s u l t s may lead t o some natural and illuminating i n t e r p r e t a t i o n s i n terms of eigenvalues and eigenvectors in dio'ids. 6.1

Jobshop scheduling

( Cuninghame-Green 1960, 1962)

The determination of the steady s t a t e behaviour of a set of machines (processing jobs with precedence c o n s t r a i n t s only) reduces t o finding an eigenvector i n t h e dioid (R, Max, +) ("schedule algebra") 6.2

Routing problems

The determination of a minimum r a t i o (cost/time) c i r c u i t i n a graph ( c f . Dantzig B l a t t n e r & Rao 1967) i s equivalent t o finding the eigenvalue and t h e associated eigenvector f o r t h e c o s t matrix i n the dioid ( R U {+-I, Min, + ) .

6.3

Hierarch i cal c 1us t e r i ng

Let A be the distance matrix of n o b j e c t s t o be c l a s s i f i e d . ( c f . Gondran 1977) t h a t i n the d i o i d ( R U {+-I, Min, Max):

.

Then i t can be shown

each level of the c l a s s i f i c a t i o n (simple-linkage c l u s t e r i n g ) corresponds t o a n

eigenvalue of A;

.

the p a r t i t i o n of t h e n o b j e c t s obtained a t any given level X , corresponds t o a

generator of the set 6.4

J (? ) .

Preferences analysis (Gondran 1979a)

Let A = ( a . .) be a preferences matrix: a i j i s equal, f o r example, t o t h e number 1J

of judges who prefer i t o j . Now, l e t us i n t e r p r e t eigenvalues and eigenvectors of A in some dio'ids. I f t h e dio'id i s (R', +, x ) , the eigenvector associated w i t h the l a r g e s t eigenvalue gives an average order, and we f i n d again the well known method of Berge 1958. If t h e dioid i s (R'U !+-I, max, x ) , the matrix A admits a unique eigenvalue which corresponds t o a c i r c u i t yo o f G such t h a t :

Linear algebra in dioids 1

1

m

n(v,) x

= W(Y0)

161

= max W ( Y ) Y

where n(Y) and w(y) a r e r e s p e c t i v e l y t h e number o f a r c s and t h e p r o d u c t o f t h e a r c s v a l u a t i o n s o f t h e c i r c u i t Y.

T h i s c i r c u i t determines t h e s e t o f o b j e c t s f o r

which t h e consensus i s t h e w o r s t . When G i s s t r o n g l y connected, we o b t a i n A by a v a r i a n t i n O(nm) o f an a l g o r i t h m o f KARP ( c f f o r i n s t a n c e Gondran Minoux 1979, Chap. 3 ) .

I f t h e d i o i d i s (R'

u

(+-I, max, min) t h e eigenvalues and t h e e i g e n v e c t o r s o f

t h e u n i q u e base a s s o c i a t e d w i t h each e i g e n v a l u e d e f i n e a f a m i l y o f p r e o r d e r s whose e q u i v a l e n c e c l a s s e s f i t i n t o each o t h e r .

ACKNOWLEDGEMENTS We thank t h e r e f e r e e s f o r t h e i r c o n s t r u c t i v e comments which h e l p e d improve t h e f i r s t v e r s i o n o f t h i s paper.

REFERENCES

[l] A b d a l i , K.S., Saunders, D.B., T r a n s i t i v e c l o s u r e and r e l a t e d s e m i r i n g p r o p e r t i e s v i a e l i m nants" (1983) To appear. [2] Berge, C.,

La t h g o r i e des graphes e t ses a p p l i c a t i o n s (Dunod, P a r i s ) .

[3]

Berge, C.,

Balanced M a t r i c e s , Mathematical Programming 2, 1 (1972) 19-31.

[4]

Carre, B.A., An Algebra f o r network r o u t i n g problems, 7 (1971) 273-294.

[5]

Cruon, R., Herve Ph. Quelques probl&nes r e l a t i f s 3 une s t r u c t u r e a l g 6 b r i q u e e t a son a p p l i c a t i o n au problgme c e n t r a l de l'ordannancement, Revue F r . Rech. Op. 34, (1965) 3-19.

[6]

Cuninghame-Green, R.A., Process s y n c h r o n i s a t i o n i n a s t e e l w o r k s - a problem o f f e a s i b i l i t y , in:Banbury and M a i t l a n d ( e d s . ) , Proc. 2nd I n t . Conf. on O p e r a t i o n a l Research, ( E n g l i s h U n i v e r s i t y Press, 1960) 323-328.

J. I n s t . Maths. A p p l i c s

[7] Cuninghame-Green, R.A., D e s c r i b i n g i n d u s t r i a l processes w i t h i n t e r f e r e n c e and a p p r o x i m a t i n g t h e i r s t e a d y - s t a t e b e h a v i o u r , O p e r a t i o n a l Research Q u a r t . 13, 1 (1962) 95-100. [8]

Cuninghame-Green, R.A., Minimax Algebra: L e c t u r e Notes i n Economics and Mathematical Systems ( S p r i n g e r Verlag, 1979).

[9]

D a n t z i g , G.B., B l a t t n e r , W.D., and Rao, M.R., F i n d i n g a c y c l e i n a g r a p h w i t h minimum c o s t t o t i m e r a t i o w i t h a p p l i c a t i o n t o a s h i p r o u t i n g problem", i n : T h g o r i e des graphes, Proc. o f t h e I n t . Symp., Rome, I t a l y , (Dunod, P a r i s 1967)

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162 77-83.

[lq

Ghouila-Houri , A . , C a r a c t g r i s a t i o n des m a t r i c e s totalement unimodulaires, C.R.A.S. P a r i s , tome 254 (1962) 1192.

[ll] G i f f l e r , B., Scheduling general p r o d u c t i o n systems u s i n g schedule algebra, Naval Research L o g i s t i c s Q u a r t e r l y , v o l . 10, no. 3 (1963).

[lq

Gondran, M., Path algebra and alqorithms, i n : Combinatorial Programing, (B. Roy Ed.), Reidel (1975).

[13]

Gondran, M., L ' a l g o r i t h m e g l o u t o n dans l e s algebres de c h m i n s , B u l l e t i n de l a D i r e c t i o n Etudes e t Recherches, EDF, S e r i e C, 1, (1975a) 25-32.

[14]

Gondran, M., Eigenvalues and eigenvectors i n h i e r a r c h i c a l c l a s s i f i c a t i o n , i n : Barra, J.R., e t a1 (Eds.), Recent Developments i n S t a t i s t i c s (North Holland Publishing Company, 1977) 775-781.

[15]

Gondran, M., Les Clements p-re'guliers dans l e s d i d i d e s , D i s c r e t e Mathematics, 25, (1979) 33-39.

[16]

Gondran, M., Ualeurs propres e t vecteurs propres en analyse des preferences, Note EDF HI-3199 (1979a). Gondran, M.,

Le the'orsme de Cayley-Hamilton dans l e s dio'ides, note EDF (1983)

[l8]

Gondran, M., Minoux, M. , Ualeurs propres e t vecteurs propres dans l e s s m i modules e t l e u r i n t e r p r e t a t i o n en t h e o r i e des graphes, B u l l e t i n de l a D i r e c t i o n Etudes e t Recherches, EDF, S e r i e C, 2, (1977) 25-41.

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Gondran, M., Minoux, M. , L'independance l i n e a i r e dans l e s dio'ides, B u l l e t i n de l a D i r e c t i o n Etudes e t Recherches, EDF, S e r i e C, 1, (1978) 67-90.

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[Zl]

Johnson, S.C., 241-243.

Minoux, M.,

Graphes e t Algorithmes ( E y r o l l e s , P a r i s , 1979).

H i e r a r c h i c a l c l u s t e r i n g schmes, P s y c h m e t r i c a 32, (1967)

[Z!] Kuntnann, J . , Theorie des reseaux graphes (Dunod, P a r i s , 1972). [23]

Minoux, M . , S t r u c t u r e s algebriques generalisees des p r o b l m e s de cheminement dans l e s graphes: theoremes, algorithmes e t a p p l i c a t i o n s , Revue Fr. Automatique, Infonnatique, Rech. Op., Vol. 10, 6, (1976) 33-62.

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Minoux, M., Linear dependence and independence i n l a t t i c e d i o i d s , Note I n t e r n e CNET ( 1982)-

[Zq

Peteanu. V., An algebra o f the optimal path i n networks, Mathematica 9, (1967) 335-342.

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Shimbel, A,, S t r u c t u r e i n communication nets, Proc. Symp. on I n f o r m a t i o n Networks, P o l y t e c h n i c I n s t i t u t e o f Brooklyn. (1954) 119-203.

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Yoeli, M.,

A note on a g e n e r a l i z a t i o n o f boolean m a t r i x theory, h e r . Math.

Linear algebra in dioids

Monthly 68 (1961) 552-557. [30] Zimmermann, U., Linear and combinatorial optimization i n ordered algebraic s t r u c t u r e s , Annals of Discrete Mathematics 10 (North Holland, 1981).

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Annals of Discrete Mathematics 19 (1984) 165-182 0 Elsevier Science Publishers B.V. (North-Holland)

165

ALGEBRAIC FLOWS AND TIME-COST TRADEOFF PROBLEMS

H.W. Hamacher and S. T u f e k c i Department o f I n d u s t r i a l and Systems E n g i n e e r i n g University o f Florida, Gai n e s v i 11e, F l o r i d a 32611 U.S.A. I n t h i s paper we i n t r o d u c e a p r o j e c t c r a s h i n g model where t h e problem i s f o r m u l a t e d as an a l g e b r a i c o p t i m i z a t i o n model. By e x p l o i t i n g t h e u n d e r l y i n g network s t r u c t u r e o f t h e problem, t h e model i s t r a n s f o r m e d i n t o a sequence o f a l g e b r a i c network f l o w problems. An e f f i c i e n t a l g e b r a i c maximal f l o w a l g o r i t h m i s implemented t o o b t a i n t h e a l g e b r a i c m i n i m a l c u t s a t each s t e p t o d e t e r m i n e t h e a c t i v i t i e s t o b e m o d i f i e d . Different selections o f algebraic structures y i e l d d i f f e r e n t o b j e c t i v e f u n c t i o n s which have i n t e r e s t i n g meanings i n r e a l l i f e situations.

1

INTRODUCTION

The c l a s s i c a l t i m e - c o s t t r a d e o f f problem (CTCTP) i s a w e l l s t r u c t u r e d l i n e a r programming problem.

T h i s problem has been s t u d i e d b y s e v e r a l r e s e a r c h e r s [I, 4,

5, 9, 10, 11, 12, 13, 14, 151.

I n a l l t h e s e s t u d i e s , t h e u n d e r l y i n g network

s t r u c t u r e o f t h e model i s e x p l o i t e d .

F u l k e r s o n [4]

and K e l l e y [8] f o r m u l a t e d t h e

problem as a l i n e a r p r o g r a m i n g problem where t h e d u a l o f t h e problem possesses a network f l o w s t r u c t u r e .

P h i l l i p s and Dessouky [12]

f o r m u l a t e d t h e problem as a

network f l o w problem where a t each i t e r a t i o n a minimal c u t i s sought t o d e t e r m i n e t h e a c t i v i t i e s t o b e crashed.

T u f e k c i [15]

i n d i c a t e d t h a t i n t h e f l o w network

suggested by P h i l l i p s and Dessouky, t h e l o c a t i o n o f a minimal c u t can e a s i l y be o b t a i n e d by u s i n g a l a b e l i n g a l g o r i t h m .

T h i s a l g o r i t h m u t i l i z e s t h e maximum f l o w

v a l u e s o b t a i n e d i n t h e p r e v i o u s s t e p as a s t a r t i n g f e a s i b l e f l o w i n t h e succeeding step.

A l l t h e s e a l g o r i t h m s a s s p e a s i n g l e l i n e a r o b j e c t i v e f u n c t i o n which r e p r e s e n t s t h e t o t a l a d d i t i o n a l d i r e c t c o s t based on a g i v e n s e t o f a c t i v i t y d u r a t i o n s . Moore, e t a l . ,

[lo]

suggested t h a t i n many r e a l l i f e p r o j e c t s , m i n i m i z a t i o n o f

t h e a d d i t i o n a l d i r e c t c o s t may n o t n e c e s s a r i l y r e f l e c t t h e t r u e o b j e c t i v e o f t h e management.

F u r t h e r , t h e y c l a i m t h a t t h e management i s g e n e r a l l y f a c e d w i t h

m u l t i p l e objectives.

They i n t u r n propose a goal programming approach f o r a

mu1 t i - c r i t e r i a p r o j e c t c r a s h i n g model. I n t h i s paper we show t h a t t h e c l a s s i c a l TCTP as w e l l as c e r t a i n m u l t i - o b j e c t i v e

H. W.Hamacher and S. Tufekci

I66

TCTP can be t r e a t e d w i t h i n a u n i f i e d framework, i f t h e problem i s modeled as an a l g e b r a i c o p t i m i z a t i o n problem.

By t a k i n g t h e u n d e r l y i n g network s t r u c t u r e o f

t h e model i n t o c o n s i d e r a t i o n the problem i s converted i n t o a sequence o f a l g e b r a i c network f l o w problems.

Recent a l g o r i t h m i c developments i n a l g e b r a i c f l o w s and

a l g e b r a i c minimal c u t s [6,

71

enable us t o s o l v e t h i s problem v e r y e f f i c i e n t l y .

I n s e c t i o n two we i n t r o d u c e t h e generalized TCTP w i t h a l g e b r a i c o b j e c t i v e function.

We a l s o show t h a t the c l a s s i c a l TCTP and the m u l t i o b j e c t i v e TCTP a r e

s p e c i a l cases o f t h i s g e n e r a l i z e d problem.

Section t h r e e i n t r o d u c e s t h e a l g o r i t h m

f o r the a l g e b r a i c time-cost t r a d e o f f problem.

An example problem i s provided i n

I n s e c t i o n f i v e we prove t h e v a l i d i t y o f t h e presented a l g o r i t h m .

section four.

Section s i x concludes t h i s work.

2

TIWE-COST TRADEOFF PROBLEM MITH ALGEBRAIC OBJECTIVE FUNCTION

I n what f o l l o w s we assume t h a t G i s an a c y c l i c a c t i v i t y - o n - a r c p r o j e c t network w i t h node set ties.

N, r e p r e s e n t i n g t h e events and a r c s e t A, r e p r e s e n t i n g t h e a c t i v i -

For convenience we assume t h a t node one represents t h e beginning o f the

p r o j e c t and node n represents t h e end o f the p r o j e c t . we associate two numbers aij

and bij

(aij

6 bij)

normal d u r a t i o n o f t h e a c t i v i t y , r e s p e c t i v e l y .

With each a c t i v i t y

(i,j)E

A

c a l l e d the crash d u r a t i o n and For g i v e n d u r a t i o n s dij,

(i,j) E A

we can f i n d the d u r a t i o n o f the p r o j e c t , T(d) by u s i n g a standard CPM technique

[81. I f T i s a g i v e n a l l o w a b l e p r o j e c t d u r a t i o n , then a c t i v i t y d u r a t i o n s d = (dij), ( i , j ) E A are c a l l e d a f e a s i b l e s o l u t i o n w i t h r e s p e c t t o time T, i f aij 6 dij 4 b i j f o r a l l (i,j) E A and T(d) 6 T. s e t w i t h respect t o time .T, i.e., F(T) = Idij:aij I f we d e f i n e d . . = bij, 1J

duration.

6 dij

V(i,j),

6

bij,

By F(T) we denote t h e f e a s i b l e

( i , j ) E A , T(d) c< T I .

then T(d) = Tn i s c a l l e d t h e normal p r o j e c t

Similarly, f o r d.. = a

V(i,j), T(d) = T C d e f i n e s t h e crash d u r a t i o n ij’ For an a r b i t r a r y p r o j e c t d u r a t i o n T, T C s. T < Tn some a c t i v i t i e s IJ

o f the p r o j e c t .

must be crashed t o achieve the d e s i r e d p r o j e c t l e n g t h . some a c t i v i t y d u r a t i o n s from dij

= bij

to dij

r e q u i r e s a d d i t i o n a l resources, and money.


e(y,x)

("backward a r c s " )

or if

If i n t h i s l a b e l l i n g process n is l a b e l l e d t h e n go t o s t e p f o u r .

Otherwise

go t o s t e p t h r e e .

3).

D e f i n e X t o b e t h e s e t o f l a b e l l e d nodes and

Node n cannot b e l a b e l l e d . t h e s e t o f u n l a b e l l e d nodes.

Now,

(X,x)u(R,X)

x

i s an a l g e b r a i c minimal c u t

(and f i s a n a l g e b r a i c maximal f l o w ) .

4).

Use t h e l a b e l s t o i d e n t i f y an augmenting p a t h P f r o m 1 t o n. 61 = miniu(x,y)

-

f(x,y)

62 = m i n [ f ( x , y )

-

n(x,y)

1 I

(x,y)

forward arc i n P j

(x,y)

backward a r c i n P }

Define,

63 = 1 n i n { 6 ~ , 6 ~ j ,

and change f by,

f(x,y)

=

1

f(x,y)

*

h 3 , i f (x,y)

f(x,y)

-

A3,

d3,

P

backward a r c i n P

, otherwise.

f(x,y)

Set v = v

i f (x,y)

forward arc i n

go t o s t e p one.

The presented l a b e l i n g a l g o r i t h m i s o n l y one p o s s i b i l i t y t o s o l v e a l g e b r a i c f l o w problems.

[7],

One c o u l d a l s o use g e n e r a l i z a t i o n s o f more e f f i c i e n t a l g o r i t h m s [3],

b u t a p r e s e n t a t i o n o f t h e s e a l g o r i t h m s i s beyond t h e scope o f t h i s paper. The i d e a o f t h e subsequen-

An example o f t h i s a l g o r i t h m w i l l be performed l a t e r .

t l y d e s c r i b e d a l g o r i t h m f o r s o l v i n g t h e a l g e b r a i c t i m e - c o s t t r a d e o f f problem i s

t o s o l v e a sequence of a l g e b r a i c maximum f l o w problems.

The r e s u l t i n g a l g e b r a i c

minimum c u t s a r e used t o change t h e a c t i v i t y t i m e s d. ..

Following t h e idea o f

1J

T u f e k c i [15]

t h e a l g e b r a i c maximum f l o w f a t t h e end o f i t e r a t i o n i i s used as

s t a r t i n g f l o w i n t h e beginning o f t h e n e x t i t e r a t i o n i t 1 . A l g o r i t h m f o r t h e a l g e b r a i c TCTOP.

I n what f o l l o w s we assume t h a t t h e r e a d e r i s

f a m i l i a r w i t h t h e CPM method f o r p r o j e c t networks.

L e t Ei and Li r e p r e s e n t t h e

e a r l i e s t r e a l i z a t i o n t i m e and l a t e s t r e a l i z a t i o n t i m e o f e v e n t i, r e s p e c t i v e l y . Then t h e s l a c k s S . . f o r each a c t i v i t y ( i , j ) E A a r e d e f i n e d b y Sij 1J

(see F o r d and F u l k e r s o n , [3]). Start: -

= L. J A c t i v i t y ( i , j ) i s c r i t i c a l i f f S . . = 0. 1J

d.. = b.. 1J

f(i,j)

1J

=

E

V

( i , j ) r A.

M l a r g e element o f H, e.g.,

M =

* ( i ,j )EA

c(i,j)

- E. 1

d..

1J

Algebraic flows and time-cost traa'eoff problems

173

1 ) . Use t h e CPM method t o compute f o r each a c t i v i t y ( i , j ) t h e s l a c k s Sij l e n g t h o f a l o n g e s t p a t h w i t h r e s p e c t t o t h e d u r a t i o n s dij.

(STOP).

t h a n o r equal t o t h e d e s i r e d p r o j e c t l e n g t h , 2).

and t h e

If T i s less

Otherwise go t o s t e p 2.

Define, u(i,j):

=

and L(i,j):

IE c(i,j)

i f d.. > a.

M

i f dij

= a

i f Sij

>

IJ

c(i,j)

=

ij

and Sij

= 0

and S . . = 0 ij 1J

0

i f d.. < b.. 1J 1J i f dij = b . .

1J

3 ) . Use t h e c u r r e n t a l g e b r a i c f l o w f ( i , j ) ,

Y ( i , j ) E A as f e a s i b l e s t a r t i n g f l o w

f o r f i n d i n g an a l g e b r a i c maximum f l o w f ' ( i , j ) c o r r e s p o n d i n g a l g e b r a i c minimal c u t

4).

If v ' a

M

then

(STOP).

(X,!)

w i t h f l o w v a l u e v ' and a

(J ( x , X ) .

The t o t a l c o m p l e t i o n t i m e cannot be d i m i n i s h e d w i t h -

o u t v i o l a t i n g t h e c r a s h d u r a t i o n o f a t l e a s t one a c t i v i t y , i.e., If v'
a . . )

I I

Sij

= 0 and dij

Sij

>

1J

1J




0.

Q

All

(i,j) E Apl ( i , j ) e A33

(3.6) and s e t

Redefine, [d:: dij:

-

A i f ( i , j ) E A1

d . . t A if ( i , j ) E A2

=

I d ij f(i,j)

= f'(i,j),

(3.7)

otherwise

v

(i,j)

E

A

T : = T - A . I f T i s l e s s t h a n o r equal t o t h e d e s i r e d p r o j e c t l e n g t h ,

Otherwise go t o s t e p one.

(STOP).

H.W.Hamcher and S. Tufekci

174

We note here t h a t f o r R = Z t h e a l g o r i t h m solves a t most Tn

-

f l o w problems and performs t h e same number of CPM a l g o r i t h m s .

Tc a l g e b r a i c maxThe complexity o f

t h e CPM a l g o r i t h m f o r an n node a c y c l i c network i s O(n2) and t h e complexity o f t h e best algebraic max-flow problem i s O(n3) (see [7] ) . Therefore t h e o v e r a l l complexi t y o f t h e proposed a l g o r i t h m i s O((T,

-

Tc)n3).

The v a l i d i t y o f t h e a l g o r i t h m i s

proved i n Section 5.

4 AN EXAPPLE - THE LEXICOGRAPHICAL TCTP Here we implement t h e a l g o r i t h m provided i n t h e previous s e c t i o n t o t h e a l g e b r a i c s t r u c t u r e d e f i n e d i n Example 4 of Section 2. Consider t h e p r o j e c t network g i v e n i n F i g u r e 2.

The c o s t v e c t o r c ( i , j )

i s repre-

sented as a column v e c t o r over each a r c .

The numbers b e f o r e and a f t e r c(i,j) represent t h e crash d u r a t i o n and normal d u r a t i o n o f t h a t a c t i v i t y , r e s p e c t i v e l y .

Note t h a t t h e f i r s t component o f t h e c o s t v e c t o r represents incremental d i r e c t costs.

The second component i s 1 f o r each a c t i v i t y i n d i c a t i n g t h e d e s i r e t o

minimize t h e c o n t r o l on t h e crashed a c t i v i t i e s .

c(i,j)

L a s t l y , t h e t h i r d component o f

represents t h e p r i o r i t y r a n k i n g o f each a c t i v i t y .

Note t h a t an a c t i v i t y

w i t h smaller r a n k i n g i s p r e f e r a b l e t o an a c t i v i t y w i t h higher ranking.

L e t Ei,

and Li f o r each node i E N represent t h e e a r l i e s t r e a l i z a t i o n t i m e and l a t e s t r e a l i z a t i o n time o f event i, determined by CPM method, r e s p e c t i v e l y .

F i g u r e 2 . An example p r o j e c t network f o r LTCTOP. aij, Start:

d.. = b. 1J

1J’

f = -

( V a l u a t i o n o f arcs:

g(i,j),bij)

0

Applying the CPM technique we g e t t h e l a b e l s as shown i n F i g u r e 3.

Algebraic flows and time-cost tradeoff problems

F i g u r e 3. Label (Ei,Li)

on each node by CPM.

175

(Valuation o f arcs:

d . . = b .) 1J iJ The a c t i v i t y s l a c k s may be found by u s i n g Sij = LJ. - Ei - d i j . T h i s y i e l d s S45 = 8, S58 = 10, S48 = 20 and o t h e r Sij = 0. The corresponding f l o w network and t h e upper a r c c a p a c i t i e s a r e d e p i c t e d i n F i g u r e 4 below. are L(i,j) =

0for

F i g u r e 4.

(The l o w e r c a p a c i t i e s

a l l (i,j)).

I n i t i a l Flow Network f o r LTCTOP.

Since t h e c u r r e n t f l o w i s f

E

0,we

(Arc valuations: l ( i , j ) )

can l a b e l t h e nodes 1, 2 , 5, 7, 8 and i d e n t i f y

t h e augmenting p a t h P = 11, 2 , 5, 7, 81. Ey(x,y)

Set f(1,2)

= f(2,5)

-

f(x,y)l

H. W.Hamacher and S. Tufekci

176

The n e x t augmenting p a t h w i l l be P = (1, 3, 5, 7, 8 } w i t h E = 5, = l e x m i n { g ( x , y ) -3

-

f(x,y)}

=

(X,Y)EP

Set f(1,3)

= f(3,5)

=

[I], [I, 1,I]!-[

The t h i r d augmenting p a t h i s P = {l,3, 6, 7, 81 w i t h E = Ll = l e x m i n -3

Set f(1,3)

=

I] 11, f(3,6)

=

f(6,7)

=

f(7,8)

=

[I.

=

Next f l o w augmenting p a t h i s P = (1, 4, 6, 7, 81

D u r i n g t h e f o l l o w i n g l a b e l l i n g a l l nodes g e t l a b e l e d e x c e p t node 8. lexmax i s a t hand w i t h

I![

, R

=

a c t i v i t y (7,8) by mint8-3,10,201

=

= 5.

{8}, and X

=

Thus a

N\x ( s e e F i g u r e 5 ) .

Now t h e p r o j e c t d u r a t i o n i s 21.

CPM method t h e new s l a c k v a l u e s a r e o b t a i n e d as f o l l o w s : S45

=

8, S58

Crash

By u s i n g =

5, S48 =I5

and o t h e r S . . = 0. 1J

The c o r r e s p o n d i n g f l o w network f o r t h i s i t e r a t i o n w i t h updated f l o w bounds i s d e p i c t e d i n F i g u r e 5 below.

Note t h a t t h e lexmax f l o w o f t h e l a s t i t e r a t i o n i s

used as a f e a s i b l e l e x f l o w f o r t h i s i t e r a t i o n . C o n t i n u i n g w i t h l a b e l l i n g we o b t a i n t h e p a t h P = 11, 4, 6, 7, 8 } w i t h

Algebraic flows and time-cost tradeoff problems

Figure 5

Maximal l e x f l o w i n i t e r a t i o n 1 w i t h updated f l o w bounds. (Arc v a l u a t i o n & ( i , j ) ,

For each ( i , j ) E P s e t f ( i , j )

f(4,6)

=

11,

f(6,7)

=

= f(i,j)

I]

and

t

!(i,j),

u(i,j))

s3. Now -f(1,4)

f(7,8)

=

r:].

=

Next a t t e m p t of l a b e l l i n g s t o p s s h o r t of l a b e l l i n g node 8. t h e nodes i n

177

1 a r e 7 and 8. A = Al

Thus X = 11, 2, 3, 4, 5, 6,},

= min{d57 = min(5-2,

-

a57' d67

8-5,

-

a67' '58'

When l a b e l l i n g s t o p s

1

= {7,

8},

'48'

5, 151 = 3.

S i n c e t h e d e s i r e d p r o j e c t l e n g t h i s 19 reduce t h e p r o j e c t l e n g t h by 2 more u n i t s by r e d u c i n g a c t i v i t i e s (5,7) d67 = 8

-

2 = 6.

and (6,7) by two u n i t s .

Set d57 = 5 - 2 = 3 and

Since t h e d e s i r e d p r o j e c t l e n g t h T = 19 i s achieved, t h e

algorithm terminates.

I f we were t o c o n t i n u e c r a s h i n g t h e n we should have

crashed t h e two a c t i v i t i e s above by t h r e e t i m e u n i t s .

I n such a case, t h e bounds

on t h e network must be updated and t h e l a b e l i n g a l g o r i t h m must c o n t i n u e .

5

VALIDITY OF

THE ALGORITHN

We prove t h e v a l i d i t y of t h e a l g o r i t h m by i n d u c t i o n on t h e number o f i t e r a t i o n s . F o r e v e r y p r o j e c t t i m e T and a c t i v i t y d u r a t i o n s d G(T) = { ( i , j )

I

Sij

=

ij

we denote

0).

(5.1)

R W. Hamacher and S. Tufekci

118

G(T) contains a l l c r i t i c a l paths w i t h r e s p e c t t o p r o j e c t t i m e T, i.e.,

a l l paths

P satisfying

1

d(P) =

(5.2)

d 1J .. = T

( i ,j )eP S t a r t = I t e r a t i o n 0: L e t T = Tn be t h e normal p r o j e c t time.

Obviously d .

ij

= bij

m i n i m i zes t h e o b j e c t i v e f u n c t i o n

* (i,j)EA I t e r a t i o n I: L e t T = T~

-

( b . .-d. . ) 0 c 1J

1J

6, 6 s u f f i c i e n t l y small.

I n order t o achieve a p r o j e c t time of Tn G(Tn) by 6 time u n i t s .

ij

-

6 we have t o crash a l l paths P i n

This can be done by f i n d i n g an a r b i t r a r y c u t ( X , j )

and crash a l l a c t i v i t y d u r a t i o n s by 6, i.e.,

G(Tn)

in

by d e f i n i n g

* C... Therefore a c u t The c o s t increase i s 6 o c ( X , R ) where c(X,R) = (i,j)e(X,i) IJ w i t h minimal value c(X,x) w i l l y i e l d a minimal c o s t increase. Such a c u t can be found by applying a maximal a l g e b r a i c flow a l g o r i t h m t o the lower and upper capacities I.(i,j) "6 sufficiently

minimal c u t ( X , x )

= 0 and u ( i , j )

E

1J

s m a l l " means t h a t d . .

1J

-

6 >, a.

1.j

Note t h a t

f o r a l l ( i , j ) E (X,X).

I f the

c o n t a i n s an a r c ( i , j ) w i t h d . - = a . . then c(X,?) > M and a l l 1J

c u t s i n G(T) have a v a l u e >c M. (i,j)

= c . . Y ( i , j ) f G(T), r e s p e c t i v e l y .

1J

Then we can f i n d a p a t h P w i t h dij

P which shows t h a t Tn = TC and t h e a l g o r i t h m stops.

= aij

for a l l

On t h e o t h e r hand i t

should be guaranteed t h a t no p a t h becomes c r i t i c a l which i s n o t contained i n G(Tn).

0
01. Therefore t h e a l g o r i t h m y i e l d s f o r a l l S w i t h IJ i j where c. i s d e f i n e d by ( 3 . 6 ) , optimal a c t i v i t y d u r a t i o n s d . ..

6 4 A,

1J

Iteration i

+

I t e r a t i o n ( i t l ) : Let T


T - 6 .

T

-

6,

A l l paths P i n G(T) s a t i s f y

6 s u f f i c i e n t l y small.

Therefore we have t o crash a t l e a s t one a c t i v i t y f o r each o f

d ! . .c< T - 6 . Again, t h e 1J ( i .J )Q problem i s t o f i n d a way o f c r a s h i n g which w i l l y i e l d a minimum c o s t increase i n

these c r i t i c a l paths i n o r d e r t o achieve d ' ( P ) =

the objective function. 6 c A3 = T

-

2

I f we choose d(P)

P n o n c r i t i c a l paths w i t h r e s p e c t t o T

(5.4)

(or e q u i v a l e n t l y , a3 = m i n I S - .IS.. > O } , as defined i n (3.6)), then t h e r e i s no 1 3 13 need t o crash d . f o r a r c s ( i , j ) which a r e n o t contained i n G(T). iJ

Algebraic flows and time-cost tradeoff problems

Ifwe f i n d i n G(T) a c u t (X,R)

such t h a t dij

-

6 3 aij

V(i,j)

179 E

(X,f)

then the

a c t i v i t y d u r a t i o n s d ! . defined by (5.3) w i l l y i e l d a p r o j e c t t i m e o f T ( d ! . ) 6 T 1J

S i n c e p a t h s i n G(T) can use a r c s of t h e c u t (X,R) can even become l e s s t h a n T

Example:

= (IR,+,c),

(H,*,&)

- 6.

1J

-

more t h a n once t h e l e n g t h d ' ( P )

6 y i e l d i n g an u n n e c e s s a r i l y h i g h c o s t i n c r e a s e :

T = 10, 6 = 1, G(T) c o n t a i n s t h e p a t h

P

= (1, 2, 5 ,

3, 4, 6 )

10 = T

dij

d(P)

d;j

d ' ( P ) = 8

aij

M

i f dij

= aij

cij

i f dij


a* y 6 6

*

W a,B

Y

y

e H.

I n a d d i t i o n we assume t h e v a l i d i t y o f t h e reduction r u l e :

(2.4)

a c< $ => 3 y e H : a*y = 6 W a,B E H

and the (2.5)

weak c a n c e l l a t i o n r u l e a

*

$ = n

*

y

-i>

6[

= y Or a

Moreover we extend H by a symbol

-

c(C) =

*

8 = a]

satisfying a

We r e q u i r e t h a t c(x,y) a 0 f o r a l l (x,y) o f any c u t C E C by

(2.6)

*

e A.

-

W

cH.

a,B,y

and a

*

- -*

V a E H. Then we can d e f i n e t h e e v a l u a t i o n
0, t h e n add ( s , v a ) t o A. Set CLt

= C(V,,t)

*

=

6 A and

I f (v,,t)

Caj.

&No CLt

(3.6)

>

0, t h e n add (v,.t)

O m i t a l l v Q w i t h II

If ( s , t )

E

E

NoU N1 -

I s , t } and t h e a r c s i n c i d e n t t o such nodes.

A, t h e n d e l e t e a r c ( s , t ) f r o m G.

F o r each c h a r a c t e r i s t i c v e c t o r

*

t o A.

(zs(l-zL))D

WN' =

csa

*

=

1a s s o c i a t e d *

a€"

*

aeN ' i e N =

*

w i t h a c u t d n t h i s network we g e t

((l-ZaP ( l - Z R ) 0 Cia

*

ieN1

Cia)

(by the d e f i n i t i o n o f El)

1

*

ieNl LeN'

(1-Za)

17 Cia ( b y t h e c o m m u t a t i v i t y o f * )

H. W. Hamacher et al.

I90 and, analogously,

Hence the b i n a r y q u a d r a t i c a l g e b r a i c problem w i t h f i x e d v a r i a b l e s can be solved by f i n d i n g an a l g e b r a i c minimal c u t i n a m o d i f i e d network. (H,*, c,(D(a))

= c ( D ( a ) ) (because

second-best c u t - s e t .

a

-

D.

Choos? is

For every c u t - s e t

S i n c e a $ D, c(D) = ca(D) >/

D ( a ) ) and t h i s shows t h a t D ( B ) i s a

T h i s approach r e q u i r e s s o l v i n g a t most O(m) minimum

a l g e b r a i c c u t problems i n networks o f t h e same s i z e as G, and r e s u l t s i n an O(mn3) a l g o r i t h m . I n o r d e r t o improve t h e c o m p l e x i t y o f t h i s a l g o r i t h m we analyze t h e s i t u a t i o n i n which we f i n d a minimal a l g e b r a i c c u t D ( a ) w i t h r e s p e c t t o t h e c a p a c i t y f u n c t i o n

Suppose f i s a maximal a l g e b r a i c f l o w w i t h r e s p e c t t o c a p a c i t y c. f e a s i b l e w i t h r e s p e c t t o ca.

Obviously f i s

I n o r d e r t o f i n d D ( a ) we apply, f o r i n s t a n c e , a n

a l g e b r a i c l a b e l i n g a l g o r i t h m t o f and (G,ca), i.e., we i d e n t i f y s u c c e s s i v e l y R augmenting elementary paths p', ...,p f r o m s t o t where f o r a l l i = l , . . . , ~ (x,y)

(4.2)

(y,x)

forward a r c i n p

i

=>

f(x,y)

i

backward a r c i n p =>f(y,x)


0

Claim: A l l augmenting p a t h s c o n t a i n a r c a and do n o t c o n t a i n any

(4.3)

o t h e r a r c a ' e D1 , a ' # a.

Proof:

S i n c e f i s a maximal a l g e b r a i c f l o w i n (G,c),

f ( a ' ) = c(a') = ca(a') f o r

a l l a ' E D1, a ' # a. Hence a ' cannot be a f o r w a r d a r c i n some augmenting p a t h .

Ifa ' # a were a backward a r c o f an augmenting p a t h P t h e n h ( a ' ) labeled.

T h i s i s o n l y p o s s i b l e i f a r c a precedes a r c a ' i n P.

E

j1had been

Since P does n o t

c o n t a i n o t h e r f o r w a r d a r c s i n D1 than a and s i n c e t h e t e r m i n a l node o f P i s t

E

1, P would c o n t a i n a r c a f o l l o w i n g a r c a ' .

twice i n P

-

T h a t means t h a t a r c a would o c c u r

a c o n t r a d i c t i o n t o t h e d e f i n i t i o n o f an elementary path.

F i n a l l y , s i n c e D1 i s a c u t , any p a t h has t o c o n t a i n a t l e a s t one a r c o f D1, i . e . , a E P f o r a l l augmenting paths P. By (4.3) we can w r i t e each pi as

H.w.Hamacher et a[.

194

p

(4.4)

i

i

= (P,J’YSP

i Y

1

i i where x = t ( a ) , y = h(a), and p, and p are augmenting paths from s t o x and from Y y t o to, r e s p e c t i v e l y . Paths p i and pi can now be found independent from each o t h e r i n m o d i f i e d networks Y Gx and Gy. The node s e t o f G, is V x = tXIU{rt)} and t h e a r c s e t is Ax = Ca r A : t ( a )

E

X1} where we r e d e f i n e h(a) = t f o r a l l a € A x w i t h h ( a ) f X1.

A ) where V = {I, U { s l l and Ay = { a E A : h ( a ) E. Y’ Y Y where t ( a ) = s f o r a l l a E A w i t h t ( a ) E. X1 (see F i g u r e 4 ) . Y Analogously, Gy = ( V

F i g u r e 4:

2,)

Construction o f G, ( f o r x = v1 o r x = v2) and G ( f o r x = v3 o r v = v4) Y

By the c o n s t r u c t i o n o f Gx and Gy each augmenting p a t h p

i corresponds t o augmenting paths (px,x,t)

i = (px,x,y,py) i i i n (G,ca)

i n (Gx,ca) and (s,y,py)

i Conversely, each p a i r o f augmenting paths (p,,x,t)

i

i n (G ,c ) . Y a

i n (Gx,ca) and (s,y,py)

i ( G ,c ).corresponds t o an augmenting p a t h pi = (p,,x,y,p;). Y a i amount o f flow which can be sent along (px,x,t) and (s.y,p’), Y i i i E = min(rx.c:) can be sent along p

cl

i

in

and ci is t h e Y r e s p e c t i v e l y , then

If

.

Using an i n d u c t i v e argument we can t h e r e f o r e f i n d t h e value f a ( t , s ) o f a maximal a l g e b r a i c f l o w i n (G,ca) by computing maximal f l o w values f x ( t , s )

i n (Gx,ca) and

Ranking the cuts and cut-sets of a network fy(t,s)

i n (Gy,ca),

195

and = min(fx(t,s),f Y (t,s)).

fa(t,s)

(4.5)

The advantage o f t h i s procedure i s t h a t we can use f x ( t , s ) f o r a p o s s i b l e o t h e r a r c a ' E D,, a"

E

a ' # a, w i t h t f a ' ) = x and f (t,s) Y If

f o r a possibly e x i s t i n g arc

D1, a" f a, w i t h h ( a " ) = y.

t ( D 1 ) = {x E. X1 : x = t ( a ) f o r some a

h(D1) = { y e

x,

v

x E t(D,),

and f (t,s), Y

V y E h(D1),

In t h i s way we i d e n t i f y B by

Y a E D ] , by (4.5).

I

f a ( t y s ) = min{fa(t,s)

(4.8)

D1l

: y = h(a) f o r some a E D1}

then we have t o compute a l l values f,(t,s), i n order t o compute fa(t,s),

E

a E DII,

Then we can f i n d the 2nd best c u t - s e t by s o l v i n g an a d d i t i o n a l maximal a l g e b r a i c f l o w problem i n (G,cz). An a l t e r n a t i v e approach can use the a l g e b r a i c minimal cuts (X,R) o r (Y,p) i d e n t i f i e d w h i l e computing f x ( t , s ) and f ( t , s ) (where x = t ( i ) , y = h ( a ) ) . Y Following (4.5) we consider the f o l l o w i n g case a n a l y s i s .

Case 1:

fg(t,s) = fX(t,s), i.e.,

fx(t,s)

(X,x) corresponding a l g e b r a i c minimal c u t - s e t i n G,,

= c(X,R).

Define

x,

(4.9)

=

x, R,

=

RU

x1

Then the d e f i n i t i o n o f G, y i e l d s c(X2,12) = c(X,R) = f x ( t , s )

That i s , D2 = (X2,X,)

= fg(t,s).

i s a 2nd best cut-set. Case 2:

f-(t,s) a

= f (t,s)

set i n Gy, (4.10)

Y

i.e.,

< f

X

(t,s),

fy(t,s)

xp

=

(Y,p) corresponding a l g e b r a i c minimal cutDefine

= c(Y,Y).

vux,, x,

=

8.

Then, again, we conclude from t h e d e f i n i t i o n o f G Y c(X2,X2) = c(Y,p) = f (t,s) = f - ( t , s ) , i.e., D2 = (X2,R2) Y a i s a 2nd best cut-set.

196

H. W. Hamacher et al.

Since we can d e f i n e a 2nd b e s t c u t - s e t by (4.9) o r (4.10) we g e t t h e f o l l o w i n g result. (4.11)

I f t h e network G contains two o r more cut-sets, then t h e r e

Theorem:

e x i s t two best c u t - s e t s D1 = (X,,X,)

x1 n R,

crossing ( i . e . ,

=

(I o r ji,n

and D2 =

x2

=

(X2,f2)

t h a t are non-

0).

I n both a l t e r n a t i v e s we solve a t most It(Dl) U h(D1)l 6 n a l g e b r a i c f l o w problems t o f i n d a 2nd best c u t - s e t .

Hence t h i s procedure i s o f o r d e r O(n'+) compared w i t h

O(m.n3) = O(n5) of t h e f i r s t v e r s i o n . D1 = [(~,3),(2,3),(1,6)1, fs(t,s)

= ft(t,s)

f2(t,s)

=

f3(t,s)

= 9,

=

(Example see F i g u r e 5 ) .

h!D1) = ~ ~ , 2 , 1 1t,( D 1 ) = I 3 9 6 t I

m

5, f 1 ( t , s ) = 5 =

9

= minIf,(t,s),

f3(t,s)1 = 9

f(2,3)(t,s)

= minIf2(t,s),

f3(t,s)}

= 5

f(1,6)(t3s)

= minIfl(t,s),

fg(t,s)]

=

f

(t,s)

fg(t,s)

(5-3)

i

Hence

= (2,3) o r

a

= (1,6)

A minimal c u t - s e t i n (G,ca) F i W e 5:

5

i s i n b o t h cases D2 = I ( s , l ) , ( s , 3 ) 3

F i n d i n g t h e 2nd b e s t c u t - s e t i n t h e network o f F i g u r e 1.

In the f o l l o w i n g we show how t o g e n e r a l i z e t h e c a p a c i t y m o d i f i c a t i o n (4.1) t o t h e case o f f i n d i n g t h e K b e s t c u t - s e t s . Assume t h a t we have found K-1 best c u t - s e t s D1,...,DK-l {Dl

,. . .

I.

and l e t DK = 2)

Any c u t D E DK must be such t h a t , f o r each i = 1,.

e x i s t s an arc ai

E

Di

-

D, f o r otherwise D would n o t be minimal.

of arcs a minimal covering o f D1

,..., DK- 1 i f Rn Di

#

and i f no proper subset o f R s a t i s f i e s t h i s p r o p e r t y .

..,K-1

there

We c a l l a s e t R

(I f o r a l l i = 1

,...,K-1

There e x i s t a f i n i t e number number, say L, o f such minimal coverings. C l e a r l y L 4 6 mK-'. so we now assume t h a t we have a l i s t R1,R 2,...,RL o f a l l such minimal coverings. L e t D

Dn

(tl

Kj

such t h a t R . = 0, f o r j = 1, ..., L. K J I C l e a r l y OK = U DK,. although t h e DK. a r e i n general n o t d s j o i n t . L e t D K be a j=l J J j

denote t h e s e t of a l l c u t - s e t s D E D I

197

Ranking the cuts and cut-sets of a network minimum c u t - s e t i n DK,.

Then a K-th b e s t c u t - s e t DK i s s i m p l y a b e s t c u t - s e t

,...,

D F i n d i n t DK amounts t o f i n d i n g a minimal a l g e b r a i c c u t among OK ,D 1 Kp KL. T h i s can be achieved by f i n d i n g a m i n i s e t t h a t does n o t c o n t a i n any a r c i n R j* ma1 a l g e b r a i c c u t - s e t i n a network (G,cR.) where J (4.12)

Hence a K-th b e s t c u t - s e t can be found by s o l v i n g a t most 0 ( m K - l ) minimum

. . ,DK-l

a l g e b r a i c c u t - s e t problems, p r o v i d e d D1,.

are available.

This r e s u l t s i n

a 0(mK-l n 3 ) a l g o r i t h m f o r f i n d i n g t h e K b e s t c u t - s e t s i n a network w i t h n nodes and m a r c s . Note t h a t t h e above a l g o r i t h m , f o r f i x e d K, i s polynomial i n m and n.

However,

i t appears q u i t e i m p r a c t i c a l f o r l a r g e v a l u e s o f K, as i t i s e x p o n e n t i a l i n K. It

is unknown whether an a l g o r i t h m polynomial i n m, n

erroneous [14].

exists f o r finding

Note a l s o t h a t t h e O(K.n4) a l g o r i t h m proposed i n [12]

K best cut-sets.

is

A polynomial a l g o r i t h m i n m, n, and K would be a v a i l a b l e p r o v -

i d e d we can s o l v e t h e f o l l o w i n g problem: F i n d a b e s t c u t - s e t c o n t a i n i n g a g i v e n s e t I o f arcs. We now p r o v e t h a t t h e problem o f f i n d i n g Dl,..-,DKml

- is

2

K-th b e s t c u t - s e t

-

w i t h o u t knowing

That i s , an a l g o r i t h m p o l y -

= (IR,+, (orLAJ) 4 x, hence

1 ( P + ) 2 (1 - + 1 ) x d contradicting again h jt x d by induction.

Remark 1 .

1 (P').

The proof then follows

Beineke and Harary [Z] have introduced t h e concept of marked graphs,

i . e . graphs i n which each vertex xk has a s i g n s k e { + , - I . Let us consider a signed and marked graph and look a t the nroblem of finding signed paths from x 1 to a l l x . , the sign o f a path being the product of t h e signs of both i t s J arcs and i t s v e r t i c e s (including end v e r t i c e s ) . This problem can be reduced to the previous one by the following transformations, which leave the signs of a l l

paths unchanged: I f s 1 = - s e t s 1 = t and reverse t h e signs of a l l a r c s with x 1 as i n i t i a l vertex ; b ) For a l l x . such t h a t s = - s e t s = + and reverse the signs of a l l a r c s with J j j x as t e r n i n a l vertex; j c ) Erase a l l signs of v e r t i c e s . a)

Remark 2. structures.

The double-label algorithm can be implemented w i t h various data For sparse graphs and D/d moderate, asjmptotically the b e s t implement-

?, mod LD/d] and ation seems to obtain by u s i n g a t a b l e o f possible values of chained l i s t s of indices o f v e r t i c e s with Lxfimod LD/d] o r L?,:]mod LD/dl equal

J

J

to such values. A very e f f i c i e n t device of Van Emde Boas, Kaas and Z i j l s t r a €291 [30] and Johnson 0 5 1 allows t o imolewnt t h e usual p r i o r i t y queue operations on such a table in time proportional t o the double algorithm o f , and snace

proportional t o , i t s length. T h i s y i e l d s an O(m log log D/d) O ( n t m t D/d) space imoTementation (see Karlsson a n d Poblete ion of the application o f these data s t r u c t u r e s t o D i j k s t r a ' s some values of n, m, D a n d d, o t h e r s t r u c t u r e s such as binary

time and [16] f o r a discussa1 gori t h m ) . For countinq t r e e s [lo]

o r heaps could y i e l d f a s t e r imnlementations. Remark 3. When a l l arcs have p o s i t i v e signs the problem reduces t o the usual s h o r t e s t path problem with non-negative weights; the normalization of lengths i m p l i c i t i n the algorithm a n d the use o f buckets solves, to some e x t e n t , the problem raised by Gallo and Pallotino [7] of having very long t a b l e s o f possible values f o r the labels when verv p r e c i s e data i s used.

P Hansen

206 Remark 4.

I n the case o f graphs i n R2, as e.g. road networks, the v e r t i c e s

selected i n step b ) belong t o d i s j o i n t ring-shaped regions i n c r e a s i n g l y f a r from

x,;

using buckets allows t o replace t h e sum and comparison o p e r a t i o n s by

less time consuming l o g i c a l t e s t s f o r arcs b o t h e n d v e r t i c e s o f which belong t o the same such r e g i o n . Remark 5.

When a l l d . are equal t o 1, an O(m) implementation i s e a s i l y Jk

obtained [9] . 3.

ELEMENTARY SHORTEST PATHS

A signed s h o r t e s t path between x, and some v e r t e x x . mav c o n t a i n a c i r c u i t , w i t h J negative s i g n . Such a p a t h may c o n t a i n a p o s i t i v e c i r c u i t o r several p o s i t i v e

and negative ones o n l y i f a l l arcs o f a l l these c i r c u i t s , except p o s s i b l y a negative one, have w e i g h t 0; should t h i s be t h e case, d e l e t i o n o f redundant c i r c u i t s y i e l d s a s h o r t e s t path w i t h a s i n g l e and n e g a t i v e c i r c u i t .

I f , however, i t i s r e q u i r e d t h a t t h e p a t h c o n t a i n no c i r c u i t a t a l l t h e Droblem becomes NP-complete.

Indeed, as very r e c e n t l y mentioned by Johnson [16],

LaPaugh and Papadimitriou L21] have shown t h a t t h e e x i s t e n c e nroblem f o r an elementary path w i t h an even number o f arcs j o i n i n g a v e r t e x x 1 t o a v e r t e x xn i n a graph G = (X,U)

i s NP-complete.

Now, i f a l l arcs o f G are given negative

signs, any such path i s a p o s i t i v e elementary p a t h from x, t o xn and conversely; i t s existence would be d e t e c t e d by a signed elementary s h o r t e s t oath a l g o r i t h m . Note t h a t t h e corresponding s h o r t e s t p a t h problem on an u n d i r e c t e d graph i s polynomial and has been solved by Edmnds (see

091)through

an e l e g a n t r e d u c t i o n

t o matching. Problems o f moderate s i z e may be solved by t h e f o l l o w i n g a l g o r i t h m , based upon decomposition and branch-and-bound ( d u r i n g t h e w s o l u t i o n the graph G =

(X,U)

w i l l be m o d i f i e d and may c o n t a i n two arcs from x, t o a v e r t e x x., w i t h signs J + and -; t h e corresponding weights w i l l be noted d+ and d- ) . 1J 1j a)

Initin2izatii.n

G

t h e t e r m i n a l v e r t e x o f which i s x,.

a.1)

Suppress a l l arcs o f

a.2)

Determine by d e p t h - f i r s t search (see T a r j a n [28])

= (X,U)

t h e s t r o n g components

o f G and then t h e b l o c k s (subgraphs w i t h o u t c u t - v e r t i c e s ) o f these s t r o n g components. a.3) Rank the v e r t i c e s o f G and re-index them i n such a way t h a t i ) v e r t ces o f the same s t r o n g b l o c k have consecutive indices;

belong t o d i f f e r e n t s t r o n g blocks (x,,x,)

g

u

=>

k

ii) if >

1.

x k and x

207

Shortest paths in signed graphs

b)

SeZection o f a subgraph

Consider t h e subgraph G' = ( X I , U x , ) o f G where X ' i s composed o f t h e v e r t i c e s o f G i n i n c r e a s i n g o r d e r o f i n d i c e s up t o and i n c l u d i n g those o f a s t r o n g b l o c k w i t h more than one v e r t e x ( o r up t o n i f no such b l o c k remains). c)

S h o r t e s t signed path and recognition of non-elementary ones

Apply t h e d o u b l e - l a b e l a l g o r i t h m t o t h e subgraph G ' .

Determine f r o m t h e p o i n t e r s

p?, p: t h e s i g n e d s h o r t e s t p a t h s f r o m x 1 t o x . f o r j = l , Z , , . . , I X ' l . N o t e which J J J o f these a r e non-elementary. I f t h e r e a r e no p a t h s w i t h c i r c u i t s go t o e ) ; o t h e r w i s e go t o d ) . d)

Branch-and-bound algorithm

Determine t h e s h o r t e s t elementary s i g n e d p a t h s f o r a l l j and s i g n s

t

or

-

such

f

corresponds t o a non-elementary p a t h b y a p p l y i n g f o r each o f them t h a t A . o r :A J J i n t u r n a branch-and-bound method. Such a method c o u l d use as s e p a r a t i o n p r i n c i p l e t h e e x c l u s i o n o f one o f t h e arcs o f t h e n e g a t i v e c i r c u i t w h i c h i s n o t used t w i c e i n t h e p a t h and as bounding r u l e

the f a c t t h a t t h e length o f the s h o r t e s t

s i g n e d p a t h g i v e n by t h e d o u b l e - l a b e l a l g o r i t h m i s , o f course, a l o w e r bound on t h e l e n g t h o f t h e elementary s h o r t e s t s i g n e d path. Update t h e values of 1' and :X the absence o f elementary s i g n e d p a t h b e i n g j J' n o t e d by an i n f i n i t e Val ue. e)

Modification o f G e.1)

Suppress a l l a r c s j o i n i n g v e r t i c e s o f X '

e.2)

I f x. Q X', J

1

I f x. E X', J

1j

If I X ' I


N f o r each o r i g i n ) .

A graph G i s path-balanced ( o r chain-balanced) i f and o n l y i f f o r e v e r y x . , x k € X J a l l elementary chains from x t o x k a r e o f t h e same s i g n . A f t e r r e p l a c i n g each j edge o f G by a p a i r o f o p p o s i t e a r c s w i t h t h e same s i g n , one can u s e t h e G i s l o c a l l y balanced a t x . J i f and o n l y i f a l l elementary c y c l e s passing by x . a r e balanced. A d i r e c t e d J g r a p h G = (X,U) i s c i r c u i t balanced i f and o n l y i f a l l elementary c i r c u i t s a r e

a l g o r i t h m o f s e c t i o n 3 t o t e s t G f o r path-balance.

p o s i t i v e and l o c a l l y c i r c u i t balanced a t x . i f t h i s i s t r u e o f a l l elementary J c i r c u i t s passing by x . . These p r o p e r t i e s can a l s o be checked w i t h t h e a l g o r i t h m J o f s e c t i o n 3.

4.2.

TRANSIENT BEHAVIOUR OF COMPLEX SYSTEMS

Roberts [24]

has proposed t o s t u d y s o c i e t a l i s s u e s (energy, p o l l u t i o n e t c ) i n

canplex systems by m o d e l i z i n g them w i t h signed graphs : v e r t i c e s a r e a s s o c i a t e d t o r e l e v a n t v a r i a b l e s , arcs t o d i r e c t i n t e r a c t i o n s between them and p o s i t i v e o r n e g a t i v e s i g n s t o t h e augmenting o r i n h i b i t i n g e f f e c t o f an i n c r e a s e o f t h e v a l u e o f t h e i n i t i a l v a r i a b l e on t h e t e r m i n a l one.

Then p o s i t i v e c i r c u i t s a r e d e v i a t i o n

amp1 i f y i n g and n e g a t i v e c i r c u i t s d e v i a t i o n - c o u n t e r a c t i n g .

Roberts and Brown [26]

have s t u d i e d t h e s t a b i l i t y o f complex systems under p u l s e processes, i n which

P. Hansen

210

d e v i a t i o n s a r e t r a n s m i t t e d a t equal i n t e r v a l s i n t i m e . Problems o f t r a n s i e n t b e h a v i o u r o f t h e systems a r e a l s o o f i n t e r e s t and c o u l d be L e t us mention, among o t h e r s ,

s t u d i e d w i t h t h e a l g o r i t h m s o f s e c t i o n s 2 and 3.

t h e f o l l o w i n g q u e s t i o n s : i ) what i s t h e f i r s t e f f e c t o f an i n c r e a s e i n v a l u e o f a v a r i a b l e upon i t s e l f ? when does i t o c c u r ( d e l a y s f o r i n t e r a c t i o n b e i n g equal o r not)

i i ) which v a r i a b l e s can most i n f l u e n c e , t h r o u g h a s i n g l e p a t h , a g i v e n

one by augmenting ( i n h i b i t i n g ) i t ?

i i i ) which v a r i a b l e s have o n l y an augmenting

( i n h i b i t i n g ) e f f e c t on a g i v e n one?

4.3.

S I G N SOLVABILITY

OF

SYSTEMS OF QUALITATIVE EQUATIONS

L e t AX = b denote a square system o f l i n e a r e q u a t i o n s and assume t h a t t h e s i g n s , b u t n o t t h e magnitudes, o f a l l c o e f f i c i e n t s o f A and b a r e known.

AX = b i s

s i g n - s o l v a h h i f and o n l y i f t h e e x i s t e n c e o f a s o l u t i o n and t h e s i g n s o f a l l components o f X a r e determined b y t h e s i g n s o f t h e c o e f f i c i e n t s o f A and b

A X = b i s strongly sign-solvable i f and o n l y i f i t i s

( c f . Samuelson [ Z ; ) ;

s i g n - s o l v a b l e and no component of X i s e q u a l o f 0 ( c f . Klee and Ladner [17-1). F o l l o w i n g a d i s c u s s i o n o f Samuelson i n ?oun&tions of Economic A n a l y s i s , Lancaster

IZ . d asked f o r necessary and s u f f i c i e n t c o n d i t i o n s f o r AX . ,

sign solvable.

= b t o be

He a l s o n o t e d t h a t s i g n s o l v a b i l i t y i s n o t a f f e c t e d b y permuta-

t i o n or rows o r columns o f A o r by m u l t i p l i c a t i o n o f a r o w o r v a r i a b l e by -1. Any s i g n s o l v a b l e system can b e m o d i f i e d i n o r d e r t o have aii bi

\
a=O and b=O

A10)

a.b=O*a=O

or

b=O

We w i l l c a l l a semiring p o s i t i v e , i f i t s a t i s f i e s axiom A9.

W e c a l l a semiring 5 ordered, i f t h e r e i s a p a r t i a l o r d e r 6 d e f i n e d on S such t h a t t h e f o l l o w i n g axiom i s s a t i s f i e d : All)

If a 4 b and c 6 d i n S, then a + c 6 b + d.

S i s t o t a l l y ordered, i f 6 i s t o t a l o r d e r on S .

Iteration and summabilit>3in semirings

233

Often a n a t u r a l o r d e r i n g on a s e m i r i n g S i s g i v e n by t h e f o l l o w i n g d i f f e r e n c e r e l a t i on: A12)

a 6 B t h e r e i s x e S w i t h a t x = b

We c a l l S o r d e r e d b y t h e d i f f e r e n c e r e l a t i o n i f t h e o r d e r i n g on S s a t i s f i e s axiom A12).

Then S i s a l r e a d y p o s i t i v e .

The d i f f e r e n c e r e l a t i o n i s a r e f l e x i v e and t r a n s i t i v e r e l a t i o n on any s e m i r i n g , and s a t i s f i e s axiom A l l ) . An i m p o r t a n t c l a s s o f s e m i r i n g s which can be o r d e r e d b y t h e d i f f e r e n c e r e l a t i o n i s t h e c l a s s o f idempotent s e m i r i n g s .

We c a l l a s e m i r i n g S idempotent, i f i t s a t i s -

f i e s t h e f o l l o w i n g axiom A13)

ata=a

Idempotent s e m i r i n g s b e a r w i t h r e s p e c t t o a d d i t i o n a s e m i - l a t t i c e s t r u c t u r e and have a+b as s m a l l e s t upper bound o f a and b . Idempotent semi r i n g s a r e f r e q u e n t l y used i n t h e 1it e r a t u r e and c a l l e d " p a t h a1 geI,

b r a i n /Ca 79/.

An i m p o r t a n t s u b c l a s s o f idempotent s e m i r i n g s i s t h e c l a s s o f

s i m p l e s e m i r i n g s , where a s e m i r i n g S i s c a l l e d s i m p l e i f i t s a t i s f i e s t h e f o l l o w i n g axiom A14)

lta=l

Simple s e m i r i n g s were f i r s t s t u d i e d i n /Yo 61/ under t h e name Q - s e m i r i n g s .

A f u r t h e r n a t u r a l subclass o f idempotent s e m i r i n g s i s formed by t h e e x t r e m a l semir i n g s , where a s e m i r i n g S i s extremal, i f i t s a t i s f i e s t h e axiom A15)

atb

E

{a,bl

A s e m i r i n g S i s extremal i f and o n l y if S i s idempotent and t o t a l l y o r d e r e d ( b y the difference relation). Many examples o f extremal semirings, l i k e t h e s e m i r i n g s [Min and Max a r e d e r i v e d from t o t a l l y o r d e r e d monoids M = (M,t,O,c). = (Mu{z), 6

min,t,z,O)

Converting

o r Max(M) = (Mu{z),max,+,z,O)

M i n t o a s t r u c t u r e Min(M)

by d e f i n i n g min and max t h r o u g h

on M, and a d j o i n i n g z, one o b t a i n s e x t r e m a l s e m i r i n g s which, i n a d d i t i o n , a r e

positive. I f i t i s f u r t h e r assumed t h a t 0 i s s m a l l e s t ( o r l a r g e s t ) element i n M, then Min(M),

and Max(M) r e s p e c t i v e l y , a r e simple. c a l l e d Oijkstra-semirings

i n /Le 77/.

Semirings, which a r e e x t r e m a l and s i m p l e a r e They have been g i v e n t h i s name, because

B. Mahr

234

they a r e r i c h enough t o y i e l d correctness o f D i j k s t r a ' s a l g o r i t h m f o r p a t h problems over such semirings. F i n a l l y i n t h i s s e c t i o n we d e f i n e t h e semiring o f m a t r i c e s o v e r some semiring. An nxn-matrix over a semiring S i s a mapping M: [n]

with [n]:={l

,...,n l .

can d e f i n e on M(n,S) L e t f o r %M(n,S) i , j - e n t r y o f A.

x

[n]

-+

S

L e t M(n,S) denote t h e s e t o f nxn-matrices over S, then we a semiring s t r u c t u r e i n the well-known way:

and i , j 6 n t h e element A ( i , j )

be denoted by A . 1j

We then d e f i n e f o r nxn m a t r i c e s A and B

and c a l l e d t h e

a d d i t i o n C=A+B by C..:=A..+B. IJ

multiplication

C=A.B

1~ I j by C..:=Ail.B 1.J

zero unit -

.+...+ Ain-Bnj

13

0 by O..:=O 1J

1 by 1..:=1

i f i = jand lij:=O

otherwise.

1J

It i s easily verified that

(M(n,S),+,.,O,l)

i s a semiring.

I n /Le 77/ a semi-

r i n g i s defined l i k e i n d e f i n i t i o n 1.1, except t h a t t h e zero r u l e A8 i s n o t assSince t h e zero r u l e i s needed t o show t h a t I i s u n i t i n

umed t o be s a t i s f i e d .

the semiring M ( n , S ) , we have i n c l u d e d t h i s axiom i n o u r d e f i n i t i o n i n c o n t r a s t t o t h e a x i o m a t i z a t i o n i n /Le 77/. F o r n 3 2, M(n,S)

can never s a t i s f y A10.

F o r n = 1 we have M(n,S) isomorphic t o S .

Obviously, ifS i s idempotent, then so i s M(n,S). It also i s true, t h a t i f

i s Pl(n,S).

S is ordered b y t h e d i f f e r e n c e r e l a t i o n (axiom A I Z ) , so

To show t h i s , one d e f i n e s an o r d e r i n g on M(n,S), which i n a n a t u r a l

way i s induced from S: A g B A.. 6 Bij 1J

f o r i,j 6 n

we c a l l t h i s o r d e r i n g the p o i n t w i s e o r d e r on matrices. o r d e r on matrices can never b F t o t a l be simple i f

n B 2.

on M(n,S).

For n

>2

the p o i n t w i s e

And a l s o no semiring M(n,S)

can

Consequently m a t r i c e s over a semiring cannot form an extremal

o r D i j ks tra-semi r i n g .

2.

ITERATION IN SEMIRINGS

I n t h i s s e c t i o n we study s o l v a b i l i t y o f f i x e d p o i n t equations o f t h e form x=b+ax, We w i l l discuss i t e r a t i o n i n t h e semiring o f m a t r i c e s over some semiring. We c a l l a

c a l l e d i t e r a t i o n , f o r a r b i t r a r y elements a and b o f a g i v e n semiring.

23 5

Iteration and summabi1it.v irr semirings

s e m i r i n g 5 c l o s e d , i f f o r a l l a E S t h e r e i s a s o l u t i o n o f x = l t a x i n S. phisms p r e s e r v e s o l v a b i l i t y o f i t e r a t i o n , i . e .

h:S

+

Homomor-

i f z s o l v e s x=b+ax i n S, and

S ' i s a homomorphism, then h ( z ) s o l v e s x=h(b)+h(a)x i n S ' .

2.1 P r o p o s i t i o n L e t S be a s e m i r i n g .

The f o l l o w i n g statements a r e e q u i v a l e n t

(1)

S i s closed, i . e .

(2)

x=a+ax i s s o l v a b l e f o r a l l a c S x=b+ax i s s o l v a b l e f o r a l l a,b E S.

(3)

x=l+ax i s s o l v a b l e f o r a l l a E S

The s i m p l e p r o o f o f t h i s propos t i o n shows more: t h e r e a r e c e r t a i n r e l a t i o n s between s o l u t i o n s . 2.2 P r o p o s i t i o n L e t S be a s e m i r i n g , then (1)

i f z s o l v e s x=l+ax, t h e n z.b s o l v e s x=btax

(2)

i f z s o l v e s x=a+ax, t h e n l + z s o l v e s x = l t a x

(3)

i f z s o l v e s x=b+ax, and t s o l v e s x=O+ax, t h e n z+t.c s o l v e s x=b+ax f o r a l l

C E

s.

I t i s n o t t h e case t h a t a l l s o l u t i o n s o f x=b+ax a r e o f t h e f o r m z.b f o r some s o l u t i o n z o f x=l+ax. T h i s i s seen i n an example t o 2.6. One cannot e x p e c t much t o know about s o l v a b i l i t y o f i t e r a t i o n i n g e n e r a l s e m i r i n g s More i n t e r e s t i n g r e s u l t s a r e o b t a i n e d i n p a r t i c u l a r s e m i r i n g s o r i n s e m i r i n g s which s a t i s f y a d d i t i o n a l axioms.

2.3 Examples the f o l l o w i n g i s true: ( 1 ) I n t h e semiringINo = (lNo,+,.,O,l) x=b+ax i s s o l v a b l e i n No i f and o n l y i f b = 0. I n t h i s case 0 i s t h e o n l y solution. the following i s true: ( 2 ) I n t h e s e m i r i n g IR = (lR,+,.,O,l) x=b+ax i s s o l v a b l e i n IR f o r a l l b and a l l a f 1. I n t h i s case

i s the only

solution. T h i s i s a l s o t r u e i n any f i e l d .

I f a s e m i r i n g i s ordered, we may ask f o r a minimum s o l u t i o n o f i t e r a t i o n . However, i t seems t o be d i f f i c u l t t o answer t h i s q u e s t i o n i n any g e n e r a l i t y (see

236

8. Mahr

a l s o the l a s t paragraph i n t h i s s e c t i o n ) . Notation: L e t S be a semiring and a and

an:=an-’.a

E

S , then

for n > 0

(1)

aO:=l

(2)

acn>:=ao+. . .an f o r n >, 0.

Using t h i s n o t a t i o n we can s t a t e

2.4 P r o p o s i t i o n Let S be a semiring which i s ordered by t h e d i f f e r e n c e r e l a t i o n , then

(1)

i f z solves x=b+ax, then f o r a l l n >, 0.a

n

.b -s z and a.b c: z

( 2 ) i f a.b solves x=b+ax f o r some n t 0, then a.b=a/ n,

Idempotent semi r i n g s a r e ordered by the d i f f e r e n c e r e l a t i o n , which then takes the form a 6 b a+b=b.

I n t h i s case we have

2.5 P r o p o s i t i o n L e t S be an idempotent semiring, then

+ z2

(1)

i f z1 and z2 s o l v e x=b+ax, then so does z1

(2)

i f f o r some n

(3)

a 4 1 i f and o n l y i f 1 i s minimal s o l u t i o n o f x=l+ax.

>

0 an=an+’,

then as

c a l l e d generalized a d d i t i o n , such t h a t the followinq axioms a r e s a t i s f i e d

D1)

1 i s t o t a l l y defined f o r a l l f i n i t e I and I=@ I

D2) :a u = a1. :iEJ i \

D3)

L ai i s defined by

CL

i f f for all k S

ie1 7 ( b a i ) i s defined by ba

ie I

and

24 1

Iteration and summability in semirings C

(sib)

i s d e f i n e d by ab

icI 04)

Given

{ai

C ai

I

i a I 1 and decomposition I =

i s d e f i n e d by

CI I jeJ j’

i f f for a l l j e J

a

then ai

C

is

i€1

icI

j

d e f i n e d by, say, aj

,

and

i s d e f i n e d by a

C a jeJ

D5 )

a. (b. c ) = (a. b ) c

D6)

a.l=l-a=a

I We c a l l an element 66s an I - i n d e x e d f a m i l y and denote i t by

6={ailicI}.

I

is

c a l l e d an i n d e x s e t .

A p a r t i a l complete s e m i r i n q i s c a l l e d complete, i f f o r a l l

countable index sets

I the generalized a d d i t i o n

i s a t o t a l l y defined

operation.

A p a r t i a l complete s e m i r i n g i s an a l g e b r a w i t h p a r t i a l l y defined i n f i n i t a r y o p e r a t i o n s f o r each i n d e x s e t I. While t h i s l o o k s l i k e a monstrous o b j e c t , i t i s v e r y c o m f o r t a b l e t o work w i t h .

E s s e n t i a l l y , t h e r e i s o n l y one i n f i n i t a r y p a r -

t i a l l y defined operation:

L e t k:J

+

I be b i j e c t i v e , t h e n f o r a l l indexed f a m i l i e s { a i l i zai= id

E

I]

z a jrJ k(j)’

Thus g e n e r a l i z e d a d d i t i o n i s i n v a r i a n t under renaming and rearrangement o f i n d e x s e t s , which i s a consequence o f axioms D2 and D4.

F u r t h e r i d e n t i t i e s can b e

o b t a i n e d which a r e f r e q u e n t l y needed:

3.3

Fact

I n a p a r t i a l complete semi-ring

(1) (2) (3)

( C ai) i aI

.

( 1 b.) jeJ

=

C .a..bj ( i ,j )~IXJ

’

iE1

jcJ ”

z ( c a..) j e J ioI

c

( , c aij)

=

ieI

J ~ J

c

(

c a. . )

=

the following i d e n t i t i e s are true

’’

c

aij

( i ,j k I x J

and one s i d e o f these i d e n t i t i e s i s d e f i n e d whenever t h e o t h e r s i d e i s d e f i n e d .

B. Mahr

242

We say t h a t a p a r t i a l complete semiring S=(S,c,.,l) p a r t i a l ordering

Q

i s ordered, i f t h e r e i s a

d e f i n e d on S such t h a t the f o l l o w i n g axiom i s s a t i s f i e d :

07) f o r a l l indexed f a m i l i e s { a i l i c I l and t b i l i c I l for a l l i t 1

ai-< bi

=)

C a . c c bi irI

provided

C ai

and

'

C bi

ipI

a r e defined.

icI

ipI

And we c a l l a p a r t i a l complete semiring idempotent, i f f o r a l l acS the sum i s d e f i n e d and equals a.

3.4

C a

i cI

Proposition

I f S i s an idempotent p a r t i a l complete semiring, then S i s ordered by

and t h e f o l l o w i n g i s t r u e

(1)

L a . i s supremum of I a i l i c I i

iLI

c

(2)

1

a

i

i f i t i s defined

i s smallest s o l u t i o n o f x=l+ax i f i t i s d e f i n e d . n

i e No

Obviously, t h e r e i s a c l o s e r e l a t i o n between semirings and p a r t i a l complete semir i n g s which we w i l l use subsequently t o d e f i n e summability i n semirings. L e t S=(S,c,.,l)

be a p a r t i a l complete semiring, then the semiring r e s t r i c t i o n o f

- i s defined t o be t h e unique s e m i r i n g S=(S,+,.O,l) 0:= al+a2:=.

with

c ai ia$

c 1E{l ,Z}

ai

Note t h a t the semiring r e s t r i c t i o n o f a p a r t i a l complete s e m i r i n g i s p o s i t i v e

[GR83].

We a l s o c a l l

determined by S.

5

a p a r t i a l completion of S , which o f course i s n o t uniquely

However, t h e r e i s a unique'lninimal" p a r t i a l completion o f a

p o s i t i v e semiring S which i s obtained by d e f i n i n g f o r each f i n i t e index s e t I the sum .C ai as i t e r a t e d b i n a r y a d d i t i o n f o r an a r b i t r a r y f i x e d o r d e r i n g o f I , and 161 l e a v i n g C ai f o r i n f i n i t e I undefined. We c a l l t h i s p a r t i a l completion o f S the id g e n e r a l i z a t i o n o f S.

243

Iteration and summability in semirings

3.5

Definition

Given a s e m i r i n g S=(S,+,-,O,l) and an indexed f a m i l y 6 ={ai l i e 1 1 i n S.

We c a l l 6

C ai e x i s t s i n S, i f t h e r e i s a p a r t i a l i eI i s defined.

a d d i t i v e and e q u i v a l e n t l y say t h a t c o m p l e t i o n o f S, i n which

C ai

icI Given a s e t F= {6.ljaJ} o f indexed f a m i l i e s 6 J j' We say F i s a d d i t i v e and e q u i v a e x i s t s f o r a l l & = { a . m . ) , i f t h e r e i s a p a r t i a l complel e n t l y say t h a t ai J-1 Jie1 j t i o n o f S i n which a l l 6 . a r e a d d i t i v e . J Semirinqs may have d i f f e r e n t p a r t i a l completions, so t h a t t h e mere e x i s t e n c e o f C ai irI

does n o t guarantee t h a t a unique s e m i r i n g element i s t h e r e b y denoted.

We

w i l l t h e r e f o r e i n a d d i t i o n t o e x i s t e n c e say how C ai can be d e f i n e d , and c a l l ieI

2 ai=b c o n s i s t e n t i n S, i f t h e r e i s a p a r t i a l c o m p l e t i o n o f S i n which C ai i s it1 icI d e f i n e d by b.

3.6

Examples

(1) L e t INo be t h e s e m i r i n g o f n a t u r a l numbers, t h e n t h e f o l l o w i n g i s t r u e : c ai=b i s c o n s i s t e n t i n INo i f and o n l y i f t h e r e i s a f i n i t e s e t F d such t h a t id ai=O f o r i r I \ F and b= C ai iEF

( 2 ) L e t d b e t h e s e m i r i n g o f p o s i t i v e r e a l numbers, then t h e f o l l o w i n g i s t r u e : A p a r t i a l completion o f p i s o b t a i n e d by d e f i n i n g f o r each c o u n t a b l e index s e t I C ai:=b icI

i f t h e r e i s an o r d e r i n g o f v : I N o > I such t h a t

TR'with

C a i s convergent i n jENo v ( j )

v a l u e b.

F u r t h e r examples a r e d e r i v e d f r o m t h e f o l l o w i n g more g e n e r a l s i t u a t i o n s , which i n c o n n e c t i o n w i t h t r a n s i t i v e c l o s u r e a r e most i m p o r t a n t : 3.7

Definition

L e t S be a s e m i r i n g .

An indexed f a m i l y {ai l i e 1 1 i n S i s c a l l e d s t a t i o n a r y , if

f o r any KcI and t h e induced s u b f a m i l y { a i l i e K } such t h a t f o r a l l f i n i t e R w i t h F(K)GRsK

t h e r e i s a f i n i t e s e t F(K)cK

B. Mahr

244

z

a

~a~ ieR

=

ieF(K) holds.

3.8 Let

We c a l l F(K) a c h a r a c t e r i s t i c index s e t o f K.

Fact {ai l i

E. I } be a s t a t i o n a r y indexed f a m i l y w i t h c h a r a c t e r i s t i c index sets

F o ( I ) and F 1 ( I ) , then ai =

X irFo( I )

1

ai

ieFl ( I )

3.9 P r o p o s i t i o n L e t S be a p o s i t i v e semiring and I a i l i E I } a s t a t i o n a r y indexed f a m i l y i n S, then f o r any c h a r a c t e r i s t i c index s e t F ( 1 ) o f I

Z ai ieI

=

a.

1

'

iaF(1)

i s c o n s i s t e n t i n S. Proof: 7: ai

5 of

We d e f i n e a p a r t i a l completion

S as f o l l o w s :

i s defined i f f i a i ) i o D i s s t a t i o n a r y , and i n t h i s case we d e f i n e

ieI

Z ai:= ieI

Z

i r F ( 1 ) ai

f o r c h a r a c t e r i s t i c index s e t F ( I ) o f I .

By f a c t 3.8 t h i s y i e l d s a w e l l - d e f i n e d

expansion o f S which we have t o show t o be a p a r t i a l completion o f S. the semiring r e s t r i c t i o n o f ?iequals S.

(D2) and (03) are s t r a i g h t forward.

Obviously

Definedness c o n d i t i o n s i n axioms ( D l ) ,

Definednesr c o n d i t i o n s i n (04) a r e d e r i v e d

as f o l l o w s : Let K

I and K

=

U jcJ

K . be d i s j o i n t , then by d e f i n i t i o n the f a m i l y { a i ( i E K:) J

s t a t i o n a r y f o r any j EJ.

I t i s l e f t t o show t h a t a l s o { C a i I j iaK j

s t a t i o n a r y : Given J '

We then consider the f a m i l y

c_

J.

6 = {aili

r

U j c J ' Kj}

6 i s a subfamily o f l a i l i r I}, and thus s t a t i o n a r y . L e t X be a c h a r a c t e r i s t i c index s e t of 6.

We then show t h a t

F ( J ' ) = { j E J ' I K j n X # $I}

E

J} i s

is

Iteration and summability in semirings i s a c h a r a c t e r i s t i c index s e t o f the family {

a . [ j c J ' } which completes t h e ieK. J

proof. __ Fact:

245

Given f o r a l l j o J ' f i n i t e s e t s Q . such t h a t J F( K j

€9 .sK i

RsJ'

then f o r a l l f i n i t e

c

j

we have

c

a.

'

itF(K.) J

jrR

C

C

jcR

ieQ.

=

a 1.

J

We have t o v e r i f y t h a t f o r a r b i t r a r y f i n i t e R w i t h F ( J ' ) r R sJ' the equation

c

C

jeF(J')

a.=

'

ieF(Kj)

C

1

a.

i c F ( KJ. )

jeR

F o r t h e l e f t s i d e of (1) we d e r i v e from t h e f a c t by choosing

holds t r u e .

R=F(J' ) and Q . = F ( K . ) u ( K.nX) J J J

C

C

jGF(J')

icF(K.) J

a. =

'

C ai icX

For t h e r i g h t s i d e o f (1) we d e r i v e f r o m t h e f a c t by choosing R a r b i t r a r y and Qj=F(Kj)u(K.nX) J

c

C

i r F ( KJ. )

jeR

Since X =

U

jeF (J ' )

a . = C jcR

C a i

(3)

iaQi

(KjnX) by d e f i n i t i o n o f F ( J ' ) , and

and c o n s e q u e n t l y a l s o

we o b t a i n by d i s j o i n t n e s s o f t h e Q j X a i = X a i ieX id). J

C joR

(1) then i s deduced from ( 2 ) and t h e combination o f z a 1. l j r J } i s s t a t i o n a r y . ieK

Thus a l s o {

j

(3)

and

(4).

B. Mahr

246

I t i s now s t r a i g h t forward t o complete t h e p r o o f .

3.10 Examples

( I ) Let 6 = { a i \ i

f o l 1owing

E I ) be an indexed f a m i l y i n a semiring S w i t h the

property : t h e r e i s a f i n i t e s e t F(K) s I such t h a t ai

= 0 for a l l

E

I V ( 1 ) then

i s called finitary. Any f i n i t a r y f a m i l y i s s t a t i o n a r y w i t h c h a r a c t e r i s t i c index s e t F ( I ) , and thus additive.

(2)

Let 8 =

I1 be an indexed family i n an idempotent semiring S w i t h

E

the f o l l o w i n g property: there i s a f i n i t e s e t F(1) c_ I such t h a t f o r a l l i E I t h e r e i s J E F ( 1 ) w i t h ai = a

j

then 6 i s c a l l e d weakly f i n i t a r y . Any weakly f i n i t a r y f a m i l y i n an idempotent semiring i s s t a t i o n a r y with characteri s t i c index s e t F ( I ) , and thus a d d i t i v e . ( 3 ) L e t 6 = t a i l i e I } be an indexed f a m i l y i n an idempotent semiring S w i t h t h e fallowing property: for all K

G

I t h e r e i s a f i n i t e s e t F(K) such t h a t f o r a l l i

E

K there

i s j E F(K) w i t h ai L a .

J

then 6 i s c a l l e d f i n i t e l y bounded.

Any f i n i t e l y bounded f a m i l y i n an idempotent semiring i s s t a t i o n a r y w i t h character i s t i c index s e t F ( I ) , and thus a d d i t i v e . I n s p e c i f i c semirings, such as f r e e semirings w i t h elements represented by formal power series, one can f i n d f u r t h e r examples o f a d d i t i v e indexed f a m i l i e s (see /Ma 821). For t h e semiring o f m a t r i c e s over some semiring S we o b t a i n t h e f o l l o w i n g r e s u l t : 3.11 _Proposition L e t S be a semiring and n a r b i t r a r y .

Then an indexed f a m i l y { A k l k E I } i n M(n,S)

i s a d d i t i v e i f and o n l y i f f o r a l l i,j c n t h e s e t o f f a m i l i e s { A k ( i , j ) l k additive i n S.

E

I} i s

Iteration and summability in semirings

4.

247

SEMIRINGS

TRANSITIVE CLOSURE I N POSITIVE

I n t h i s s e c t i o n we apply t h e theory developed i n the previous sections t o t r a n s i t i v e c l o s u r e o f matrices over semirings. t r a n s i t i v e closure

There are several ways t o d e f i n e

o f matrices over general semirings.

We t h e r e f o r e i n v e s t i g a t e

t h e i r r e l a t i o n f i r s t , and then study existence. I f S i s a p o s i t i v e semiring and A e M(n,S),

then we might simply ask f o r a solu-

t i o n T(A) o f t h e i t e r a t i o n X=l +Ax T(A), however, i s n o t uniquely determined and we do n o t

i n t h e semiring M(n,S).

know much about the minimal s o l u t i o n s , i f S i s ordered.

Another drawback o f t h i s

s p e c i f i c a t i o n o f t r a n s i t i v e c l o s u r e i s t h a t i t t e l l s l i t t l e about u n d e r l y i n g a p p l i c a t i o n s , a t l e a s t i n the c o n t e x t o f path problems. We g i v e t h r e e ways of d e f i n i n g matrices T(A) from a m a t r i x A and show t h a t i n a c e r t a i n way they are a l l equivalent, and, whenever they e x i s t , solve t h e above iteration: L e t S be a p o s i t i v e semiring and A

E

M(n,S).

Then matrices A',

A* and A

T

are

f o r m a l l y d e f i n e d as f o l l o w s :

A':

A'

. = C Aijk f o r i,j 4 n

ij-

kaO

A*:

A* : = 1

+

A(n) where A(n) i s d e f i n e d i n 2.10

w i t h a* denoting the formal element

1 ai

irNo AT: ATj:

=

C

v a l A ( p ) f o r i,j 6 n

pep..

1J

Here P.. denotes the s e t o f a l l paths from i t o j i n the complete S - l a b e l l e d graph 1.I

which has A as adjacency m a t r i x .

valA(p) denotes the l a b e l l i n g o f path p which

i s d e f i n e d by valA(p) = A(el). f o r p = el. ..er

and ei

Q

...

.A(er)

[n]x[n].

These d e f i n i t i o n s do n o t i m p l y existence o f the matrices A', f o r e i n t r o d u c e t h e f o l l o w i n g convention: Let

5 denote a p a r t i a l completion o f

S.

Then we say:

A* o r AT.

We there-

B. Mahr

248

A'

i s d e f i n e d i n M(nS) i f f o r a l l i , j r n t h e sum k A..

Z kc INo

A* i s d e f i n e d i n rl(n,S)

i s defined i n

s

lJ

i f f o r a l l k h n t h e sum

Z

(ALt-l))i

i s defined i n

i e No

AT i s d e f i n e d i n M(m,S) i f f o r . a l l i , j a the sum valA(p)

C

i s defined i n

5

pep..

1J

The f o l l o w i n g r e s u l t shows t h a t a l l t h r e e concepts o f t r a n s i t i v e c l o s u r e are e q u i valent: 4.1 P r o p o s i t i o n Let S be a p o s i t i v e semiring and

be a p a r t i a l completion o f S.

Then are

equivalent: (1)

A'

(2)

A* i s d e f i n e d i n M(n,s)

(3)

AT i s d e f i n e d i n M(n,S)

i s d e f i n e d i n M(n,S)

and i n case a l l t h r e e matrices are d e f i n e d i n M(n,S),

then A'

= A*

=

A'.

The p r o o f o f t h i s p r o p o s i t i o n i s based on two lemnata which g i v e w e l l known i n t e r p r e t a t i o n s o f t h e m a t r i c e s A'

and A(r) which d e f i n e A

and A* r e s p e c t i v e l y .

4.2 Using the n o t a t i o n

P P..

L

”

k A..

c

t h a t i s t o say t h a t f o r a l l i , j s n

kaNo

i s a decomposition o f P . .

lJ

i t follows

1.J’

from axiom D4 t h a t

5.

i s consistent i n

Consequently

Since f o r a l l r = o ,..., n axiom 04 t h a t f o r a l l r =

z

P.:)(

A

EP..

1J 0,

T e x i s t s and AT and

1J

..., n

P (. n ) = 1J

= A’,

P..\{k},

we deduce f r o m

1J

valA(p)

pep!:) 1J

exists i n

and t h e r e f o r e by lemma 4 . 3 ( 1 ) f o r a l l k = 1, ... ,n

5, i it

N

0

5.

i s defined i n A?.

=

Thus a l s o A* e x i s t s .

Usino lemma 4 . 3 ( 2 ) we o b t a i n

1 . . + A (. 1 )

1J

=

1J

1.1

5

valA(P)

pep!!) 1J

-- -

valA(p) = AT.

~

PEP.

1J

’

1J

thus

A

*

I-

= A-.

So we h a v e shown:

(2)

A‘

e x i s t s = A* e x i s t s and A T e x i s t s and A‘

Assume k ’ e x i s t s i n M(n,S),

is defined i n 5 .

Since Pij

=

*

= A

T

= A

then f o r a l l i , j c n

u

#ij 7 i s

r40

a decomposition o f P . . we o b t a i n f r o m 1J

Iteration and summability in semirings

253

axiom D4 and lemma 4.2 t h a t f o r i,j 6 n

c re

AYj

No

5.

i s defined i n (3)

Assume A

*

i s defined i n

exists i n

5.

Thus a l s o A

c

exists.

e x i s t s i n M(n,S),

then by d e f i n i t i o n f o r a l l k s n

Using lemma 4 . 3 ( 2 ) , we o b t a i n t h a t f o r a l l r = 1, ..., n

5.

Since P!n)

1 j

=

P..\thl 1J

we deduce e x i s t e n c e o f A

T

and i , j s n

i n N(n,S).

T h i s completes t h e p r o o f o f p r o p o s i t i o n 4.1. We may f u r t h e r ask, how t h e m a t r i c e s AT,

C

A',

A' and A* a r e r e l a t e d w i t h t h e m a t r i x

which i s expressed i n terms o f an i n f i n i t e sum i n t h e s e m i r i n q Fl(n,S)

i e No r a t h e r t h a n S.

b l i t h o u t p r o o f we s t a t e t h e f o l l o w i n g r e s u l t which i s c l o s e l y

r e l a t e d t o p r o p o s i t i o n 3.11:

4.4 P r o p o s i t i o n L e t S be a s e m i r i n g and AEM(n,S).

Then

Z

A' e x i s t s i n M(n,S)

i f and o n l y i f

i a No

there i s a p a r t i a l completion

C

5

o f S such t h a t A'

i s d e f i n e d i n M(n,S) and

Ai = A'

i e No

i s consistent i n M(n,Sl F i n a l l y we o b t a i n

4.5 C o r o l l a r y L e t S be a s e m i r i n g and Ae M(n,S). s o l v e X = 1 + AX i n M(n,S),

C

Then any of t h e t h r e e m a t r i c e s A ,. A

*

and A

T

p r o v i d e d t h e r e i s a p a r t i a l c o m p l e t i o n o f S i n which

they a r e defined.

We now t u r n t o t h e q u e s t i o n o f e x i s t e n c e of t r a n s i t i v e c l o s u r e and s t a t e a number

B. Mahr

254

o f r e s u l t s , which are well-known i n the l i t e r a t u r e (see e.g. never been s t a t e d i n t h i s g e n e r a l i t y .

/ Z i 81/), b u t have

Also these r e s u l t s c o n f i r m our concept o f

sumnabi lity. I f S i s an idempotent semiring and A e M(n,S).

Then A i s c a l l e d semipositive, i f

f o r a l l cycles q i n A valA(q) 4 1

A i s c a l l e d p o s i t i v e , i f Aij

6 1 f o r a l l i , j 4 n.

I n t h e semiring M = (Ru(-l,min,t,m,O), i s semipositive,

which i s idempotent, a m a t r i c A E M(nJMin)

i f a l l i t s cycles q have non-negative value, i . e . v a l A ( q ) bRO

w i t h respect t o the n a t u r a l o r d e r i n g

on

IR. A i s

positive i f a l l i t s entries

are non-negative. 4.6 P r o p o s i t i o n L e t S be an idempotent semiring and A

E

M(n,S) semipositive.

Then we conclude:

n-1 . A = C A ’ i=O i s consistent, and AT i s s m a l l e s t s o l u t i o n o f X=ltAX ( a c c o r d i n g t o the p o i n t w i s e order on m a t r i c e s ) .

T Moreover, A =A*,

t h e Kleene-matrix w i t h r e s p e c t t o ( A i k - ’ ) ) *

=

1 f o r k = 1,

...,n.

Since a m a t r i x over a simple semiring i s always p o s i t i v e , and thus a l s o semip o s i t i v e , we have

4.7 Corol 1a r y The conclusion i n 4.6 i s t r u e i f S i s simple and A a r b i t r a r y . For extremal ( t o t a l l y ordered idempotent) semi r i n g s we o b t a i n 4.8 P r o p o s i t i o n I f S i s extremal and A t M(n,S)

semipositive, then the conclusion i n 4.6 i s t r u e .

Moreover, f o r a l l i,j 4 n t h e r e i s a simple p a t h po

B

P . . such t h a t 1J

ATj = valA(po).

For m a t r i c e s over D i j k s t r a - s e m i r i n g s we o b t a i n the conclusion o f 4.8 g e n e r a l l y . T /Le 77/ has shown t h a t f o r t h e computation o f A over a D i j k s t r a - s e m i r i n g the

Iteration and summability in semirings

255

well-known D i j k s t r a - a l g o r i t h m i s c o r r e c t . 4.9 Remark Summability i n a r b i t r a r y and n o t j u s t p o s i t i v e semirings can be s t u d i e d by a d i f f e r e n t a x i o m a t i z a t i o n o f p a r t i a l complete semirings, which i s obtained by chang i n g t h e " i f f " i n axiom D4 t o " o n l y i f " . This change a f f e c t s f a c t 3.3 and propos i t i o n 4.1: Existence o f AT i m p l i e s e x i s t e n c e o f A c and A*, b u t n o t v i c e versa. ( T h i s was p o i n t e d o u t t o t h e author by G. Rote /GR 83/).

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/BC 75/

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/Br 74/

Brucker, P., Theory o f m a t r i x algorithms, Mathematical Systems i n Economics 13 (Anton Hain, 1974).

/Ca 71/

Carre, B.A., An a l g e b r a f o r network r o u t i n g problems, J . I n s t . Maths. A p p l i c s . 7 (1971).

/Ca 79/

Carr6, B.A.,

/CG 791

Cuninghame-Green,

/CH 65/

Cruon, R. and HervC, P., Reone fr. Rech. oper. No. 34 (1965) 3-19.

/Co 71/

Conway, J.H., 1971).

/ E i 74/

Eilenberg, S.,

/ E l 77/

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/GM 82/

Gondran, M., and Minoux, M., unpub 1ished paper.

/HR 68/

Hammer, P.L. and Rudeanu, S., (Springer, 1968).

/Le 77/

Lehmann, D., A l g e b r a i c s t r u c t u r e s f o r t r a n s i t i v e closure, Theoretic. Computer Science 4 (1977).

/Ma 82/

Mahr, B., Semirings and T r a n s i t i v e Closure, TU B e r l i n , FB 20, B e r i c h t NO. 82-5 (1982).

/Mar 76/

M a r t e l l i , A Gaussian e l i m i n a t i o n a l g o r i t h m f o r t h e enumeration o f c u t sets i n a graph, J . ACM 23 (1976).

/Mo 611

M o i s i l , G.C.,

Graphs and Networks (Clarendon Press, 1979).

R.A.,

Minimax algebra, (LNEMS 166, Springer, 1979).

Regular Algebra and F i n i t e Machines (Chapman and H a l l , Automata, Languages and Machines (Academic Press, 1974).

D i o i d Theory and i t s A p p l i c a t i o n s , Boolean Methods i n Operations Research

Comnunicarile Acad. R e p u b l i c i i Populare Romine, 10, 647-652.

256

B. Mahr

/Ta 81a/

Tarjan, R . E . , (1981).

A unified approach t o path problems, J . ACM, 28 No. 3 ,

/Ta 81b/

Tarjan, R . E . , 3, (1981).

Fast algorithms f o r solving path problems, J . ACM, 28, No.

/Wo 79/

Wongseelashote, A . , Semirings and path spaces, Discrete Mathematics 26 (1979).

/Yo 61/

Yoeli, M . , A note on a generalization of Boolean matrix theory, Americ. Math. Monthly 68, 1961.

/Zi 81/

Zimnermann, U., Linear and combinatorial optimization i n ordered algeb r a i c s t r u c t u r e s , book t o appear, 1981.

/GR 83/

Rote, G . , Personal communication, (1983).

Annals of Discrete Mathematics 19 (1984) 257-356 0 Elsevier Science Publishers B.V. (North-Holland)

251

SUBSTITUTION DECOMPOSITION FOR DISCRETE STRUCTURES AND CONNECTIONS WTTH COMB1NATOR I A L OPTIMIZATION* R.H. MtJhring

T e c h n i c a l U n i v e r s i t y o f Aachen Lehrstuhl f u r Informatik I V 51 Aachen West Germany

F. J

. Radermacher

U n i v e r s i t y o f Passau L e h r s t u h l f u r I n f o r m a t i k und Operations Research 839 Passau West Germany

I n t h i s paper we deal w i t h t h e s u b s t i t u t i o n decomposition as known f o r Boolean f u n c t i o n s , s e t systems and r e l a t i o n s . I t i s shown how c o m b i n a t o r i a l o p t i m i z a t i o n problems, p a r t i c u l a r l y on graphs, p a r t i a l o r d e r s , p r o j e c t networks, ( i n - ) dependence systems and c l u t t e r s , n a t u r a l l y l e a d , under weak assumptions,to t h i s k i n d o f decomposition v i a c e r t a i n uniqueness r e s u l t s . We t h e n g e n e r a l i z e t h e comnon f e a t u r e s o f t h i s t y p e o f decomposition from t h e s p e c i a l cases t o a q u i t e g e n e r a l a l g e b r a i c l e v e l , c o v e r i n g i n f i n i t e cases, and deduce a g e n e r a l Jordan-Holder theorem as w e l l as uniqeness r e s u l t s f o r t h e associated composition t r e e i n t h i s general s e t t i n g . These r e s u l t s a r e t h e n r e i n t e r p r e t e d f o r t h e s p e c i a l cases, and t h e c o m p u t a t i o n a l aspects o f t h e s u b s t i t u t i o n decomposition a r e d i s c u s s e d . Throughout, a p p l i c a t i o n s t o many problems concerning d i s c r e t e s t r u c t u r e s a r e i n c l u d e d , as a r e connections w i t h o t h e r approaches i n these f i e l d s , i n p a r t i c u l a r t h e s p l i t decomposition. CONTENl INTRODUCTION SUBSTITUTION DECOMPOSITION FOR BOOLEAN FUNCTIONS, SET SYSTEMS AND RELATIONS:

I

BASIC RESULTS AND COMMON PROPERTIES 1. 2. 3. 4. 5.

6.

I1

Aspects o f common b e h a v i o u r S u b s t i t u t i o n decomposition f o r Boolean f u n c t i o n s S u b s t i t u t i o n decomposition f o r s e t systems S u b s t i t u t i o n decomposition f o r r e l a t i o n s - A p p l i c a t i o n s o f t h e s u b s t i t u t i o n decomposition t o graphs - A p p l i c a t i o n s o f t h e s u b s t i t u t i o n decomposition t o p a r t i a l o r d e r s Interfaces - Coherent systems: Boolean f u n c t i o n s vs. c l u t t e r s - Conformal c l u t t e r s vs. graphs - P a r t i a l o r d e r s vs. c o m p a r a b i l i t y graphs Connections w i t h t h e s p l i t decomposition

UNIQUE CHARACTERIZATION OF THE SUBSTITUTION DECOMPOSITION FOR RELATIONS AND SET SYSTEMS FROM THE VIEW-POINT OF COMBINATORIAL OPTIMIZATION

1.

*

C o m b i n a t o r i a l o p t i m i z a t i o n o v e r graphs

Work was supported by t h e M i n i s t e r f u r Wissenschaft und Forschung des Landes N o r d r h e i n - N e s t f a l e n , West Germany.

R.H. Mohring and EJ. Radermacher

258

Combinatorial o p t i m i z a t i o n over p a r t i a l orders - d e t e r m i n i s t i c p r o j e c t networks time-cost t r a d e - o f f i n p r o j e c t networks - one machi ne scheduling s t o c h a s t i c p r o j e c t networks 3. Combinatorial o p t i m i z a t i o n over (in-)dependence systems and c l u t t e r s 4. Summary and h i n t s on some o t h e r approaches t o decomposition o f c e r t a i n combinatorial o p t i m i z a t i o n problems

2.

-

111 AN ALGEBRAIC MODEL OF DECOMPOSITION

1.

2. 3. 4. 5. 6. IV

The a l g e b r a i c model The system o f congruence p a r t i t i o n s The Jordan-Holder theorem f o r composition s e r i e s The composition t r e e Connections w i t h t h e s p l i t decomposition On t h e a l g o r i t h m i c complexity o f decomposition

ALGEBRAIC AND ALGORITHMIC ASPECTS OF THE SUBSTITUTION DECOMPOSITION FOR RELATIONS, SET SYSTEMS AND BOOLEAN FUNCTIONS 1. 2. 3.

Relations Set systems Boolean f u n c t i o n s

ACKNOWLEDGEMENT REFERENCES

INTRODUCTION The decomposition o f s t r u c t u r e s i s i m p o r t a n t f o r t h e development of many mathematical theories.

Extension o f fundamental r e s u l t s from group t h e o r y and r e l a t e d

f i e l d s has l e d t o q u i t e general concepts i n u n i v e r s a l algebra.

Some o f t h e

s t r o n g e s t r e s u l t s a r e those o f t h e Jordan-Holder type d e a l i n g w i t h t h e unique f a c t o r i z a t i o n o f s t r u c t u r e s i n t o prime (simple, i r r e d u c i b l e ) s t r u c t u r e s and being r e l a t e d t o t h e m o d u l a r i t y o f t h e congruence p a r t i t i o n l a t t i c e s . Most s t r u c t u r e s i n u n i v e r s a l algebra a r e q u i t e s t r o n g l y i n t e r n a l l y r e l a t e d ; f o r instance, a r e s t r i c t i o n t o a subset o f t h e u n d e r l y i n g base s e t does n o t i n general l e a d t o a subalgebra.

However, f o r many s t r u c t u r e s i n d i s c r e t e mathematics,

operations research and computer science, such behaviour is o f t e n the case; e.g. Boolean f u n c t i o n s , s e t systems (e.g.

(in-)dependence systems, c l u t t e r s and mat-

r o i d s ) and r e l a t i o n s (e.g. graphs, p a r t i a l orders and p r o j e c t networks) a r e o f t h i s type, as are submodular f u n c t i o n s and matrices.

The decomposition o f such

d i s c r e t e s t r u c t u r e s i s o f i n t e r e s t from t h e o r e t i c a l , computational as w e l l as p r a c t i c a l p o i n t o f view.

However, t h e e x i s t i n g t h e o r e t i c a l framework i s n o t

259

Substitution decomposition for discrete structures r e a l l y adequate f o r h a n d l i n g a l l t h e s e d i s c r e t e cases.

I n particular, the

u n i v e r s a l a l g e b r a r e s u l t s a r e n o t e x t e n d a b l e t o t h i s s i t u a t i o n w h i l e many ad-hoc concepts a r e r e s t r i c t e d t o s p e c i a l d i s c r e t e s t r u c t u r e s and o f t e n do n o t y i e l d deeper s t r u c t u r a l i n s i g h t . T h i s paper aims t o overcome t h i s s i t u a t i o n by p r e s e n t i n g a g e n e r a l i z e d v e r s i o n o f an i n t e r e s t i n g approach known f o r Boolean f u n c t i o n s , s e t systems and r e l a t i o n s (and o t h e r s t r u c t u r e s such as u t i l i t y f u n c t i o n s ) and t h e i r a p p l i c a t i o n s .

It

shows a s i m i l a r b e h a v i o u r i n a l l t h r e e c l a s s e s and t h e n a t u r a l l y i n v o l v e d subs t i t u t i o n o p e r a t i o n appears t o be one o f t h e e s s e n t i a l common f e a t u r e s . I n S e c t i o n I we g i v e a s h o r t survey o f e x i s t i n g r e s u l t s f o r t h e s u b s t i t u t i o n decomposition t o g e t h e r w i t h h i n t s on a number o f a p p l i c a t i o n s .

I n t h i s context,

we a l s o d i s c u s s some c o n n e c t i o n s w i t h t h e s p l i t decomposition.

While more general

and f a r - r e a c h i n g i n i t s r e s u l t s , t h i s decomposition l a c k s t h e s i m p l e a p p l i c a b i l i t y o f t h e s u b s t i t u t i o n decomposition.

Such aspects become p a r t i c u l a r l y c l e a r i n

S e c t i o n 11, where i t i s shown how a g e n e r a l f a c t o r i z a t i o n problem i n c o m b i n a t o r i a l o p t i m i z a t i o n o v e r graphs, p a r t i a l o r d e r s , p r o j e c t networks, (in-)dependence systems and c l u t t e r s n a t u r a l l y l e a d s e x a c t l y t o t h e s u b s t i t u t i o n decomposition f o r a1 1 t h e s e c l a s s e s . I n S e c t i o n I 1 1 we p r e s e n t a g e n e r a l and r a t h e r a b s t r a c t a l g e b r a i c s t r u c t u r e t h e o r y f o r such weakly i n t e r n a l l y r e l a t e d d i s c r e t e s t r u c t u r e s as Boolean f u n c t i o n s , s e t systems and r e l a t i o n s . c o u n t e r p a r t s here.

It t u r n s o u t , t h a t many u n i v e r s a l a l g e b r a r e s u l t s have

T h i s i s even t r u e f o r theorems o f t h e Jordan-Holder t y p e ,

a l t h o u g h t h e congruence p a r t i t i o n l a t t i c e s a r e h e r e g e n e r a l l y o n l y upper semimodular ( i n t h e f i n i t e case), i . e . t h e y have weaker p r o p e r t i e s t h a n t h o s e f a m i l i a r from u n i v e r s a l algebra.

I n a d d i t i o n , t h e s i t u a t i o n h e r e t u r n s o u t t o be

n i c e f o r a p p l i c a t i o n s , due t o t h e presence o f c o m p o s i t i o n t r e e s .

Some h i n t s on

t h e use o f t h i s i n s t r u m e n t f o r a l g o r i t h m i c approaches t o decomposition, t o g e t h e r w i t h some c o m p l e x i t y a n a l y s i s f o r t h e s p e c i a l s t r u c t u r e s considered, f o l l o w i n Section I V . I n view o f t h e g e n e r a l i t y o f t h e d i f f e r e n t concepts presented, we hope t h a t t h e paper w i l l s e r v e as a s t e p on t h e way t o an a l g e b r a i c decomposition t h e o r y i n d i s c r e t e m a t h m a t i cs.

I SUBSTITUTION DECOMPOSITION FOR BOOLEAN FUNCTIONS, SET SYSTEMS AND RELATIONS: BASIC RESULTS AND COMMON PROPERTIES I n t h i s p a r t we w i l l g i v e an i n t r o d u c t i o n i n t o a v a i l a b l e i n s i g h t i n t o t h e s u b s t i These

t u t i o n decomposition f o r Boolean f u n c t i o n s , s e t systems and r e l a t i o n s .

R.H. Mohring and F.J. Radermacher

260

t h r e e classes o f d i s c r e t e s t r u c t u r e s have been q u i t e thoroughly s t u d i e d by d i f f e r e n t authors i n various, seemingly u n r e l a t e d contexts, and p r e s e n t l y c o n s t i t u t e

p5],

( a p a r t f r o m u t i l i t y theory

which w i l l n o t be covered here) t h e main f i e l d s

o f a p p l i c a t i o n s o f t h e s u b s t i t u t i o n decomposition.

[39]

There i s some evidence [3g,

p o i n t i n g towards p o s s i b l e f u t u r e i n c l u s i o n o f o t h e r s t r u c t u r e s , e.g.

modular f u n c t i o n s and systems o f l i n e a r equations. r e s u l t s w i l l be done w . r . t .

sub-

The p r e s e n t a t i o n of t h e

the intended g e n e r a l i z a t i o n , and t h e terminology,

d i f f e r e n t i n a l l t h r e e f i e l d s , w i l l be u n i f i e d .

The r e s u l t s are o f t e n reformula-

t i o n s , m o d i f i c a t i o n s , o r extensions o f e a r l i e r , versions, b u t some cdses, i n part i c u l a r Tor i n f i n i t e s e t systems, appear here f o r t h e f i r s t time. I n 1.1 we w i l l l i s t some common features, w i t h examples, which a c t as a guide f o r the treatment o f t h e s u b s t i t u t i o n decomposition i n a l l t h r e e classes o f s t r u c t u r e s (1.2

-

1.4) t o g e t h e r w i t h a d i s c u s s i o n o f t h e major a p p l i c a t i o n s .

We c o n t i n u e

i n 1.5 w i t h h i n t s on i n t e r f a c e s between these classes o f s t r u c t u r e s , which f a c i l i t a t e understanding o f t h e analogous behaviour observed. 1.6 some remarks on s i m i l a r i t i e s and d i f f e r e n c e s w . r . t .

F i n a l l y , we i n c l u d e i n t h e more general, b u t

c l o s e l y r e l a t e d s p l i t decomposition [3g.

1.1

ASPECTS OF COMMON BEHAVIOUR

We g i v e a l i s t o f s i x groups o f p r o p e r t i e s ( P l )

-

(P6), which d e s c r i b e i n t e r e s t i n g

aspects o f the s u b s t i t u t i o n decomposition, comnon t o a l l t h r e e classes o f s t r u c t u r e s considered, and j o i n t l y demonstrated i n Example 1.1 .l. Subsequently, ( P l ) (P5) w i l l be r e f e r r e d t o i n 1.2

-

-

1.4, where h i n t s on t h e p r o o f s f o r t h e respec-

t i v e cases are given, treatment o f (P6) being postponed t o S e c t i o n I V .

We mention

here t h a t any s t r u c t u r e S o f t h e t h r e e types i s d e f i n e d on an u n d e r l y i n g s e t

A = As ( t h e base s e t o f A ) . i.e.

a f u n c t i o n F: (0,1ln

variables o f

F.

-f

For example, if t h e s t r u c t u r e i s a Boolean f u n c t i o n , {O,ll,

t h e base s e t AF i s t h e s e t

I f i t i s a s e t system T, i.e.

{X

l,...,~n}

of

a c o l l e c t i o n o f subsets o f some

s e t A, then t h i s s e t A i s t h e base set. F i n a l l y , i f R i s a k-ary r e l a t i o n on A , k i.e. R G A = A x .. x A ( k - t i m e s ) , then again AR = A. Note t h a t f o r s e t systems

.

and r e l a t i o n s , t h e base s e t may be i n f i n i t e . of these s t r u c t u r e s , each subset

B

Due t o t h e weak i n t e r n a l coherence

o f t h e base s e t induces a new s t r u c t u r e S I B by

simply r e s t r i c t i n g t h e g i v e n s t r u c t u r e t o t h i s subset (sanething n o t t r u e f o r a l g e b r a i c s t r u c t u r e s such as groups e t c . ) .

(Pl)

SUBSTITUTION OPERATION

I n each of the t h r e e cases, g i v e n a s t r u c t u r e S ' on a base s e t A ' and f o r each B c A ' a s t r u c t u r e S,

on a base s e t A

6

with

ABnA 8 '

=

0

f o r 8 # B ' , there i s a

26 1

Substitution decomposition for discrete structures

u n i q u e l y d e f i n e d s t r u c t u r e S on t h e base s e t A = the structures S

into S' for B E A'.

B s a i d t o be o b t a i n e d by s u b s t i t u t i o n .

u

AB, o b t a i n e d by s u b s t i t u t i n g $€A I S i s denoted b y S = S'[S,, 8 e A ' ] and i s

T h i s b a s i c o p e r a t i o n , which i s usuaSly m o t i v a t e d f r o m h i g h e r l e v e l problems, and which g i v e s t h e s u b s t i t u t i o n decomposition i t s name, means i n t h e s p e c i a l cases e i t h e r s u b s t i t u t i o n o f Boolean f u n c t i o n s i n t o a n o t h e r i n t h e sense o f e.g. (41, [40], o r s u b s t i t u t i o n o f s e t systems i n t h e sense o f e.g. [lo], [38], [13q, o r e.g.

t h e s o - c a l l e d X - j o i n o f graphs [132]

o r t h e o r d i n a l sum o f p a r t i a l o r d e r s

P4I. S t r u c t u r e s a r e c a l l e d decomposable, i f t h e y have a r e p r e s e n t a t i o n S = S ' [S B €A']

w i t h I A S a / > 1 and

lABl

1 f o r a t l e a s t one 6

>

E

A ' , and

prime

B' otherwise.

T h i s s t r u c t u r a l decomposition occurs n a t u r a l l y i n a p p l i e d problems such as d e t e r m i n a t i o n o f maximal c l i q u e w e i g h t s o r m i n i m a l c o l o u r i n g s i n graphs (compare t h e uniqueness r e s u l t s i n c o m b i n a t o r i a l o p t i m i z a t i o n g i v e n i n S e c t i o n 1 1 ) .

The a p p l i -

c a t i o n s s t i m u l a t e i n t e r e s t i n t h o s e p a r t i t i o n s IT = { A 6 I B a A ' } o f t h e base s e t 8 e A'], where S a r e A o f a s t r u c t u r e S which a l l o w a r e p r e s e n t a t i o n S = S'[S 6' 6 structures over A B Q A ' and S ' i s t h e q u o t i e n t s t r u c t u r e S ' = S/IT on A ' . These B'

p a r t i t i o n s a r e c a l l e d congruence p a r t i t i o n s o f S.

They can e q u i v a l e n t l y be i n t e r -

p r e t e d by means o f t h e s u r j e c t i v e homomorphism 0,: A

+

A ' d e f i n e d by nIT(a)= B

a aA f r o m S o n t o S/n. So qIT maps elements o n t o t h e same element i f t h e y 5 a r e i n t h e same c l a s s o f IT, which i s h e n c e f o r t h denoted by CY II B o r a E

:

where

[BIT i s t h e c l a s s o f

[BIT,

IT

c o n t a i n i n g B.

Congruence p a r t i t i o n s a r e c l o s e l y r e l a t e d w i t h t h e i r c l a s s e s ( c f . ( P 3 ) below) These s e t s can e s s e n t i a l l y b e i d e n t i f i e d w i t h A (a) = CY f o r a l l a E. AB, i n j e c t i v e homomorphisms i n c f AB A, d e f i n e d by i n c A 6' 8 from t h e u n i q u e l y determined autonomous s u b s t r u c t u r e S ( = S I A ) o f S i n t o S. B B

which a r e c a l l e d autonomous s e t s . 9

-f

Note t h a t i n t h e s p e c i a l cases c o n s i d e r e 4 a s t r u c t u r e S i s u n i q u e l y determined b y s p e c i f y i n g a congruence p a r t i t i o n IT,a q u o t i e n t s t r u c t u r e S ' on A ' , and a u t o n m u s substructures SB, B E A ' .

However, i n t h e g e n e r a l a l g e b r a i c Jpproach i n P a r t 111,

t h i s uniqueness o f S ( w h i c h may be viewed as t h e p o s s i b i l i t y t o r e c o n s t r u c t S f r o m t h e d a t a s p e c i f i e d above b y means o f a " g e n e r a l " s u b s t i t u t i o n o p e r a t i o n ) i s n o t required.

In f a c t , i t t u r n s o u t t h a t t h e weak assumptions o f t h e g e n e r a l We w i l l , however, i n c l u d e h i n t s on those

model p e r m i t such a non-uniqueness.

properties o f a general s u b s t i t u t i o n operation t h a t are s u f f i c i e n t f o r i t s i n t e g r a t i o n i n t o t h e g e n e r a l model; t h i s a p p l i e s p a r t i c u l a r l y t o t h e s u b s t i t u t i o n o f Boolean f u n c t i o n s , s e t systems and r e l a t i o n s .

R.H. Mohring and EJ. Radermacher

262 (P2)

AUTONOMOUS SETS

Autonanous sets a r e t h e classes o f congruence p a r t i t i o n s and are r e f e r r e d t o i n t h e l i t e r a t u r e as e.g. bound sets [40], committees [144l,

[lo],

[41],

closed s e t s [58],

11381, e x t e r n a l l y r e l a t e d s e t s [27l,

p a r t i t i v e sets [64],

[153]

[68],

and s t a b l e sets "1411.

[20],

clumps [3],

modules [14],

[9q,

I n t h e f i n i t e case,

autonomous s e t s p r o v i d e a very easy and n a t u r a l way o f v i z u a l i z i n g t h e s u b s t i t u t i o n decanposition, s i n c e they already determine t h e congruence p a r t i t i o n s ( c f .

I n t h e i n f i n i t e case, however, t h i s may no longer be

(S2) and (S2)* below).

t r u e , and congruence p a r t i t i o n s f o r m t h e n a t u r a l n o t i o n f o r v i z u a l i z i n g t h e decanposi t i o n . Given a s t r u c t u r e S we denote t h e system o f autonanous sets by A ( S ) .

While f o r

s e t systems and r e l a t i o n s t h e autonomous s e t s are q u i t e n i c e l y i n t e r n a l l y described, t h i s i s u n f o r t u n a t e l y n o t t h e case f o r Boolean f u n c t i o n s .

However, A ( S )

w i l l f u l f i l some o f t h e f o l l o w i n g p r o p e r t i e s , where t h e standard p r o p e r t i e s ( A l ) , (A2) and (A3) h o l d i n a l l t h r e e cases. (Al)

ia}

E

A(S) for a l l

(A3) L(A4)

CL

a AS, AS

Q

Bn

c

Bn c + I

( ~ 2 ) B,CC A(s), [(A2)* Bi E A ( S ) f o r i

E.

B,C e A ( S ) , B\C f B,C E A ( S ) , 6%

+

I,

cI =>

Bi # fl

A ( S ) ( s o c a l l e d t r i v i a l autonomous s e t s )

-n E:A(S) iEI

and Bi

0, B n C # 0, C\B # 0 0, Bn C 6 , c\B # k3

+

B Uc E A ( S )

E

A(S),

gI Bi

E

A(S)]

B\C E A ( S ) and C\B

E

A(S)

B A C : = ( B \ C ) U (C\B) E A ( S ) ]

For r e l a t i o n s , (A2)* i s a l s o t r u e , which i s n o t g e n e r a l l y t h e case f o r Boolean I f (A4) holds f o r A ( S ) , then A ( S ) i s s a i d t o be

f u n c t i o n s o r set systems. symmetrically closed.

This i s t r u e e.g. f o r Boolean f u n c t i o n s , s e t systems and

symnetric r e l a t i o n s , b u t n o t i n general f o r p a r t i a l orders ( c f . Theorem 4.1.1). Note t h a t ( A 3 ) w i l l n o t be r e q u i r e d f o r the general model discussed i n Section

111; i t w i l l , however, become e s s e n t i a l when i n t r o d u c i n g t h e composition t r e e i n 111.4. For a l l classes o f s t r u c t u r e s considered, autonomy i s a t r a n s i t i v e property. Even stronger, t h e autonanous s e t s o f a s u b s t r u c t u r e SIB o f S are:

(Sl)

A ( S 1 B ) = !C E A ( S )

I

CSB)

f o r each B

EA(S).

Note t h a t w . r . t. the a l g e b r a i c i n t e r p r e t a t i o n o f autonany v i a i n j e c t i v e hanomorA A phisms i n c g , (S1) means t h a t i n c B induces a b i j e c t i o n between t h e autonomous s e t s o f t h e s u b s t r u c t u r e S I B and t h e autonomous sets o f S contained i n B .

(P3)

CONGRUENCE PARTITIONS

We have already h i n t e d a t t h e f a c t t h a t i n u n i v e r s a l algebra congruence p a r t i t i o n s

Substitution decomposition for discrete structures

determine t h e decanpositions.

263

The s e t of a l l such p a r t i t i o n s i s denoted by V ( S )

and i s a subset o f t h e p a r t i t i o n l a t t i c e Z(A) o f AS ordered by f i n e r than -

or

IT'

IT'

c o a r s e r than

o f IT'.I n p a r t i c u l a r ,

c o a r s e s t p a r t i t i o n o f A.

i f each c l a s s o f

IT)

I T :=' {{a)

I

a

EA}

IT

4 IT'

(read

TI

i s c o n t a i n e d i n some c l a s s

IT

i s t h e f i n e s t and n 1 := {A1 i s t h e

I n f a c t , f o r A f i n i t e , i t t u r n s o u t t h a t V ( S ) i s an

upper semimodular s u b l a t t i c e o f Z(A), and t h u s i n p a r t i c u l a r f u l f i l s t h e JordanDedekind c h a i n c o n d i t i o n [13],

[154];

compare Theorem 3.2.5.

A v e r y t y p i c a l f e a t u r e o f t h e s u b s t i t u t i o n decomposition, w i t h no c o u n t e r p a r t i n e.g.

g r o u p t h e o r y , i s t h e s p e c i a l c o n n e c t i o n between congruence p a r t i t i o n s and t h e

autonomous s e t s . out t h a t

IT

I n f a c t , i n t h e f i n i t e case o r f o r a r b i t r a r y r e l a t i o n s i t t u r n s

i s a congruence p a r t i t i o n i f f a l l c l a s s e s of

c o n d i t i o n ( S 2 ) * below).

IT

I

= iLi

i E I}E V ( S )

'TI

( ~ 3 )

(S3):

Li E. A(S) f o r a l l i

=>

6

I

= {Li

I

i

r L i n L . = I f o r i # j= > n = t L 1 ,...,L r , i a l ( r r a A \ U Ljl€V(S) J j=l I} E V ( S ) Li 6 A(S) f o r a l l i E. I]

= iLi

I ~I

i

I}EV(S),

L1 ,... , L rE A ( S ) , [(S2)*

a r e autonomous ( s e e

I n t h e i n f i n i t e case, t h e c o n n e c t i o n i s , i n g e n e r a l ,

somewhat weaker and i s d e s c r i b e d by ( S Z ) , (S2)

TI

[ ( ~ 3 ) *71 = { L

G

:=

:=

U

I T * E V ( S ~ L)~ --3

ioEI,

I

IT*UIL~

0

i e I}

E.

v ( s ) , TI^

G

E v ( s I L ~ )=>

E

IT

icI\{ioIIEV(S)

v(s)]

ie1 i Thus t h e c l a s s e s o f congruence p a r t i t i o n s a r e autonomous s e t s (S1) and f i n i t e l y many d i s j o i n t autonomous s e t s can be extended t o a congruence p a r t i t i o n by means o f singletons (S2).

Also, a l o c a l r e f i n e m e n t o f a congruence p a r t i t i o n by a con-

gruence p a r t i t i o n a s s o c i a t e d w i t h one o f i t s c l a s s e s r e s u l t s i n a new congruence The i n f i n i t e c o u n t e r p a r t s ( S 2 ) * and ( S 3 ) * a r e g e n e r a l l y t r u e f o r

p a r t i t i o n (S3).

O f course, ( S 2 ) g e n e r a l l y means t h a t

r e l a t i o n s , b u t n o t i n t h e o t h e r cases. q u e s t i o n s concerning autonomous s e t s B

A(S) can be t r a c e d back t o q u e s t i o n s con-

E

c e r n i n g congruence p a r t i t i o n s o f t h e form

IT^

:=

\

IB,Ia)

a E A\BI E V ( S ) .

The importance o f congruence p a r t i t i o n s i n a l g e b r a i c t h e o r i e s l i e s i n t h e f a c t t h a t t h e y d e s c r i b e t h e decompositions o f a s t r u c t u r e S, which may be i d e n t i f i e d w i t h t h e q u o t i e n t s t r u c t u r e s S/TI on A ' := A/IT = IBi c l a s s e s o f t h e p a r t i t i o n n = {Bi

I

mentioned s u r j e c t i v e homanorphisms

I

i E I}( i . e . on t h e s e t o f

i e I)E V ( S ) ) o r e q u i v a l e n t l y w i t h t h e above qTI:

A

+

A'.

I n a l l three classes o f structures

we have t h e f o l l o w i n g c o u n t e r p a r t t o ( S l ) : (S4) ( i ) (ii)

IT,U

E

01e

V(S),

IT 4

v ( s / ~ => )

n I T ( u ) := InIT(B) n-'(u'):=rn~'(B')lB't u

=>

I

B u11

6 E

V(S/n)

v(s),

71

4

nIT-1 ( u s )

The f i r s t p a r t o f (54) i s sometimes r e f e r r e d t o as t h e "Theorem o f Induced Homomorphisms".

I t t e l l s us t h a t , g i v e n

IT,U E

V(S) with

TI




E A(S/n)

C ' E A ( s / ~ )=> ~ , ' ( c ' I

(ii)

=A(s)

I n the f i n i t e case ( 8 4 ) ' i s e q u i v a l e n t t o (S4), even w i t h o u t t h e f o l l o w i n g property (S5), which i s needed i n t h e i n f i n i t e case f o r t h i s equivalence and holds f o r the s t r u c t u r e s considered here. C

(85)

E

A(S),

TI

EV(S)

-

[C]a

E A ( S ) , where [C]T

u

:=

[a]a denotes t h e

a d n-completion o f C .

F i n a l l y , we mention two o t h e r p r o p e r t i e s which h o l d ( t r i v i a l l y ) i n the t h r e e classes.

The f i r s t deals w i t h t h e r e s t r i c t i o n T ) B o f a congruence p a r t i t i o n t o an

autonomous s e t B, w h i l e the second i s t h e analogue o f t h e " F i r s t Isomorphism Theorem" [30],

[66]

i n u n i v e r s a l algebra. = i Li

I

i a I } E V ( S ) ==, n l B := i m L i

(S6)

BeA(S),

(S7)

( S I B ) / ( r l B ) i s isomorphic t o ( S ( [B]T)/(T(

(P4)

PRINCIPLES OF INVARIANCE

TI

I

i E I,BnLi#

PIeV(S1B)

NT).

I t turns o u t t h a t t h e systems o f autonanous s e t s remain i n v a r i a n t under c e r t a i n

operations, e.g.

d u a l i s a t i o n , b l o c k i n g o r complementation w i t h i n t h e r e s p e c t i v e

classes o f s t r u c t u r e s . (P5)

ALMOST ALL STRUCTURES ARE PRIME

Contrary t o the s i t u a t i o n f o r important a l g e b r a i c s t r u c t u r e s (e.g. groups), i t t u r n s o u t t h a t the m a j o r i t y of d i s c r e t e s t r u c t u r e s are prime (indecanposable), i.e.

although t h e number o f decomposable s t r u c t u r e s on A = 11 ,.. .,nl may grow

e x p o n e n t i a l l y w i t h n, t h e i r r e l a t i v e frequency w . r . t .

{l, ..., n) tends t o zero when n goes t o i n f i n i t y .

a l l s t r u c t u r e s on A =

This behaviour was observed i n

[99] f o r many d i f f e r e n t classes, such as k-ary r e l a t i o n s , parametric r e l a t i o n s ( i n c l u d i n g graphs, tournaments), c l u t t e r s and p a r t i a l orders, and a l s o f o r Boolean functions, cf.

[136].

Furthermore, i n most o f these cases t h i s behaviour holds

f o r both the l a b e l e d ( d i f f e r e n t s t r u c t u r e s are d i s t i n g u i s h e d ) and unlabeled case (isomorphic s t r u c t u r e s are i d e n t i f i e d ) .

So i f one randomly p i c k s o r generates a

265

Substitution decomposition for discrete structures

s t r u c t u r e on A = 11 ,.

. . ,n}

w.r. t. t h e u n i f o r m d i s t r i b u t i o n on t h e s e t o f s t r u c -

t u r e s on A, t h e n i t w i l l almost c e r t a i n l y b e a prime s t r u c t u r e f o r l a r g e n.

This

b e h a v i o u r t h a t " n i c e " p r o p e r t i e s have an a s y m p t o t i c a l l y v a n i s h i n g f r e q u e n c y i s q u i t e canmon f o r d i s c r e t e s t r u c t u r e s , c f . f o r example t h e i n v e s t i g a t i o n s on prop e r t i e s o f almost a l l graphs [21].

I t does, however, n o t n e c e s s a r i l y mean much

f o r t h e p r a c t i c a l a p p l i c a b i l i t y o f t h e s u b s t i t u t i o n decomposition ( n o t e e.g. almost a l l p a r t i a l o r d e r s ( n e t w o r k s ) have a l e n g t h o f a t most t h r e e [86]). f a c t , e x p e r i e n c e w i t h a p p l i c a t i o n s (e.g.

that In

i n s w i t c h i n g t h e o r y o r network t h e o r y )

i n d i c a t e s t h a t t h e u n i f o r m d i s t r i b u t i o n i s an u n r e a l i s t i c measure, s i n c e t h e The reason may be t h a t

s t r u c t u r e s encountered a r e v e r y f r e q u e n t l y decomposable.

i n s w i t c h i n g t h e o r y , Boolean f u n c t i o n s o c c u r r i n g a r e o f t e n generated b y symmetric i n t e r n a l c o m p o s i t i o n laws which f a v o u r t h e e x i s t e n c e o f autonomous s e t s [40)

,

w h i l e i n a p p l i c a t i o n s o f e.g. p a r t i a l o r d e r s ( p r o j e c t n e t w o r k s ) , h i e r a r c h i c a l p l a n n i n g techniques, proceeding from one l e v e l t o another, n a t u r a l l y i n v o l v e substitution.

(P6)

UNIQUE FACTORIZATION RESULTS

The s t r o n g e s t r e s u l t s f o r t h e s u b s t i t u t i o n decomposition (which i n most aspects a l s o e x t e n d t o t h e more g e n e r a l s p l i t decomposition, c f . 1.6 and 111.5) have been "uniqueness" r e s u l t s o f c e r t a i n f a c t o r i z a t i o n s , which i n c l u d e those o f t h e JordanH o l d e r t y p e , and i m p l y e.g. t h e independence o f m u l t i - s t e p d e c a n p o s i t i o n f r o m t h e o r d e r and s t a r t i n g p o i n t o f t h e s t e p s .

These r e s u l t s a r e n o t o n l y t h e h i g h l i g h t s

o f the t h e o r e t i c a l treatment b u t are e q u a l l y important f o r p r a c t i c a l applications.

A t y p i c a l s i t u a t i o n ( l a t e r t h e b a s i s f o r t h e c o m p o s i t i o n t r e e B(S)), i s : e i t h e r t h e r e i s a c o a r s e s t n o n - t r i v i a l congruence p a r t i t i o n , i . e . V(S)\{vl}

a g r e a t e s t element i n

meaning t h e r e i s a u n i q u e way o f silmultaneously c o n t r a c t i n g a l l maximal

n o n - t r i v i a l autonomous s e t s , o r t h e s t r u c t u r e S has q u o t i e n t s which a r e e x t r e m e l y s p e c i a l , i . e . "degenerate" o r " l i n e a r " i n t h e f o l l o w i n g sense:

D e f i n i t i o n : A s t r u c t u r e S on A i s c a l l e d degenerate, i f each non-empty subset o f i f A ( S ) = P(A)\{P)}, where P(A) denotes t h e power s e t o f A.

A i s S-autonomous, i . e .

S i s c a l l e d l i n e a r i f t h e r e e x i s t s a l i n e a r o r d e r < on A such t h a t A ( S ) i s t h e

s e t A(,C

1 for a l l T

.r< 1 f o r a l l T

TI is is

These d e f i n i t i o n s are s l i g h t l y d i f f e r e n t from the

usual ones f o r f i n i t e c l u t t e r s T, i n which one considers b[T]

:= b[TImin

and

t o be t h e b l o c k e r and a n t i b l o c k e r o f T, r e s p e c t i v e l y , c f . [47],

a[T] := a[T]""

[57].

[5q, [56],

E

E TI

We use t h i s sonewhat d i f f e r e n t n o t i o n here, as i t i s q u i t e

n a t u r a l i n t h e framework o f a r b i t r a r y s e t systems and a l s o avoids non-existence, which can occur f o r i.[T]

i n t h e i n f i n i t e case.

Indeed, f o r i n f i n i t e T , b[TImin

may n o t e x i s t , whereas a[TImaX always e x i s t s , since a[T]

i s the independence

system o f the c l i q u e s o f the canplementary graph G(T)' o f G(T), where G(T) has node s e t A and edges Obviously, b[T] = b[Fi3

if

dence system.

(a,B)

f o r a l l a , E~ A w i t h { a , B } c T f o r some T E T.

i s convex and normal, if T i s normal.

Fin6 T.

Also, T c b[b[T]],

I n particular,

Fin=

Furthermore, b[T]

=

where e q u a l i t y holds i f T i s a depen-

b[b[TImiTmin

f o r f i n i t e s e t systems, which

i s the well-known i d e n t i t y C = b [ b ( C ) ] f o r b l o c k e r s o f f i n i t e c l u t t e r s c f . [47], With regard t o the decomposition p o s s i b i l i t i e s o f b[T], we note: [El].

219

Substitution decomposition for discrete structures

L e t T b e a (normal) s e t system on A w i t h T = b[b[T]].

Theorem 1.3.7: V(b[T]),

i.e.

i n p a r t i c u l a r A(T) = A(b[T]).

t i o n r u l e s b[TlB]

= b[T]l

B for B

E

Furthermore, we have t h e t r a n s f o r i n a -

A(T) and ~ [ T / I T ] = b[T]/a

F o r f i n i t e c l u t t e r s T and b[TImin,

Proof:

Then V ( T ) =

Pl],

cf.

for

[Sl].

IT

e V(T).

The p r o o f methods can

be extended t o t h e case considered here, s i n c e t h e c r u c i a l p r o p e r t i e s t h a t t o each T e T ( U

E

b[T])

and a E T ( a E. U ) t h e r e e x i s t s U

= {a} remain v a l i d f o r normal s e t systems w i t h

5 = b[b[T]]

= a[PaXJ

i f T 6 Tmax.

Thus T = a[a[T]]

i f f T i s conformal, t o o .

Theorem 1.3.8:

L e t T be a s e t system on A.

C_

A(a[T]).

I

Even s t r o n g e r , a[T]

Then V ( T ) c V(a[T]),

E q u a l i t y h o l d s i f T i s conformal.

t h e t r a n s f o r m a t i o n r u l e s a[TIB] 77 E

*

Tn U

with

i s convex

is a

may be i n t e r p r e t e d as t h e system o f c l i q u e s o f a graph.

conformal s e t system, i.e.

u l a r A(T)

(T e T)

b[T]

t i s obvious t h a t a[T]

From t h e above remarks on t h e a n t i b l o c k e r a[T], and normal and t h a t a[T]

E

= a[T]

IB for B

E

i.e.

i n partic-

Furthermore, we have

A(T) and a[T/a]

= ~[T]/II

for

V(T), i f T i s conformal. F o r f i n i t e c l u t t e r s c f . [81].

Proof:

s i d e r e d here.

See a l s o t h e r e l a t i o n s h i p between T and G ( T ) i n 1 . 5 . a

Finally, w.r.t.

(P5), i t was shown t h a t " a l m o s t a l l " l a b e l e d

systems a r e p r i m e [99]. and c l u t t e r s .

The method c a r r i e s o v e r t o t h e case con-

independence

T h i s r e s u l t extends i m e d i a t e l y t o dependence systems

The p r o o f o f t h e theorem i s b y p u r e c o m b i n a t o r i a l arguments and

uses s t r o n g bounds on t h e number o f independence systems on an n-element base s e t [69], (i.e.

[85].

As t h e c l a s s o f independence systems t u r n s o u t t o be r i g i d r99]

almost a l l s t r u c t u r e s have a t r i v i a l automorphism g r o u p ) , t h i s a s y n p t o t i c

behaviour also c a r r i e s over t o t h e unlabeled structures are identified.

below) and m - c l u t t e r s ( I T 1 = m for a l l T cases a r e proved i n [99] t i o n s [Zl],

1.4

case, i . e .

also holds i f i s m o r p h i c

The same i s t r u e f o r conformal c l u t t e r s ( s e e graphs

via 0

-

E

T).

The r e s u l t s i n these s p e c i a l

1 laws induced b y f i r s t - o r d e r l o g i c c o n s i d e r a -

[51].

SUBSTITUTION DECOMPOSITION FOR RELATIONS

k F i n i t e and i n f i n i t e k - a r y r e l a t i o n s ( i . e . subsets o f A ) were t h e f i r s t s t r u c t u r e s f o r which t h e i n v e s t i g a t i o n o f t h e s u b s t i t u t i o n decomposition has l e d t o r e s u l t s o f t h e Jordan-Holder t y p e and t o t h e c h a r a c t e r i z a t i o n o f t h e a s s o c i a t e d congruence These i n c l u d e d t h e i n f i n i t e case, due t o t h e [122]. p a r t i t i o n l a t t i c e s [98],

280

R.H MGhring and EJ. Radermacher

f a c t that rather strong properties are v a l i d f o r relations. The discussion of aspects (Pl) - ( P 5 ) therefore i s much easier t h a n f o r s e t systems. The situation here i s rich f o r applications, since undirected (simple) graphs (which can be identified w i t h symnetric, irreflexive binary relations) and p a r t i a l orders ( i . e . reflexive, asymnetric and t r a n s i t i v e binary relations) are covered. The l a t t e r play, i n the f i n i t e case, a n important role in the description of the Below we technological structure underlying p r o j e c t networks [48], [77] , [80]. give a l i s t of structural aspects related t o the substitution decomposition, such as clique determination and perfectness of graphs and dimension or Moebius function computation in partial orders. Higher level problems o f that s o r t in COMBINATORIAL OPTIMIZATION over graphs and partial orders (such as detennination o f the shortest project duration i n networks), will be considered in Section 11. Another area of application involving k-ary instead of binary relations comes fran canputer science and concerns the decomposition of non-deterministic automata, cf. [149] , [150] , [151] . All these applications have - in a long historical developnent - led t o the same concept of substitution (X-join El321 or ordinal sum [74]) and t o the same autonomous s e t s (temed e.g. closed s e t s [58] , clumps [3] , [20] , externally related sets [ Z q , p a r t i t i v e s e t s [64] and stable s e t s [141]), g i v e n below. This concept o f s u b s t i t i t u t i o n r e s u l t s from replacing elements of a given relation by other relations, where elements from d i f f e r e n t relations are related t o each other i n the same way in which the replaced elements were. From the algebraic point of view, the resulting homanorphisms between k-ary relations R and R ' over base s e t s A and A ' are surjective homanorphisms h: A + A ' such t h a t , f o r a l l a l ,. .. ,ak E A with I { h ( a , ) , ...,h ( a k ) l l > 1 , ( a l ,...,ak) E R i f f ( h ( a , ) , ...,h ( a k ) ) e R' ( i . e . they are "almost" the strong relational homomorphisms, cf. [118], [llg]).

Definition: Let R' be a k-ary relation ( k a 2 ) on A' and l e t , f o r each B e A ' , R g be a k-ary relation on AB, where the s e t s A6 are non-empty and pairwise d i s j o i n t . Let A := U P u t A := I ( B , ...,6) E A t k B € A ' } and s e t

U

"

WA'

A!3'

I

A x...xA , Then R i s called the canposition (61, ..., B k ) RYL' B1 'k of R ' and the Re, 8 6 A ' , and i s denoted by R = R'[RB, B E A ' ] . R i s said t o be

R :=

BEA'

RB

obtained by substitution of the elements 5 e A ' by the relations R B in R ' . The canposition i s proper i f \ A ' \ > 1 and [ A B [ > 1 f o r some 6 t A'. A relation B i s said to be decomposable i f i t has a representation as a proper composition. Otherwise, i t i s said t o b e indecanposable or prime.

28 1

Substitution decomposition for discrete structures

a) A p a r t i t i o n 'TI = {Bi I i e I } o f A i s c a l l e d a congruence p a r t i t i o n o f R i f t h e r e e x i s t k-ary r e l a t i o n s Ri on Bi, Definition:

L e t R be a k-ary r e l a t i o n on A.

i E I , and a k - a r y r e l a t i o n R ' on A/.

such t h a t R = R ' [Ri ,Bi

R ' i s c a l l e d the q u o t i e n t o f R modulo

IT

E

A/T].

and i s denoted by R/'TI.

I n t h i s case,

V(R) denotes the

system o f congruence p a r t i t i o n s o f R. b ) A subset B o f A i s c a l l e d an R-autonanous s e t i f t h e r e i s 'TI t V(R) w i t h B E P I n t h i s case, t h e r e l a t i o n R I B := Rn B k i s c a l l e d the autonomous s u b - r e l a t i o n of R induced by B.

.

A(R) denotes the system- o f a l l R-autonanous sets.

As f o r s e t systems, congruence p a r t i t i o n s and autonanous sets o f r e l a t i o n s have nice internal characterizations.

Lemma 1.4.1: a)

L e t R be a r e l a t i o n on

A p a r t i t i o n IT = IBi 1 i Q 1) o f A i s a congruence p a r t i t i o n o f R i f f each ( a 1 ,.. ,ak) E R w i t h elements a . from a t l e a s t two d i f f e r e n t classes o f T

.

i m p l i e s t h a t (B1, b)

A.

...,B k )

J

R f o r a l l B~

E

A subset B o f A i s R-autonmous i f f

f o r scme i # j i m p l i e s

(al

(a1

=

,. .. , a k )

,...,aj-l,B,aj+l

i = 1, ...,k .

[ai]r, E

,...,ak)

R, E.

aj E

B, B e B and

ai E. ?SB

R.

S i m i l a r l y t o the case o f Boolean functions, the q u o t i e n t R/'TI i s n o t u n i q u e l y detk I n order t o o b t a i n

etmined b u t o n l y up t o r e f l e x i v e t u p l e s ( a , . . . ,a) e (A/'TI)

.

uniqueness, we w i l l t h e r e f o r e i d e n t i f y r e l a t i o n s which o n l y d i f f e r i n r e f l e x i v e t u p l e s (which e.g. f o r (undirected, simple) graphs and p a r t i a l orders i s no r e s triction at all). With regard t o t h e o t h e r c o n d i t i o n s i n (P2)

Theorem 1.4.2:

-

(P3), we o b t a i n :

Relations f u l f i l ( A l ) , (A2)* and (A3), ( S l ) , (S2)*,

( S 3 ) * , (S4)

-

(S7), ( b u t i n general n o t (A4)).

Proof

( A l ) i s obvious.

( c f . [98]):

autonanous sets w i t h

n

cI

n

Bi P 8.

icI

To show (A2)*,

L e t (al

,. .. ,ak) s

l e t (Bi)icI

be a f a m i l y o f R-

R and, w.1 .o.g.,

a e B := 1

Bi I a2 $ B, . In order t o show t h a t B 6 A(R), we must (because o f i€1 Lemna 1.4.1 j show t h a t (B,a2 ,... ,ak) e R f o r any B E B. Since a2 6 B, t h e r e e x i s t s io E I w i t h e 2 6 Bi But al, ~ E 5 B B . and B i E A(R). Hence (B,a23.*.,"k)

&

0

R.

To show t h a t C :=

u

1QI

.

'0

Bi E A ( R ) , assume again t h a t (a1

0

,...,a k ) E

R,

al E

Bi

1

,

282 a2

R.H. Mohringand EJ. Radermacher

# C , and fl

d

Bi

2

.

Let

obtain t h a t (Y,a2 , . . . , a k ) that

( 6 , a2 , . . . , a k ) E

Y E E

R.

n

i CI

Bi.

Since B i

Similarly, B i z €

1

6

A(R), a l , Y e B i

A(R), f l , y e B i

2'

1'

a2

a2 B B i

# Bi 2

1

we

yields

R.

(A3) i s shown s i m i l a r l y .

A(R). I f C .e A(R) and C c B, then obviously C E A(R1B). I n In the opposite d i r e c i t o n , l e t C e A ( R l B ) , ( a l ,..., a k ) 6 R , w.1.o.g. a1 E C, C , and 6 E C . If some a j + B ( j = 2 ,..., k ) , ( 6 , a 2 ,..-,a k ) e R because of a2 the R-autonomy of B. Otherwise, ( a l ,..., a k ) e R I B and we obtain ( 6 , a 2 , ...,a k ) e R f r m the RIB-autonomy of C. This shows t h a t C c A ( R ) . To show ( S l ) , l e t B

(S2)* follows immediately from the c h a r a c t e r i z a t i o n of congruence p a r t i t i o n s and autonomous sets in Lemma 1.4.1.

(S3)* follows from (S2)* and ( S l ) (54) i s e a s i l y v e r i f i e d with Lemma 1.4.1. F i n a l l y , (S5) follows immediately from (S2)* and (A2)*,

since [C].

=

U (BU C).w

Bell

W+b Going over t o (P4) there a r e two p r i n c i p l e s of invariance. For the complement R C := A k\R of a k-ary r e l a t i o n , we have A(R) = A(Rc) and V(R) = V(Rc) and the f o r any 6 € A(R), and RC/* = (R/T)' f o r any transformation r u l e s R C I B = IT Q

!AR).

The o t h e r p r i n c i p l e i s r e s t r i c t e d t o p a r t i a l orders and means t h e v a l i d i t y o f A ( R ~ ~= ) (A(R))sy, where RSy := RU R-l ( w i t h R-l : = {(y,x) I (x,y) 6 R } ) denotes t h e symnetric closure o f a r e l a t i o n (here: of a p a r t i a l order) and ASY the symmetr i c closure of the s e t system A(S) ( i n the sense of condition (A4)). The given i d e n t i t y , which c h a r a c t e r i z e s t r a n s i t i o n from p a r t i a l orders t o comparability graphs, i s not e a s i l y obtained and i s treated a s an i n t e r f a c e i n 1.5.

Finally, w.r.t. (P5), we mention r e s u l t s on t h e r e l a t i v e frequency of prime r e l a t i o n s . In [YY] i t i s shown t h a t i n each non-trivial c l a s s of parametric k-ary r e l a t i o n s [114], "almost a l l " members a r e prime. This includes as special cases t h a t "almost a l l " binary antisymnetric r e l a t i o n s , tournaments and p a r t i c u l a r l y , graphs, a r e prime. The r e s u l t follows form the f a c t t h a t i n these cases primeness i s a consequence of 0 - 1 laws f o r f i r s t order l o g i c p r o p e r t i e s f o r these s t r u c A l l these c l a s s e s a r e again rigid, so t h a t these t u r e s ; c f . [21], [51], [115]. r e s u l t s extend t o the unlabeled case, too. T h e s i t u a t i o n f o r t r a n s i t i v e relations,

283

Substitution decomposition for discrete structures

quasi-orderings and p a r t i a l orders (which are a l l n o t parametric) i s much harder t o deal w i t h .

Here r e s u l t s f o l l o w from pure combinatorial considerations [99],

using strong bounds on the number o f s t r u c t u r e s o f the r e s p e c t i v e types [49],

[86]. As i t i s n o t c l e a r whether these classes a r e r i g i d , the u n l a b e l e d case i s here n o t y e t s e t t l e d , b u t we conjecture t h a t here, too, almost a l l unlabeled s t r u c t u r e s are prime.

APPLICATIONS OF THE SUBSTITUTION DECOMPOSITION TO GRAPHS

-

Determination o f cliques, independent sets and o f the blocker [47]

o f these

sets, as w e l l as o f c l i q u e coverings and c o l o u r i n g s by means o f decomposition, c f . [28]

-

and Section 11.

Determination o f (maximal) matchings by means o f decomposition. Determination o f the automorphism group o f a graph v i a the associated automorphism groups f o r subgraphs and q u o t i e n t graph [72].

-

The classes o f e.g. p e r f e c t p8] graphs [64] Interval

w. r.t

-

,

[ZZ]

, superperfect,

t u r n o u t t o be closed w . r . t .

chordal and c o m p a r a b i l i t y

decomposition and composition.

graphs (as w e l l as proper i n t e r v a l graphs) [60]

are closed ( o n l y )

. decanposi t i o n .

For c o m p a r a b i l i t y graphs G(o) o f a p a r t i a l order o ( c f . Example 1.1.1 f o r t h e d e f i n i t i o n ) i t is known t h a t G ( o ) = G(0')[G(oi),

i

E

I],f o r B = o'[Oi,i

E

A'].

Furthermore, the uniquely p a r t i a l l y orderable graphs (UP0 gra hs [l],i.e.

rp } ) , are e s s e n t i a l l y

c a n p a r a b i l i t y graphs f o r which G(o*) = G ( o ) i f f o* E CO,O prime [141],

[158].

This has the i n t e r e s t i n g consequence t h a t "almost a l l "

Comparability graphs are prime

[loll.

A b a s i c observation i n t h i s c o n t e x t i s

t h a t i n v e r t i n g the o r i e n t a t i o n on some classes o f a congruence p a r t i t i o n o f G and/or on the associated q u o t i e n t leads again t o a ( a p a r t from t r i v i a l cases new) o r i e n t a t i o n o f G.

Another i n t e r e s t i n g consequence i n the case o f V ( G )

being f i n i t e i s t h a t G(o) = G(o') i m p l i e s t h a t o and O' have the same dimension [107],

[158],

a r e s u l t r e c e n t l y also shown i n t h e i n f i n i t e case

[Z], 0651.

APPLICATIONS OF THE SUBSTITLITION DECOMPOSITION TO PARTIAL ORDERS

-

Counting p a r t i a l orders, i t e r a t i v e l y

b u i l t up from c e r t a i n prime p a r t i a l orders

( i n p a r t i c u l a r s e r i e s - p a r a l l e l networks [129]),

-

c f . [lll].

For p r o j e c t networks, where the technological s t r u c t u r e i s i n t e r p r e t e d as a p a r t i a l order, i t i s known [77]

t h a t f o r the t o p o l o g i c a l s o r t i n g o f t h e a c t i v i -

t i e s , as w e l l as f o r the c o n s t r u c t i o n o f a c t i v i t y - o n - n o d e and a c t i v i t y - o n - a r c

R.H. Mohrmg and F.J. Radermacher

284

diagrams, the substitution decomposition may be used. Due t o the d i f f i c u l t i e s i n finding such diagrams and t o t h e i r frequent use in applications the l a s t case i s particularly interesting. The basis for the use of decomposition i s the easy identification of autonomous s e t s in such diagrams, a s was described i n connection with Example 1.1.1.

-

Concerning the so-called dimension (dim(oj) [46], [74] of partial orders, i t i s known that partial orders of dimension l e s s than some fixed cardinal number are closed w.r.t. to decomposition and composition. In particular, f o r partial orders with V ( G ) of f i n i t e length, dim(@) = maxIdim(o/n),dim(olLi), i = l , . . . , r J f o r any TI = { L 1 , ...,L r } E V ( O ) , i . e . dim(o) i s j u s t the maximum o f the dimension of a l l factors (canpare Section 111) of 8 [74], [107]. This implies t h a t (dim)-irreducible partial orders are prime. O f course, w i t h regard t o the reversibility of partial orders o 1461, which i s equivalent to dim(o) 6 2 [5], [46], t h i s implies that r e v e r s i b i l i t y i s also closed w.r.t. decanposition and composition. (1 i f a = B -c u(a,Y) if a , 2 ,

aG(X) : =

max 1 x ( v ) , where I(G) denotes t h e system o f ( + m a x i m a l ) independent UeI(G) veU

sets i n G and x i s again a w e i g h t i n g f u n c t i o n . Other i n t e r e s t i n g f e a t u r e s are ( i n the i n t e g e r case) t h e c l i q u e covering number

G ( x ) ( i . e . the s m a l l e s t nunber o f c l i q u e s t h a t cover each v c V a t l e a s t x ( v ) times) and the c h r a n a t i c number x G ( x ) ( i . e . t h e s m a l l e s t number o f independent

p

s e t s covering each v

t

V a t l e a s t x ( v ) times).

Note t h a t the d e f i n i t i o n s o f wG and Further

aG

=

wGC

aG may

can r e s t r i c t ourselves w.1 .o.g.

t o the c o n s i d e r a t i o n of t h e weighted c l i q u e number.

There i s a s i m i l a r connection between

x.

be extended t o i n c l u d e n e g a t i v e weights

holds, where Gc denotes t h e complementary graph o f G, i . e . we p

and x; here we w i l l r e s t r i c t ourselves t o

Note t h a t the weighted case may be traced back t o t h e case x

each node v E V by x ( v ) i d e n t i c a l copies.

=

1 by r e p l a c i n g

O f course, we have aG(x)

e.g. perfectness would y i e l d e q u a l i t y [64].

6

xG(x), where

Even then, t h e covering problem i s o f

s p e c i a l i n t e r e s t i n the sense t h a t an optimal c l i q u e does s t i l l n o t s t r a i g h t forwardly y i e l d a b e s t covering. I n t h e f o l l o w i n g , we w i l l deal w i t h s o l v i n g the problem o f determining wG, which i s (even f o r x : 1) i n general an NP-canplete problem [59],

by deccmpos

Subsequently, sane h i n t s on t h e covering problem a r e added. F a c t o r i z a t i o n Problem: w i t h fi : R!LiI-lRk

A partition

71

= {Ll,

...¶Lml, a

and a normal c l u t t e r C ' over {B1,

function f = ( f l

...,@,1 c o n s t i t u t e

s o l u t i o n t o the f a c t o r i z a t i o n problem f o r wi; w i t h G = ( V , E ) w

( x ) = max

1

y(6)

iff

i = 1,

w i t h ~ ( 6 := ~ )f i ( x l L i ) ,

...

TeC' 6eT

holds. Note t h a t t h i s approach i s q u i t e n a t u r a l f o r a step-by-step computation o f graph parameters v i a decomposition. Gi

The idea i s t o p a r t i t i o n G i n t o d i s j o i n t subgraphs

:= G / L i and t o a l l o w an a r b i t r a r y computation on G(Li,

f u n c t i o n x / L i there.

using t h e weighting

The r e s u l t i s a r e a l number, v i z . f i ( x I L i ) ,

serves as the weight o f an a r t i f i c i a l l y introduced element gi G/Li.

Concerning fi,

which then

t h a t w i l l replace

n o t h i n g i s r e q u i r e d o t h e r than t h a t i t be independent fran

29 1

Substitution decomposition for discrete structures the weights o u t s i d e Li.

This r e f l e c t s a step-by-step computation w i t h o u t the use

o f (and need f o r ! ) e x t e r n a l i n f o r m a t i o n (LOCALITY o f INFORMATION TRANSFER).

i s then assumed t h a t on the s e t V / T = { B ~ ..., , 6l,

It

o f these a r t i f i c i a l l y i n t r o -

duced elements a n o t too complicated f u n c t i o n ( a l l o w i n g e.g. the weighted c l i q u e number w . r . t .

any graph on t h i s s e t ) determines the o b j e c t i v e wG(x). This i s n e.g. i n order t o cover t h e x E Rb,

r e q u i r e d t o h o l d n o t j u s t f o r one, b u t f o r s t o c h a s t i c case (canpare Theoren 2.2.6).

all

I n f a c t , a l l these c o n d i t i o n s mean a

r e p r e s e n t a t i o n o f wG i n a s o r t o f f a c t o r i z a t i o n , d e f i n e d on s m a l l e r domains, f o r one other.

substituting certain functions,

Now, as the f o l l o w i n g theorem shows,

the s o l u t i o n s t o t h i s problen are very s p e c i a l and c l o s e l y r e l a t e d t o the s u b s t i t u t i o n decunposi t i o n .

Theorem 2.1.1:

f = (f,,

71,

PROBLEM f o r 6, i f f

...,fm)and

C ' g i v e a solu-tion t o t h e FACTORIZATION

i s a congruence p a r t i t i o n o f G, fi = w

TI

C(G') w i t h G ' = G/T.

Proof :

1.

" ”.

I n f a c t , i t extends t o t h e more g e n e r a l s i t u a t i o n o f m i n i m i z i n g any ( w e l l behaved) r e g u l a r c o s t f u n c t i o n ( r e g u l a r measure o f performance [93] ) K : R: which i s a m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n o f t h e c a n p l e t i o n t i m e s tl the a c t i v i t i e s a,

,. . . ,a,

and g i v e s t h e performance c o s t

w i t h t h e s e c a n p l e t i o n times. K(@;x) := ~(Es,[x](a~) scheduling

o

+ x(al)

K(

tl ,.

. . ,tn)

+

W,,1

,. .. ,tn o f

associated

It i s e a s i l y obtained t h a t

,...,ESo[x](an)

f o r a c t i v i t y durations x

E

+ x ( a n ) ) gives the lowest c o s t f o r

Wr. Note t h a t p r o j e c t d u r a t i o n i s

i n c l u d e d h e r e as a s p e c i a l case b y p u t t i n g

= max.

K

However, f o r t h i s s p e c i a l

o b j e c t i v e , a f u r t h e r , d u a l r e p r e s e n t a t i o n can be g i v e n , v i z . A0(x) :=

max KcC( 0)

c x ( a ) , where C(O) denotes t h e system ofG-maximal c h a i n s i n o ( r e g a r d e d as s e t s a€ K

r a t h e r than l i n e a r orders). Thus t h e s h o r t e s t p r o j e c t d u r a t i o n equals t h e l e n g t h o f a l o n g e s t c h a i n , t h e soc a l l e d c r i t i c a l path length.

G i v e n t h e f a c t t h a t t h e s m a x i r n a l c h a i n s i n EI a r e

e x a c t l y t h e S-maximal c l i q u e s i n t h e a s s o c i a t e d c a n p a r a b i l i t y g r a p h G(O), i . e . C ( o ) = C(G(o)), we may e q u a l l y w e l l i n t e r p r e t A,(x)

t o t h e s i t u a t i o n i n 11.1.

Theorems 2.1.1 and 2.1.2

as w

( x ) , i . e . we a r e back G(0) t h u s extend t o t h e problem o f

f a c t o r i z a t i o n o f t h e s h o r t e s t p r o j e c t d u r a t i o n f u n c t i o n o f a network and t e l l us t h a t , g i v e n t h e l o c a l i t y c o n d i t i o n as d e s c r i b e d , t h e congruence p a r t i t i o n s o f t h e a s s o c i a t e d c o m p a r a b i l i t y graph d e s c r i b e

fl e x i s t i n g

i n t e r f a c e r e s u l t s i n 1.5 (Theorem 1.5.1),

decanpositions.

Given the

these a r e e s s e n t i a l l y (up t o c e r t a i n

symmetric d i f f e r e n c e s ) t h e congruence p a r t i t i o n s o f t h e u n d e r l y i n g p a r t i a l o r d e r . I t i s , however, p o s s i b l e t o g i v e a s t r o n g e r f o r m u l a t i o n , which l e a d s e x a c t l y t o

t h e congruence p a r t i t i o n s o f t h e p a r t i a l o r d e r O; compare [77],

0211.

This i s i n

f a c t an immediate consequence o f Theorem 2.1.1.

C o r o l l a r y 2.2.1:

L e t 0 = ( A , O ) be a poset,

IT

= IL1,

.... Lr)

be a p a r t i t i o n o f A

296

R. H. Mohring and F. J. Radermacher

and A ' := A/T := { ~ ~ , . . . , 6 ~ ] . Then: 1 ' 1,

f = (fl

,...,f m )

and 0' = ( A ' , O ' )

I 0'1,

(cx&) e 0 i m p l i e s ( B . , B . ) J

h g ( x ) = n,,(y)

iff

TI

1

[where o ' i s such t h a t a e Li,

i = 1 ,..., r, f o r a l l x ER'

w i t h ~ ( € 3 : ~= )f i ( x I L i ) ,

i s a congruence p a r t i t i o n f o r 0, fi =

i # j,

a ' e L., 3

y i e l d s the e q u a l i t y

>/

(xlLi). i

i = 1,

...,r

and o ' = o / n .

Corol1ar.y 2.2.1 s t i m u l a t e d study o f the s u b s t i t u t i o n decomposition f o r p a r t i a l orders ( p r o j e c t networks), c f . a l s o [42], e.g.

[145],

a l l t h e more so as i t extends t o

s h o r t e s t path canputation, d e t e r m i n a t i o n o f t h e weighted independence number

(which d e f i n e s f e a s i b i l i t y i n scheduling problems p r o j e c t networks and many other s u b j e c t s [793, i n 11.3.

[loo],

[127];

[125]),

f l o w problems i n

see a l s o the g e n e r a l i z a t i o n

For p r o j e c t networks C o r o l l a r y 2.2.1 means t h a t a t l e a s t one b a s i c net-

work parameter, v i z . s h o r t e s t p r o j e c t d u r a t i o n , can be handled v i a s u b s t i t u t i o n decanposition.

Fortunately,

i t turned o u t t h a t t h i s i s e q u a l l y t r u e f o r most

other aspects o f p r o j e c t networks ( w i t h t h e exception o f many problens i n v o l v i n g resource c o n s t r a i n t s ) .

For instance, we have already p o i n t e d o u t t h a t a c t i v i t y -

on-node and a c t i v i t y - o n - a r c diagrams as w e l l as t h e d e t e r m i n a t i o n o f a t o p o l o g i c a l sorting, can be obtained t h i s way.

The same i s t r u e f o r t h e c h a r a c t e r i s t i c a c t i v i t y

times (such as e a r l i e s t s t a r t and l a t e s t f i n i s h ) and the r e s u l t i n g f l o a t s ( o r s l a c k s ) (such as t o t a l f l o a t (TF), f r e e f l o a t (FF), backward and independent f l o a t and path f l o a t ) , canpare 0 2 4 1 . congruence p a r t i t i o n n = {L, ESoILi[x/Li](a)

f o r a e Li,

For exanple, i t t u r n s o u t t h a t , given a

,..., L r } o f o i = 1, ...,r .

and y = f ( x ) , ESo[x](a)

= ESolGY](~i)

O f course, t h i s y i e l d s the p o s s i b i l i t y

o f determining l e a s t p r o j e c t costs K ( O ; X ) v i a s u b s t i t u t i o n decomposition. very analogous behaviour, v i z . TFo[x] ( a ) = TF,CS;] total float.

t

( Bi) t TF,l

Li

[XI

We see

Li] (a), f o r t h e

This has the consequence t h a t t h e c l u t t e r o f c r i t i c a l paths over

the c r i t i c a l elements i s j u s t the s u b s t i t u t i o n o f t h e corresponding c l u t t e r s belonging t o the r e s t r i c t i o n o f n t o t h e s e t o f c r i t i c a l elements. f o r the o t h e r f l o a t s i s more canplicated. FF&]

The s i t u a t i o n

For instance,

FFo,[y](~i)+FFolLi[x(Li](a)

a maximal i n olLi

FFolLi C x I L i l ( a )

otherwise.

(a) =

This l a s t r e s u l t shows t h a t the f r e e f l o a t o f an a c t i v i t y a e A i s already d e t e r mined by any (s-minimal) autonanous s e t , c o n t a i n i n g a as a non-maximal element. A l t o g e t h e r , these observations show t h a t the c l a s s i c a l time a n a l y s i s o f p r o j e c t networks as a whole allows t h e employment o f the s u b s t i t u t i o n decanposition. f a c t , t h i s extends ( c f . [62])

even t o t h e VPM case [131]

In

and i t s g e n e r a l i z a t i o n s

291

Substitution decomposition far discrete structures

where a r b i t r a r y time c o n s t r a i n t s between s t a r t i n g time and c a n p l e t i o n t i m e o f any two a c t i v i t i e s a r e allowed.

These g e n e r a l i z a t i o n s imply a s t r a i g h t f o r w a r d t r a n s -

i t i o n t o r e l a t i o n a l systems, each r e l a t i o n d e s c r i b i n g another type o f time cons t r a i n t s i n the MPM-network.

TIME-COST TRADE OFF I N PROJECT NETWORKS The s i t u a t i o n considered here i s s t i l l d e t e r m i n i s t i c , i.e. a c t i v i t y d u r a t i o n s are n o t random.

However, durations may be v a r i e d t o some e x t e n t by f i n a n c i a l i n p u t s ,

where s h o r t e r d u r a t i o n s r e q u i r e a higher i n p u t .

A general model f o r such a s i t u -

a t i o n i s g i v e n by p r o j e c t networks w i t h costs systems (o,K) where

:= (A.O,(ka)aeA),

ka, f o r any a E A , i s a R ' , where IaC Wl g i v e s t h e p o s s i b l e

= (A,O) describes t h e p r o j e c t s t r u c t u r e , w h i l e

EI

m o n o t o n i c a l l y decreasing f u n c t i o n k a :

Ia

-*

d u r a t i o n s f o r a , a n d k , ( x ( a ) ) f o r x ( a ) E I(a) denotes the associated c o s t ( r e q u i r e d financial input).

Given (A,O,(ka)aeA),

the main i n t e r e s t concerns the f o l l o w i n g

two problems: Determination o f t h e minimal c o s t f u n c t i o n H ( t ) : Given a f i x e d time l i m i t t

1.

f o r t h e p r o j e c t d u r a t i o n , what i s the l e a s t c o s t H ( t ) f o r achieving t h i s task? Determination o f the minimal time f u n c t i o n H*(k): Given a f i x e d budget k,

2.

what i s the s h o r t e s t p r o j e c t d u r a t i o n H*(k) obtainable w i t h t h i s budget? From now on we w i l l concentrate on H, although H* behaves s i m i l a r l y .

under s u f f i c i e n t l y strong assumptions (e.g.

a l l ka s t r i c t l y monotonically decreas-

ing and convex on closed i n t e r v a l s Ia, canpare [12]), of H.

X Ia and n,(x)

6

H* i s the i n v e r s e f u n c t i o n

More d e t a i l e d , H i s d e f i n e d on t h e s e t o f possible s h o r t e s t p r o j e c t

d u r a t i o n s J := { A ~ ( X )

x

I n fact,

6

x E X I a l and i s g i v e n by H ( o , K ) ( t ) := i n f I c k a ( x a ) 1 @A CLeA t), t Q J. Note t h a t the infimum may n o t be a t t a i n e d and

I

aeA

t h a t canputation o f H ( t ) may be a d i f f i c u l t problem.

I n f a c t , f o r piecewise

a mixed l i n e a r o p t i m i z a t i o n l i n e a r cost f u n c t i o n s ka on closed i n t e r v a l s Ia, problem w i t h b i n a r y i n t e g e r c o n s t r a i n t s has already t o be solved.

However, i f t h e

k a are a l s o convex, a l i n e a r d e s c r i p t i o n i s p o s s i b l e t h a t does n o t r e q u i r e i n t e g e r

variables, f a c i l i t a t i n g e f f i c i e n t canputation.

As, furthermore, the l a r g e class

of a l l convex c o s t f u n c t i o n s can be (smoothly) approximated t h i s way, t h e r e are good reasons f o r concentrating on t h i s class o f piecewise l i n e a r , convex cost functions.

An a d d i t i o n a l argument i n t h i s d i r e c t i o n f o l l o w s below (Theorem 2.2.3).

But f i r s t , we w i l l t r e a t the decanposition problem f o r t h i s case.

I n f a c t , one

obtains the f o l l o w i n g theorem as a g e n e r a l i z a t i o n o f Theorem 2.1.1

and C o r o l l a r y

2.2.1;

cf.

[lz]:

R.H Mohringand F.J. Radermacher

298 Theorem 2 . 2 . 2 :

Given ( o , K ) = (A,O,(ka)acA), a p a r t i t i o n

TI

p a r t i a l order 0' on A ' = C13~,...,6~1, and c o s t f u n c t i o n s k have H ( 0 , K ) = H(Oi , K ' ) i f f

TI

o f A, a

= { L l,...,Lr} = fi((ka)aeLi)

1 3 i

we

i s a congruence p a r t i t i o n o f G(o) ( o r even o f EI i n

the stronger v e r s i o n ) , G ( o ' ) = G ( o ) / i r ( o r even 8 ' = @/IT)and ksi = H(o,K),Li. So again, o n l y s u b s t i t u t i o n decanposition a l l o w s i t e r a t e d d e t e r m i n a t i o n o f the

minimal c o s t f u n c t i o n under t h e above assumptions, and t h i s i s done by canputing f i r s t the minimal c o s t f u n c t i o n s o f t h e associated autonanous suborders and then u s i w these f i i n c t i o n s as c o s t f u n c t i o n s f o r t h e image elements.

Following t h i s

approach, i t i s o f course i m p o r t a n t f o r p r a c t i c a l a p p l i c a t i o n s t h a t t h e considered class o f cost f u n c t i o n s i s closed w . r . t .

time-cost t r a d e o f f , i.e.

f o r each n e t -

work w i t h a c t i v i t y cost f u n c t i o n s f r a n t h e g i v e n class, t h e minimal c o s t f u n c t i o n shculd a l s o belong t o t h e c l a s s .

For such classes o f c o s t f u n c t i o n s , decanpo-

s i t i o n w i l l then n o t l e a d t o more " c a n p l i c a t e d " c o s t f u n c t i o n s .

Fortunately,

the piecewise l i n e a r , convex c o s t f u n c t i o n s behave t h i s way; i n f a c t , an even stronger i n s i g h t i s p o s s i b l e [lZ]:

Theorem 2.2.3:

The c l a s s o f piecewise l i n e a r and convex c o s t f u n c t i o n s on

closed i n t e r v a l s i s t h e

least closed

c l a s s o f c o s t f u n c t ons c o n t a i n i n g the l i n e a r

c o s t f u n c t i o n s on closed i n t e r v a l s . Theorem 2.2.3

t e l l s us t h a t l i n e a r c o s t f u n c t i o n s do n o t form a closed class.

However, i f we r e s t r i c t ourselves t o

[.,-[- l i n e a r

k a ( x ( a ) ) := -a.x(a) + b, a,b I 0,

= [c,-[,

Ia

c o s t functions,

i.e. functions

c % 0, then a t l e a s t w i t h regard

t o the two two-element prime p a r t i a l orders, closedness i s obtained. t h i s then extends t o a l l s e r i e s - p a r a l l e l networks; i n f a c t we have

Theorem 2.2.4: systems K o f

Given a f i n i t e poset

[. ,-[- l i n e a r

or

H

(0.K)

cost f u n c t i o n s over

O f course,

PZ]:

[.,-[- l i n e a r f o r a l l c o s t o iff o i s series-parallel. is

The p r o o f o f t h i s theoren i s e s s e n t i a l l y based on t h e r e s u l t t h a t a p a r t i a l order i s s e r i e s - p a r a l l e l i f f i t does n o t c o n t a i n a suborder i s a n o r p h i c t o (A,O) w i t h A = (1,2,3,419 S o = { ( 1 9 3 ) 3 ( 1 , 4 1 3 ( z 9 4 ) } ; cf. a l s o [12], [80], D60]. Theorem2.2.4 r e f l e c t s s p e c i a l f e a t u r e s o f s e r i e s - p a r a l l e l networks, due t o t h e simple n a t u r e

o f t h e prime s t r u c t u r e s frcm which they are i t e r a t i v e l y b u i l t up.

This special

nature a l s o plays a r o l e i n another c o n t e x t which d e a l s w i t h one o f t h e few known a p p l i c a t i o n s o f s u b s t i t u t i o n decanposi t i o n t o scheduling theory.

Substitution decomposition for discrete structures

299

ONE MACHINE SCHEDULING Consider t h e NP-complete problem o f s c h e d u l i n g n j o b s w i t h a r b i t r a r y p r o c e s s i n g t i m e s on one machine s u b j e c t t o a r b i t r a r y precedence c o n s t r a i n t s among t h e j o b s such as t o MINIMIZE TOTAL WEIGHTED COMPLETION TIME [93].

I n [92],

p44]

i t was

shown t h a t t h i s problem can be t r e a t e d v i a s u b s t i t u t i o n d e c m p o s i t i o n , d u e t o t h e f a c t t h a t any o p t i m a l sequence f o r t h e j o b s o f an autonomous s e t can be extended t o an o p t i m a l sequence f o r a l l j o b s .

Theorem 2.2.5:

T h i s has t h e f o l l o w i n g consequence:

m2

There i s an O(n ) a l g o r i t h m f o r t h e m i n i m i z a t i o n o f t h e t o t a l

weighted c o m p l e t i o n t i m e i n a one machine s c h e d u l i n g problem w i t h a r b i t r a r y p r o c e s s i n g t i m e s and precedence c o n s t r a i n t s , p r o v i d e d t h a t t h e precedence r e l a t i o n i s i t e r a t i v e l y o b t a i n e d v i a s u b s t i t u t i o n d e c a n p o s i t i o n from prime p a r t i a l o r d e r s

EN

w i t h a t m o s t m elements, m

fixed.

The a l g o r i t h m f i r s t decides whether t h e precedence r e l a t i o n i s o f t h e d e s c r i b e d t y p e . T h i s can be done i n O(n3) t i m e w i t h t h e methods presented i n S e c t i o n I V . Proceeding by i n d u c t i o n , we may t h e n assume t h a t t h e precedence r e l a t i o n has a congruence p a r t i t i o n w i t h a t most m b l o c k s , and t h a t each b l o c k i s a l r e a d y o p t i m a l l y o r d e r e d i n a l i n e a r o r d e r . We t h e n c o n s i d e r a l l subproblems induced by f i x e d p o s i t i o n s f o r t h e l a s t j o b i n each b l o c k and f i x e d numbers o f j o b s f r o m each b l o c k t o precede these l a s t j o b s .

F o r each such subproblem, t h e o p t i m a l sequence Since

can be determined by methods f o r p a r a l l e l c h a i n s i n O(n l o g n ) t i m e [92].

The d e t a i l s

t h e r e a r e nm(m-1)-2 such subproblems, we o b t a i n t h e above c o m p l e x i t y . o f t h i s a l g o r i t h m w i l l be p u b l i s h e d s e p a r a t e l y .

STOCHASTIC PROJECT NETWORKS There a r e good reasons f o r t r e a t i n g t h e s t o c h a s t i c v e r s i o n s o f t h e u s u a l q u e s t i o n s c o n c e r n i n g p r o j e c t networks.

We w i l l r e s t r i c t o u r s e l v e s h e r e t o t h e case o f

f i x e d p r o j e c t s t r u c t u r e s and random a c t i v i t y d u r a t i o n s . more g e n e r a l GERT-Networks [113], p o s i t i o n [113]

[120]

,

[164],

We h i n t however a t t h e

f o r which s e r i e s - p a r a l l e l decom-

as w e l l as g e n e r a l s u b s t i t u t i o n decomposition [96]

A s t o c h a s t i c p r o j e c t network i s g i v e n by (A,O,P),

where

i s possible.

P i s the j o i n t d i s t r i b u t i m

o f a c t i v i t y d u r a t i o n s , i . e . a p r o b a b i l i t y measure on ( R , ! A l , B F l ) ; where B,!Al I * \ . L e t t h e r e a l random v a r i a b l e X . denotes t h e system o f Bore1 s e t s o f R>, J

describe the duration o f

C X . and

J

a . e A , e x i s t . For any B 6- A, 3 w i t h t h e X j . a . r B. We w i l l J

assume t h a t t h e expected d u r a t i o n s E(X . ) , J

l e t PB denote t h e m a r g i n a l d i s t r i b u t i o n a s s o c i a t e d

-

except f o r sane h i n t s on bounds

g i v e n below -

R. H. Mohrimg and EJ. Radermacher

300

concentrate on t h e independence case, i . e . assume t h a t P

x Pa.

=

A main o b j e c t

a A

o f i n t e r e s t is then the d i s t r i b u t i o n o f t h e s h o r t e s t p r o j e c t d u r a t i o n P more generally, f o r measurable and l i n e a r l y bounded o f the l e a s t p r o j e c t cost). distribution function F

hO

P K(O,.)

o r s e c u r i t y q u d n t i l e s ) the needed i n f o r m a t i o n concerning This i s even more t r u e as, i n general,

,...,

E(Xn)) < E [{J,i . e . d e t e r m i n i s t i c p l a n n i n g techiiiques - as does P - s y s t e m a t i c a l l y underestimate t h e expected p r o j e c t d u r a t i o n Q7], [97], cl2].

no(E(X1)

-

the d i 3 t r i b u t i o n

(or,

These d i s t r i b u t i o n s g i v e (e.g. v i a the associated

planning, scheduling and d e c i s i o n making. PERT

K,

AO

Unfortunately, t h e treatment o f t h e s t o c h a s t i c case t u r n s o u t t o be d i f f i c u l t because o f inissiny data and r a t h e r i n v o l v e d computational requirements [75], [135].

So, a p a r t from q u i t e s p e c i a l cases [77]

,

[146]

and from bounds f o r t h e

d i s t r i b u t i o n f u n c t i o n s such as t h e ones discussed below, m a i n l y s i m u l a t i o n cdn p r a c t i c a l l y be used t o c a l c u l a t e F

A0

, i n general.

O f course, i n view of these

d i f f i c u l t i e s , t h e q u e s t i o n o f decomposition becomes even more urgent.

Again we

have the f o l l o w i n g theorem c h a r a c t e r i z i n g s u b s t i t u t i o n decomposition. Given a network 8 = ( A , O ) ,

Theorem 2.2.6: ~1

=

t L ,,..., L r i o f A, a poset

0'

0 Pa, a p a r t i t i o n Ci€A and d i s t r i b u t i o n s

a distribution P =

on A ' = tB1

,...,By),

), we have P = P'!,@,' i f f TI i s a congruence p a r t i t i o n o f G(0) ( o r Li even o f c i n the s t r o n g e r v e r s i o n ) , G(0') = G ( a ) / n ( o r even 0' = O / T ) and r P ' = .@ P w i t h P; . = (PILi), . I = I 'i 1 01 Li Pai

= fi(P

L,

Theorem 2 . 2 . 6 shows t h a t again e x a c t l y the s u b s t i t u t i o n decomposition a l l o w s determination o f the s h o r t e s t p r o j e c t d u r a t i o n as r e q u i r e d , by f i r s t doing t h i s f o r a1 1 associated suborders and subsequently for the q u o t i e n t , where marginal d i s t r i b u t i o n s are those obtained p r e v i o u s l y on the r e s p e c t i v e classes. there are more general versions [123]

Note t h a t

n o t r e q u i r i n g s t o c h a s t i c a l independence.

However, these do n o t h e l p much i n p r a c t i c a l a p p l i c a t i o n s .

Theorem 2.2.6 may

h e l p considerably i n t h e exact computation of l a r g e r examples (e.g. t e s t examples

[78],

[l46]),

and can a l s o be used when s i m u l a t i o n i s employed.

This i s s i m i l a r l y

t r u e i f bounds f o r t h e d i s t r i b u t i o n f u n c t i o n FA o f t h e p r o j e c t d u r a t i o n ( o r O

analogously o f c e r t a i n p r o j e c t costs) are t o be determined. bounds a r i s e s

-

besides from computational aspects

frcm p o s s i b l e s t o c h a s t i c dependences.

-

I n t e r e s t i n such

from m i s s i n g d a t a as w e l l a s

For b o t h cases, s u i t a b l e bounds are a v a i l -

able, which, furthermore, t u r n out t o be compatible w i t h s u b s t i t u t i o n .

We w i l l

g i v e h i n t s t o b o t h approaches and s t a r t w i t h t h e case t h a t a c t i v i t y d u r a t i o n s are independent and a l l variances V(Xa),

a E

A, e x i s t .

The aim then i s , among other

t h i n g s , t o g e t around the d a t a problem, which i s o f t e n c r u c i a l i n a p p l i c a t i o n s , by p r o v i d i n g bounds depending on the p a i r s (E(Xa),U(Xa))aeA o n l y .

To t h i s end

Substitution decomposition for discrete structures

t h e f a c t i s used t h a t t h e random v a r i a b l e s YK :=

X(a),

30 I

g i v i n g t h e random

a K

l e n g t h o f t h e maximal chains K

IZ C ( O ) , a r e a s s o c i a t e d random v a r i a b l e s [7] ,[SO]. For l a r g e p r o j e c t networks, any such v a r i a b l e Y K may ( a p p r o x i m a t e l y ) be assumed t o

t o be n o r m a l l y d i s t r i b u t e d w i t h meanE(YK) =

1

E ( X a ) and v a r i a n c e V(YK)

=

aeK

1 V(Xa) - i n d e p e n d e n t l y o f t h e r e s p e c t i v e a c t i v i t y d u r a t i o n d i s t r i b u t i o n s a€ K [ 1 4 g on a s s o c i a because of t h e C e n t r a l L i m i t Theorem. Now a b a s i c r e s u l t [50], r~

t e d random v a r i a b l e s g i v e s FA ( t ) 5

F K ( t ) , FK b e i n g t h e d i s t r i b u t i o n f u n c -

KEC(0)

0

t i o n o f YK.

I t i s worth noting t h a t i n a l l a v a i l a b l e p r a c t i c a l applications t h i s stochastic upper bound (which can t h e o r e t i c a l l y be a r b i t r a r i l y bad) proved t o be q u i t e near t o t h e d i s t r i b u t i o n f u n c t i o n FA , c f . [7:78), [146]. I t i s t h e r e f o r e , when combined 0 w i t h a s t r a i g h t f o r w a r d l y o b t a i n e d s t o c h a s t i c lower bound, a u s e f u l i n s t r u m e n t f o r h a n d l i n g a p p l i c a t i o n s i n v o l v i n g s t o c h a s t i c p r o j e c t networks. These bounds can be f u r t h e r improved, i f s u b s t i t u t i o n decomposition can be used

[i'],

i . e . i f t h e bounds o b t a i n e d f o r t h e autonomous suborders induced b y t h e

c l a s s e s o f a congruence p a r t i t i o n

V ( o ) a r e used as a c t i v i t y d u r a t i o n d i s t r i b -

II E

u t i o n s f o r t h e elements o f t h e q u o t i e n t

O/T.

F i n a l l y , we d e a l w i t h t h e q u i t e n a s t y case o f s t o c h a s t i c dependences between a c t i v i t y d u r a t i o n s . To t h i s end, l e t m a r g i n a l d i s t r i b u t i o n s Pa., a c A be g i v e n J J and l e t Q denote t h e s e t o f a l l p r o b a b i l i t y measures Q over (RR,Bn) w i t h m a r g i n a l d i s t r i b u t i o n s Q,.

= Paj,

aj E A = 11,. ..,n}.

For any t clR1 and x = (xl

J EIRn

,... ,xn)

put n

$ ( t ) :=

i n 1 {(A,(X) X€R

I t can be shown [95]

-

t)+

+

f (€(Xi) i=l

-

xi)+},

where a+ := max {O,ai.

E(Z - t ) + Z i s a convex t o a l l p r o j e c t d u r a t i o n d i s t r i b u t i o n s PA ,

t h a t t h e r e i s a r e a l random v a r i a b l e

Z

such t h a t

=

+ ( t ) f o r a l l t eIR1 and, f u r t h e r m o r e , t h a t t h e d i s t r i b u t i o n o f upper bound ( i n t h e sense o f [148])

P e p.

T h i s bound i s even t i g h t i n t h e sense t h a t t o any t

such t h a t Ept(AO

-

t)+ =

E(Z - t)'.

R1 t h e r e i s P t

0

E

2

For s e r i e s - p a r a l l e l networks (but p o s s i b l y n o t

f o r a r b i t r a r y a) t h e r e i s even sane P* j u s t P;

E

E

Q such t h a t t h e d i s t r i b u t i o n o f Z i s

. 0

Using t h i s approach, i t i s f o r example p o s s i b l e t o g i v e an upper bound f o r t h e expected p r o j e c t d u r a t i o n i n t h e s i t u a t i o n d e s c r i b e d , r e g a r d l e s s o f t h e p o s s i b l e s t o c h a s t i c dependences [84],

[162].

A l s o t h e d e t e n n i n a t i o n o f t h e f u n c t i o n ji

seems t o be e f f i c i e n t l y p o s s i b l e and shows a c l o s e s i m i l a r i t y t o t h e t r e a t m e n t of

R. H. Mohriilg arid KJ. Radermachcr

302

time-cost trade o f f i n the case o f piecewise l i n e a r convex c o s t f u n c t i o n s [log].

So a l t o g e t h e r , t h e use o f t h i s method f o r p r a c t i c a l a p p l i c a t i o n s looks q u i t e promising.

A f u r t h e r n i c e f e a t u r e i s g i v e n by t h e o b s e r v a t i o n [162]

that the

employment o f the s u b s t i t u t i o n decomposition f o r a g i v e n congruence p a r t i t i o n ~f

E V ( a ) leads t o an i t e r a t i v e method t o compute $ by f i r s t l y determining the

convex upper bounds

aB

elB, and afterwards

f o r the associated autonomous suborders

using t h e d i s t r i b u t i o n s of t h e associated random v a r i a b l e s ZB as marginal d i s t r i b u t i o n s on the q u o t i e n t e/n.

Note t h a t t h i s i s s u r p r i s i n g as t h e i n v o l v e d d i s t r i b -

u t i o n s o f the randan v a r i a b l e s

ZB may never occur as a p r o j e c t d u r a t i o n

distribution.

11.3

COMBINATORIAL O P T I M I Z A T I O N OVER (IN-)DEPENDENCE SYSTEMS AND CLUTTERS

A c m o n type o f o p t i m i z a t i o n problem over independence systems I S i s g i v e n by

ma% 1 x ( a ) , x being a weighting f u n c t i o n on the base s e t A. UaIS a d systems DS i t i s g i v e n by

1

min x(a). UEDS aeU

However, i n b o t t l e n e c k problems,

replaced by e i t h e r min o r max, w h i l e i n r e l i a b i l i t y theory, f o r "+" plays a r o l e .

Over dependence

'I.

"

"+"

is

as replacement

Obviously, e.g. f o r non-negative weights, a l l these prob-

lems may e q u i v a l e n t l y be considered on the c l u t t e r o f e i t h e r c_-maximal independent o r s - m i n i m a l dependent sets, where w.1.o.g.

n o r m a l i t y may a l s o be assumed ( i . e .

any a c A can be assumed t o belong t o sane s e t i n t h e c l u t t e r ) .

So by v a r y i n g

the e x t e r n a l composition max/min, t h e i n t e r n a l canposi t i o n +/max/min/- or the c l u t t e r (e.g. maximal c l i q u e s o r independent sets o f ( c o m p a r a b i l i t y ) graphs, c-minimal cuts i n networks, and o t h e r s ) , o p t i m i z a t i o n problems as discussed i n 11.1 and 11.2, as w e l l as s h o r t e s t p a t h problems [go],

minimal o r maximal f l o w

canputation ( v i a t h e theorem o f Ford and Fulkerson [52],

[go]),

and problems i n

re1 iabi 1it y t h e o r y are i n c l u d e d . These STANDARD CASES, as w e l l as o t h e r s [25],

[166]

are covered by considering

a r b i t r a r y normal c l u t t e r s C on s e t s A and p a i r s (m, @ ) o f e x t e r n a l (m) and i n t e r n a l ( @ ) canpositions on ( s u i t a b l e ) sets o f r e a l numbers, where m = max o r min, and

0

i s assumed t o be a s s o c i a t i v e , comnutative and m o n o t o n i c a l l y

increasing w . r . t .

t o both canponents ( i . e . x1 ,x2, y1,y2 E

i m p l i e s x1 @ x 2 6 y1

r C,m, @I ( x )

:=

m

0 y2).

W 1 , x1

6 yl,

x2

B

y2

O f i n t e r e s t i s then the o p t i m a l value f u n c t i o n

@ x ( a ) , where x belongs t o a s e t o f "admissible" r e a l

TEC ~ E T

weighting f u n c t i o n s over A.

The r e s u l t i n g problem o f decomposition discussed i s :

303

Substitution decomposition for discrete structures

A partition

F a c t o r i z a t i o n Problem:

71

= { L ly...,Lm},

c l u t t e r s Ci o v e r Li and a

normal c l u t t e r C ' o v e r A/n a r e c a l l e d a s o l u t i o n t o t h e f a c t o r i i a t i o n problem f o r

r

C,myQ

iff

rc,m,O(x)

where Y ( B ~ =)

= rcl,m,a(~),

r Ci,m,O(~ILi),

i = 1,

...,m .

Note t h a t , due t o t h e d i f f e r e n t cases covered, t h i s f o r m u l a t i o n i s somewhat more r e s t r i c t i v e t h a n those i n 11.1 and 11.2. [127]

, generalizations

Though f o r a t l e a s t some o b j e c t i v e s

pa,

o f t h e s t r o n g e r v e r s i o n s t o t h e p r e s e n t case a r e p o s s i b l e ,

t h e r e i s l i t t l e hope f o r a s i m i l a r r e s u l t i n such a g e n e r a l s e t t i n g t h a t covers a l l functions

r

C,m,O

Note a l s o t h a t , w h i l e a l a r g e s e t o f a d m i s s i b l e

as described.

w e i g h t i n g f u n c t i o n s i s wanted f o r a r e s u l t on f a c t o r i z a t i o n , i t i s t h e o t h e r way round f o r a uniqueness r e s u l t . Concerning t h e f a c t o r i z a t i o n problem mentioned, B i l l e r a and B i x b y Birnbaum and Esary [14]

showed f o r t h e s t a n d a r d cases i n network and r e l i a b i l i t y

t h e o r y t h a t t a k i n g a congruence p a r t i t i o n

71

o f A, and p u t t i n g Ci

r

v e r s i o n o f these r e s u l t s , c o n c e r n i n g a l l f u n c t i o n s

as described, was sub-

C,m,@ I n t h a t paper a uniqueness r e s u l t was a l s o f o r m u l a t e d ,

s e q u e n t l y g i v e n i n [81]. 71 E

:= CILi,

A more general

C ' := C/n, a s o l u t i o n t o t h e f a c t o r i z a t i o n theorem i s g i v e n .

which s t a t e s t h a t

El] , and

V(C), Ci

f a c t o r i z a t i o n problem i f (m,@)

= CILi and C ' = C/T g i v e t h e

only s o l u t i o n s r

t o the Cymy@ Note t h a t a l l

i s r e g u l a r ; r e g u l a r i t y meaning t h a t

f o r a l l admissible weighting functions implies C = C ' . 'cI,m,@

s t a n d a r d cases (max,t),

(max,min),

(max;),

(min,+),

(min,max),

(min;)

(even w i t h r e s t r i c t i o n t o v e r y s p e c i a l w e i g h t i n g f u n c t i o n s [Sl]), and (min,min)

are not regular.

I n fact,

rCYmaxyGx) = max x ( a )

=

ae A

f o r any normal c l u t t e r C ' o v e r A ( t h e same i s t r u e f o r (min,min)), f a c t o r i z a t i o n problem i s s o l v e d b y

each p a r t i t i o n

71

o f A.

are regular

w h i l e (max,max)

rc

1

,max

,&A

and t h u s t h e

So (max,max) and (min,

m i n ) f o r m two q u i t e i r r e g u l a r (and u n i n t e r e s t i n g ) e x c e p t i o n s ( i n f a c t , e s s e n t i a l l y t h e o n l y such i r r e g u l a r i t i e s p r o v i d e d t h a t @ induces a p o s i t i v e l y o r d e r e d semigroup p66] on t h e a d m i s s i b l e w e i g h t s ) , w h i l e i n o t h e r cases, decompositions a r e - g i v e n C - t h e same f o r glJ cases, and i n p a r t i c u l a r independent o f t h e T h i s adds t o t h e understanding o f t h e occurrence o f t h e s u b s t i t u t i o n p a i r (m,@). decomposition i n a v a r i e t y o f q u e s t i o n s i n g r a p h t h e o r y , networks, f l o w networks, r e l i a b i l i t y t h e o r y and o t h e r areas, a l l t h e more so when combined w i t h t h e i n t e r face r e s u l t s discussed i n 1.5. I n t h e f o l l o w i n g , we g e n e r a l i z e a s t e p f u r t h e r , t h e (max,+)-algebra i n [33],

M o t i v a t e d by t h e d i s c u s s i o n o f

i t t u r n e d o u t t o be u s e f u l t o view ( m , @ )

m u t a t i v e s e m i - r i n g on (a subset o f ) r e a l numbers.

as a

com-

I n fact, d i s t r i b u t i v i t y f o r

m = max (min) f o l l o w s f r o m t h e o b s e r v a t i o n t h a t , g i v e n t h e monotony o f

0,

R. H. Mohring and F.J. Radermacher

304

Q

x @ max (y,z) = max ( x (min) (min)

I f we extend t h e s e t o f admissible weight-

y, x @ z ) .

i n g f u n c t i o n s accordingly, we even g e t a semi-ring w i t h zero O* and one 1* i n a l l standard cases, e.g. by proceeding according t o Table 2.1.

Here, from an appli;

c a t i o n a l p o i n t o f view, the assumption o f a r t i f i c i a l elements

"-"

and

"-ml'

can be

replaced by l i m i t arguments o r by r e s t r i c t i o n t o "extreme" upper and lower bounds, c f . C81-J. admissible weights

zero

(max,+) (min,+)

one 0 0

( max ,mi n )

+-

(min,max)

- m

(niax, .)

1

(min, .)

1

TABLE 2.1 Note t h a t types (max,max) and (min,min)

a l s o l e a d t o semi-rings, b u t these semi-

r i n g s cannot c o n t a i n a O* and a l*simultaneously, because t h e r e i s then no way t o d i s t i n g u i s h between these two elements. a r t i f i c i a l elements +m, +m* w i t h

+-

(Note t h a t even t h e i n t r o d u c t i o n o f two

< t-*

would n o t h e l p ! )

A l t o g e t h e r the above remarks m o t i v a t e t h e c o n s i d e r a t i o n o f a r b i t r a r y f u n c t i o n s

r c,

ffJ ,@ with

cases

c,+,

$36

CY

( x ) :=

@

6

x ( a ) , thereby i n t e g r a t i n g a l l standard

k C acT

mentioned above, and a l s o t h e n a t u r a l composition *

(x

:=

1 .

( + , a ) ,

i.e. the function

x ( a ) , which i s e.g. o f i n t e r e s t i n r e l i a b i l i t y t h e o r y .

Now

TEC aeT

adapting the f a c t o r i z a t i o n problem t o t h i s general case, we have t h e f o l l o w i n g theorem:

Theorem 2.3.1:

1.

If

(8.6) forms

f a c t o r i z a t i o n problem i s given i f n

2.

If

(@,a)forms

a c o m u t a t i v e semi-ring, a s o l u t i o n t o t h e

E.

V ( C ) , Ci

= CILi

and C ' = C/n.

a c o m u t a t i v e semi-ring w i t h O* and 1*, t h e o n l y s o l u t i o n s t o

the f a c t o r i z a t i o n problem a r e those given i n l.,even i f weights a r e r e s t r i c t e d t o the set {0*,1*}. Proof: - 1. L e t 71 We w i l l show t h a t

E

r

V ( C ) , Ci = CILi and C ' = C/a, i . e . C = C '

c,o,a(x)

= rcl,O,O(~),

where ~ ( f 3 i = )

[ti, i

= 1

,..., 4 .

(xlLi)

r ci ,@ ,6

and

305

Substitution decomposition for discrete structures

x i s any a d m i s s i b l e w e i g h t i n g f u n c t i o n . d i s t r i b u t i v i t y (see s t e p

2a.

*

T h i s i s e s s e n t i a l l y a consequence o f

i n the proof).

We have:

We show t h a t ( @ , @ )

L e t t h e r e be a O* and 1* f o r ( @ , @ ) .

pair, i.e.

that %,@&l

= rC * , @ , @

w e i g h t s O* and 1* a l o n e ) .

C = C* ( t h i s i s a l r e a d y t r u e w . r . t .

Assume

rC , O , @

=

rc*,@,@

w i t h C # C*.

W.1.o.g. 1* a d o

T* $ C* f o r a l l T* s To.

P u t x o ( a ) := O*

1*, w h i l e 2b.

r

Ci,@,@

Finally l e t

IT,

t h e r e i s some T o € C such t h a t

.

Then o b v i o u s l y

"$To

C'

3

8

,@

(y). (y).

. . ,r

(xo) =

0

C i , i = 1,

...,r,

and C ' be any s o l u t i o n t o t h e f a c t o r i z a t i o n

P u t C* = C ' cCi , i = 1,. Consequently, we have r

regularity, implies

rC,$,@

( x ) = 0*, a c o n t r a d i c t i o n .

problem f o r some r e g u l a r p a i r ( Q , @ ) . Then by assumption,

r C',O,QI r

i s then a r e g u l a r

f o r c l u t t e r s C,C* on t h e base s e t A i m p l i e s

..,r] .

C,@,@

c

= C*,

i.e.

c

= C'

[Ci,

rC,Q,@(')

=

Then, by 1 ., r C*,Q,@(') = which, because o f C*,@,@' i = 1 ,r], i . e . C . = cIL., i =

= r

,...

1

1

I,..

and C' = C / IT, c o n c l u d i n g t h e p r o o f . I

Note t h a t though t h e f o r m u l a t i o n o f t h i s r e s u l t i s more general t h a n f o r t h e o r i g i n a l s t a n d a r d cases, t h e p r o o f t u r n e d o u t t o be s i m p l e r , i . e . t h e a l g e b r a i s a t i o n and t h e s e m i - r i n g i n t e r p r e t a t i o n o f (m,@)

seem t o be w o r t h w h i l e .

R.H. Mohring and F.J. Radennacher

306

We would l i k e t o add t h a t Theorem 2.3.1 extends ( f o r measurable@,@) t o t h e stochastic case completely analogously t o Theorem 2.2.6. Furthermore, again r e s t r i c t i n g ourselves t o the o p t i m i z a t i o n case (my 8 ), t h e bounding approach f o r the d i s t r i b u t i o n f u n c t i o n of

D471,

rC

,m, @

i n the independence case a l s o c a r r i e s over

due t o the f a c t t h a t because of

random weights o f elements T

6

@ being monotonically increasing, the

C are again associated random variables.

questly, l e t t i n g FT denote t h e d i s t r i b u t i o n f u n c t i o n o f the weight t h e d i s t r i b u t i o n f u n c t i o n o f t h e optimal value, we have: F, 6

n

FT

i f m = max

Conse-

o f T and F,

and

T6.C

The use o f t h i s approach has, a p a r t from stochastic p r o j e c t networks, been part i c u l a r l y f r u i t f u l i n RELIABILITY THEORY, where most o f the concepts i n v o l v e d have o r i g i n a l l y been introduced and where f u r t h e r improvements have been obtained, c f . e.g.

[llO].

Also i n t h i s more general s e t t i n g , the use o f t h e s u b s t i t u t i o n decom-

p o s i t i o n improves the bounds i n a way analogously t o 11.2, c f [7]. Me would f i n a l l y l i k e t o mention t h a t t h e r e have been obtained [84],

[162]

also

extensions o f the bounds f o r the case o f s t o c h a s t i c dependences, mentioned i n 11.2.

I n f a c t , f o r t h e case (min,+),

concave lower bound i n t h e sense o f

t h e r e i s completely analogously obtained a

p4g, w h i l e f o r

t h e cases (max,min) and (min,

max), there are s t o c h a s t i c upperjlower bounds, r e s p e c t i v e l y .

I n a l l these cases,

the bounds can be computed v i a t h e s u b s t i t u t i o n decomposition, c f . p62].

11.4 SUMMARY AND HINTS ON SOME OTHER APPROACHES TO DECOMPOSITION OF CERTAIN COMBINATORIAL OPTIMIZATION PROBLEMS W e would l i k e t o sumnarize t h a t the r e s u l t s i n t h i s p a r t i n d i c a t e t h a t using a

n a t u r a l but q u i t e strong approach t o f a c t o r i z a t i o n o f optimal value f u n c t i o n s i n combinatorial optimization, one i n e v i t a b l y a r r i v e s a t t h e s u b s t i t u t i o n decomposition.

The reason f o r t h i s l i e s i n asking f o r a simultaneous s o l u t i o n f o r a l l the

weighting functions considered, together w i t h t h e l o c a l i t y c o n d i t i o n and an i n f o r mation t r a n s f e r f o r each class c o n s i s t i n g i n j u s t one r e a l number. Given t h e indecomposability o f "almost a l l " s t r u c t u r e s (P5), such comfortable decomposition p o s s i b i l i t i e s are a q u i t e r a r e s i t u a t i o n .

However, as mentioned before, t h e s i t u a t i o n i n p r a c t i c a l a p p l i c a t i o n s o f t e n seems t o be the o t h e r way round: because o f h i e r a r c h i c a l planning, coarse models are subsequently r e f i n e d from one l e v e l t o

another, thereby n a t u r a l l y generating autonomous sets.

Furthermore, one obtains

307

Substitution decomposition for discrete structures many a d d i t i o n a l p r o p e r t i e s o r i g i n a l l y n o t r e q u i r e d , e.g.

the possibility o f hier-

a r c h i c a l computation o f many parameters o f i n t e r e s t , as shown above.

Also i t i s

p o s s i b l e t o e f f i c i e n t l y compute and r e p r e s e n t a l l autonomous s e t s i n t h e most wanted cases o f r e l a t i o n s and bounded c l u t t e r s . s i t i o n t r e e d e a l t w i t h i n S e c t i o n s I11 and I V .

T h i s i s done by u s i n g t h e compoFurthermore, t h e Jordan-Holder

theorem deduced i n S e c t i o n I 1 1 shows t h e independence o f m u l t i - s t e p decomposition f r o m t h e o r d e r and s t a r t i n g p o i n t o f t h e s t e p w i s e minimal decompositions, t h u s l e a d i n g t o a r a t h e r s i m p l e concept f o r p r a c t i c a l a p p l i c a t i o n s . N a t u r a l l y , g i v e n t h e l i m i t a t i o n s e x i s t i n g f o r t h e s u b s t i t u t i o n decompositions, o t h e r concepts have been t r i e d . and r e l i a b i l i t y t h e o r y [48],

[97],

T h i s i s p a r t i c u l a r l y t r u e f o r p r o j e c t networks p16],

@43],

F5q,

c5q].

The m a j o r i t y

o f t h e s e methods e s s e n t i a l l y aim a t r e p l a c i n g an a r b i t r a r y connected subnetwork by i t s e n t r a n c e and e x i t nodes, where an e n t r a n c e and an e x i t node a r e j o i n e d by an 2dge i f f t h e y a r e connected by a d i r e c t e d p a t h i n t h e o r i g i n a l network.

Then, f o r

i n s t a n c e , t h e d u r a t i o n o f such an a r t i f i c i a l edge w i l l be t h e l o n g e s t p a t h l e n g t h between t h e corresponding o r i g i n a l nodes.

I n t h i s way, t h e s h o r t e s t p r o j e c t

d u r a t i o n and some o t h e r parameters can be computed. depend on t h e w e i g h t s x valued functions. one.

E

However, these computations

R,” i n a c o m p l i c a t e d way which cannot be expressed by real-

I n f a c t , t h e “reduced“ network may even be l a r g e r than the o r i g i n a l

Therefore, h i g h e r - l e v e l aspects, e.g.

time-cost trade o f f s o r stochastic

g e n e r a l i z a t i o n s can s c a r c e l y be handled t h i s way.

I n t h e s t o c h a s t i c case, f o r

i n s t a n c e , s t o c h a s t i c dependences between t h e r e s u l t i n g d i s t r i b u t i o n s o f t h e new a c t i v i t y d u r a t i o n s would have t o be t a k e n i n t o account, a l s o i n t h e case o f s t o c h a s t i c a l l y independent a c t i v i t y d u r a t i o n s . We c l o s e t h i s s e c t i o n w i t h some h i n t s on t h e s p l i t decomposition, which may i n t h e l o n g r u n prove, t o some e x t e n t , t o be a f u l l c o u n t e r p a r t o f t h e s u b s t i t u t i o n decomposition i n t h e f i e l d o f c o m b i n a t o r i a l o p t i m i z a t i o n .

There a r e by now e.g.

a p p l i c a t i o n s t o t h e d e t e r m i n a t i o n o f t h e weighted independence number a G ( x ) o f

[3g.

graphs

I n f o r m a t i o n t r a n s f e r h e r e i s more i n v o l v e d , and n e c e s s i t a t e s t h e

comparison o f a t l e a s t two d i f f e r e n t a l t e r n a t i v e s , depending on t h e p o s i t i o n o f t h e marker.

I n p a r t i c u l a r , no such i n s t r u m e n t as a simultaneous t r e a t m e n t o f a con-

gruence p a r t i t i o n i s a v a i l a b l e .

S t i l l , given a proper organization, e f f i c i e n t

s o l u t i o n s may be p o s s i b l e and may perhaps extend t o some o f t h e o t h e r cases handled above f o r t h e s u b s t i t u t i o n decomposition.

111.

AN ALGEBRAIC MODEL OF DECOMPOSITION

I n t h i s s e c t i o n , we p r e s e n t a general a l g e b r a i c decomposition t h e o r y which

generalizes and u n i f i e s t h e basic p r o p e r t i e s o f t h e s u b s t i t u t i o n decomposition f o r r e l a t i o n s , s e t systems and Boolean f u n c t i o n s . Subsection 111.1 i n t r o d u c e s t h e assumptions o f t h e general model and shows how t o embed the s u b s t i t u t i o n decomposition i n t o t h i s framework.

This w i l l be done by

i n t e r p r e t i n g the n a t u r a l l y a r i s i n g n o t i o n s o f q u o t i e n t s and autonomous subs t r u c t u r e s i n an " a l g e b r a i c " way, i . e . as a l g e b r a i c q u o t i e n t s o r substructures f o r s u i t a b l y d e f i n e d homomorphisms.

The p r o p e r t i e s t h a t c h a r a c t e r i z e t h e s u b s t i -

t u t i o n decomposition then correspond ( a p a r t from the usual a l g e b r a i c p r o p e r t i e s (MI)

-

(M6)) t o a s p e c i a l s o r t o f i n t e r p l a y between these two n o t i o n s which have

no counterpart i n a l g e b r a i c t h e o r i e s ( c f . ( M 7 ) , (M8)). I n 111.2, we i n v e s t i g a t e the system V ( S ) o f congruence p a r t i t i o n s o f a s t r u c t u r e

S, when considered as a suborder o f the p a r t i t i o n l a t t i c e Z ( A ) o f t h e base s e t A

o f S.

As the main p r e p a r a t o r y step f o r t h e i n v e s t i g a t i o n o f composition s e r i e s

i n 111.3 we o b t a i n t h a t i f V ( S ) i s o f f i n i t e length, then i t i s already an upper semimodular s u b l a t t i c e o f Z ( A ) .

Furthermore, o t h e r l a t t i c e - t h e o r e t i c a l p r o p e r t i e s

o f V ( S ) such as being modular, d i s t r i b u t i v e o r complemented can a l s o be characteri z e d i n terms o f p r o p e r t i e s o f t h e given s t r u c t u r e

s.

Subsection 111.3 deals w i t h t h e f a c t o r i z a t i o n o f a s t r u c t u r e by means o f composit i o n s e r i e s , which correspond t o c h i e f s e r i e s i n u n i v e r s a l algebra.

Besides

c r i t e r i a f o r t h e i r existence, we o b t a i n a " c l a s s i c a l " Jordan-Holder theorem which s t a t e s t h a t any two composition s e r i e s o f a s t r u c t u r e have t h e same l e n g t h and, up t o isomorphism and rearrangement, a l s o t h e same f a c t o r s .

This c e n t r a l r e s u l t

on decomposition i s then r e l a t e d t o the Jordan-HUlder theorems i n u n i v e r s a l algebra. Under a d d i t i o n a l assumptions (which a r e f u l f i l l e d i n t h e a p p l i c a t i o n s considered i n 1.2

-

1 . 4 ) , i t i s p o s s i b l e t o represent t h e decompositions o f a s t r u c t u r e S i n

a t r e e , the composition t r e e

B(S) o f S.

This i s discussed i n 111.4.

The basic

methods f o r the t r e e c o n s t r u c t i o n a r e two m u t u a l l y e x c l u s i v e decomposition p r i n c i p l e s , which g e n e r a l i z e the methods observed f o r t h e s p e c i a l cases o f c l u t t e r s , Boolean f u n c t i o n s and c e r t a i n r e l a t i o n s ( c f . Section I V ) : E i t h e r t h e r e e x i s t s a maximal d i s j o i n t decomposition i n t o autonomous s e t s ( i n which case t h e q u o t i e n t s t r u c t u r e i s indecomposable), o r t h e r e e x i s t s a f i n e s t decomposition such t h a t the q u o t i e n t s t r u c t u r e belongs t o c e r t a i n , w e l l - c h a r a c t e r i z e d classes. Furthermore, t h i s t r e e represents a l l " e s s e n t i a l " decompositions of a s t r u c t u r e i n a polynomial ( l i n e a r ) number o f nodes, a f a c t which makes i t a s u i t a b l e data s t r u c t u r e f o r algorithms concerned w i t h decomposition.

Suhstitutioir decomposition fiir discrctc stnictiirc's

309

I n 111.5, we d i s c u s s t h e c o n n e c t i o n o f t h e s e r e s u l t s w i t h t h e s p l i t decomposition F i n a l l y , i n 111.6, we g i v e some h i n t s on t h e a l g o r i t h m i c d e t e r m i n a t i o n o f t h e c o m p o s i t i o n t r e e B(S) o f a s t r u c t u r e S.

It can be shown t h a t t h i s t a s k i s p o l y -

n o m i a l l y e q u i v a l e n t t o two a p p a r a n t l y weaker t a s k s , v i z . d e t e r m i n i n g t h e autonomous c l o s u r e o f a g i v e n s e t , o r (under a d d i t i o n a l assumptions) d e c i d i n g whether a g i v e n s t r u c t u r e i s decomposable o r n o t , and p r o d u c i n g a n o n - t r i v i a l autonomous set i f i t i s .

These polynomial r e d u c t i o n s f o r m t h e s t a r t i n g p o i n t f o r t h e

r e s u l t s on t h e c o m p u t a t i o n a l c o m p l e x i t y o f decomposing r e l a t i o n s and c l u t t e r s i n Section I V . A p a r t f r o m 111.2,

the presentation o f t h i s section follows that i n

[log,

so

t h a t we can r e s t r i c t o u r b i b l i o g r a p h i c a l notes and r e f e r t o D O 2 3 f o r f u r t h e r d e t a i l s on r e s u l t s n o t proved here.

111.1

THE ALGEBRAIC MODEL

I n t h e g e n e r a l decomposition model we c o n s i d e r ( c f . a l s o t h e h i n t s i n S e c t i o n I ) a " c o n c r e t e " c a t e g o r y K, whose o b j e c t s a r e c a l l e d s t r u c t u r e s and a r e denoted by S,T e t c .

"Concrete" means ( c f .

[73])

t h a t each s t r u c t u r e i s d e f i n e d on an under-

base s a t o f S), and t h a t each homomorphism ( o r morphism i n l y i n g s e t A = AS ( t h e ___c a t e g o r i c a l t e r m i n o l o g y ) f r o m S t o T i s a mapping f r o m As i n t o AT.

A s p e c i a l r o l e w i l l be p l a y e d by t h e s u r j e c t i v e and i n j e c t i v e homomorphisms which ( i n accordance w i t h t h e usual a l g e b r a i c t e r m i n o l o g y [30],

b u t d i f f e r e n t from

c a t e g o r i c a l t e r m i n o l o g y [73] ) w i 11 be r e f e r r e d t o as epimorphisms and monomorphisms, r e s p e c t i v e l y .

F o r s t r u c t u r e s S,T f r o m K, l e t Hom(S,T),

Epi(S,T),

and

Mono(S,T) denote t h e s e t s o f homomorphisms, epimorphisms, and monomorphisms f r o m S t o T, r e s p e c t i v e l y .

L e t S denote t h e c l a s s o f s t r u c t u r e s o f K.

Two s t r u c t u r e s

S and T a r e isomorphic i f t h e r e e x i s t s a b i j e c t i v e mapping f such t h a t f

E

Hom

(S,T) and f - l E Hom(T,S). I n a d d i t i o n t o t h e usual c a t e g o r i c a l p r o p e r t i e s (which e s s e n t i a l l y mean t h a t home morphisms a r e c l o s e d under composition, i . e . f, c Hom(S1,S2), f 2 E Hom(S2,S3) yields

f20

fl

E

Hom(S1,S3),

and t h a t , f o r each s t r u c t u r e S on A, t h e i d e n t i c a l

A f u l f i l l s i d A Hom(S,S) and f o i d A = di ,?, o f = f f o r each S T we impose two groups o f c o n d i t i o n s , ( M l ) - (M5) and (M6) - (M8). The f i r s t group p r o v i d e s us w i t h elementary a l g e b r a i c p r o p e r t i e s needed t o d e f i n e

mapping i d A : A f

E

+

Hom(S,T)),

q u o t i e n t s and s u b s t r u c t u r e s , w h i l e t h e second group d e a l s w i t h t h e r e l a t i o n s h i p between these n o t i o n s .

Note t h a t (Ml)

-

(M6) h o l d i n the f a m i l i a r algebraic

t h e o r i e s (e.g. t h e t h e o r y o f groups, r i n g s , e t c . ) and t h a t i t i s , i n f a c t ,

R.H. Mohring and F.J. Radermacher

310

o n l y c o n d i t i o n s (M7) and (M8) which a r e "non-algebraic".

These two c o n d i t i o n s

may thus be viewed as r e p r e s e n t i n g t h e s p e c i a l c h a r a c t e r o f t h e s u b s t i t u t i o n decomposi ti on. Each f

E

g e Epi(S,U),

h

E

-t

and f-'

there e x i s t U

S,

g i v e n a s t r u c t u r e S on A and a b i j e c t i o n

B, t h e r e e x i s t s a unique s t r u c t u r e T on B such t h a t f E

E

Mono(U,T) such t h a t f = hog

S t r u c t u r e i s a b s t r a c t , i.e., f: A

i.e.,

Hom(S,T) has an epi-mono-factorization,

E

Hom(S,T)

Hom(T,S).

Given a s t r u c t u r e S on A and a s u r j e c t i o n f from A onto a s i n g l e t o n t h e r e e x i s t s a s t r u c t u r e So on A,

such t h a t f

E

A,,

Epi(S,So).

I f h E E p i ( S , T l ) n Epi(S,T2) then T1 = T2. If g

Q

Mono(S1,T)

n Mono(S2,T)

then S1 =

s2.

D e f i n i t i o n a) A s t r u c t u r e T i s c a l l e d a q u o t i e n t s t r u c t u r e o r q u o t i e n t o f a s t r u c t u r e S on A i f t h e r e e x i s t s a p a r t i t i o n TI o f A such t h a t nTI E Epi(S,T), where n,, denotes the n a t u r a l mapping associated w i t h a l l a € A , where [a].

i s the class o f

CY

w.r.t.

TI

(i.e.,

= [a]* f o r

qTI(ct)

I n t h i s case,

n).

i s called a

IT

congruence p a r t i t i o n o f S, and t h e u n i q u e l y determined (because o f (M4)) q u o t i e n t T i s denoted by S / n .

A

V ( S ) denotes t h e system o f congruence p a r t i t i o n s o f S.

s t r u c t u r e S on A i s c a l l e d prime o r indecomposable i f V ( S ) c o n t a i n s no proper congruence p a r t i t i o n , i . e . i f C c

TI

c

V(S) implies C

=

A o r I C [ = 1.

b ) A s t r u c t u r e T i s c a l l e d a s u b s t r u c t u r e o f a s t r u c t u r e S on A, i f t h e r e e x i s t s a subset B o f A such t h a t i n c AB L Mono(T,S), where i n c AB denotes the i n c l u s i o n mapping associated w i t h B and A ( i . e . i n c i ( a ) =

CY

f o r a l l a e B).

I n t h i s case,

B i s c a l l e d an S-autonomous set, and t h e uniquely determined (because o f (M5)) substructure T i s denoted by S I B .

A ( S ) denotes the system o f S-autonomous s e t s .

Condition (142) i m p l i e s t h e ( i n a l g e b r a i c t h e o r i e s ) usual f a c t o r i z a t i o n o f epimorphisms and monomorphisms:

a) Each epimorphism h 0 Epi(S,S') has a f a c t o r i z a t i o n h = where o7 E Hom(S,T) i s the n a t u r a l mapping associated w i t h the p a r t i t i o n Lemna 3.1.1:

induced by h ( i . e . anB i f f h ( a ) = h ( B ) ) , f i s the q u o t i e n t S/TI.

E

foqT, T

o f AS

Hom(T,S) i s an isomorphism, and

T

311

Substitution decomposition for discrete structures b ) Each monomorphism h tsMono(S',S) has a f a c t o r i z a t i o n h = incg'o f, where f E Hom(S',T) i s an isomorphism, T = S I B i s t h e s u b s t r u c t u r e SIB induced by

B

= h ( A S , ) , and i n c ?

Proof:

E

Hom(T,S) i s t h e i n c l u s i o n o f T = SIB i n S.

Apply (M2) t o t h e b i j e c t i o n s gl:AS,

and g2:AS, +

B, d e f i n e d by g 2 ( a ' )

homomorphisms, nn = gloh

E

= h(a'),

-+

AS/r,

d e f i n e d by g,(h(a))

:= [a]~,

r e s p e c t i v e l y . Then, by c o m p o s i t i o n o f -1 = hog2 e Mono(T,S), which a l s o

Epi(S,T) and i n c ?

gives t h e claimed f a c t o r i z a t i o n . s So epimorphisms and monomorphisms correspond e s s e n t i a l l y t o q u o t i e n t s t r u c t u r e s and s u b s t r u c t u r e s , which i n t u r n can f o r a g i v e n s t r u c t u r e S be " i n t e r n a l l y " d e s c r i b e d by i t s system o f congruence p a r t i t i o n s V ( S ) and i t s system o f S-autonomous s e t s A ( S ) . The q u e s t i o n t h e n i s how t o embed t h e s u b s t i t u t i o n decomposition i n t o t h i s framework.

T h i s i s n o t q u i t e obvious, s i n c e , i n general, t h e r e i s no " n a t u r a l " n o t i o n

o f homomorphism.

There are, however, n a t u r a l n o t i o n s o f " q u o t i e n t " and "sub-

s t r u c t u r e " which may be used t o d e f i n e homomorphisms a p p r o p r i a t e l y .

To t h i s end,

l e t S be a s t r u c t u r e ( i . e . a r e l a t i o n , s e t system o r Boolean f u n c t i o n ) o b t a i n e d by s u b s t i t u t i o n , i . e . each B

A', S

B

S = SICS,,

B

E

A'],

where S ' i s a s t r u c t u r e on A ' and, f o r

i s a s t r u c t u r e on AB.

Then t h e r e l a t i o n s h i p between S and t h e " q u o t i e n t " S ' may be e q u i v a l e n t l y des+ A ' with h ( a ) = B i f a e A 8 induces a congruence p a r t i t i o n ( i n t h e sense o f t h e s u b s t i t u t i o n o p e r a t i o n , c f .

c r i b e d by t h e f a c t t h a t t h e s u r j e c t i v e mapping h:A

1.2

-

1.4).

S i m i l a r l y , t h e embedding o f each " s u b s t r u c t u r e " Sg i n t o S may be e q u i v a l e n t l y d e s c r i b e d by t h e p r o p e r t y t h a t t h e i n c l u s i o n i n c A s i s an isomorphism f r o m S o n t o B A6 S I A B ( i n t e r p r e t e d as t h e induced s u b s t r u c t u r e i n t h e g i v e n c l a s s , i . e . t h e r e s t r i c t i o n t o A ) and t h a t A 6

B

i s autonomous ( w . r . t .

the substitution

decomposition). These s u r j e c t i v e mappings and i n c l u s i o n s a r e t h e n t a k e n as " s p e c i a l " homomorphisms i n t h e general model and a r b i t r a r y homomorphisms a r e d e f i n e d as t h e c o m p o s i t i o n o f f i n i t e l y many o f t h e s e " s p e c i a l " homomorphisms. Based on t h e p r o p e r t i e s (Sl) - ( S 7 ) o f t h e s u b s t i t u t i o n o p e r a t i o n given a t t h e b e g i n n i n g o f S e c t i o n I , one o b t a i n s t h e f o l l o w i n g c h a r a c t e r i z a t i o n o f t h e homomorphisms d e f i n e d above.

R.H. Mohring and F.J. Radermacher

312 L e m a 3.1.2:

Each homomorphism h = hno ...Ohl has a f a c t o r i z a t i o n h = gof, where

f and g a r e a s u r j e c t i v e mapping and an i n c l u s i o n o f t h e s p e c i a l type described above , r e s p e c t i v e l y .

Proof:

I t f o l l o w s from ( S 4 ) ( i i ) t h a t t h e composition o f two s u r j e c t i v e mappings

of t h e special type i s again o f t h e special type.

Similarly, (Sl) implies t h a t

the composition o f two i n c l u s i o n s o f t h e special type i s again an i n c l u s i o n o f t h e special type. where hl e Hom(S1,S2) i s an

Thus i t remains t o be shown t h a t given f = h20hl, i n c l u s i o n and h2

Hom(S2,S3) i s s u r j e c t i v e o f t h e s p e c i a l type, t h e r e e x i s t an

i n c l u s i o n f2 and a s u r j e c t i o n fl o f t h e s p e c i a l type such t h a t f = L e t Ai be the base s e t o f Si, of S2 associated w i t h h2.

i = 1,2,3,

and l e t

be t h e congruence p a r t i t i o n

i s S2-autonomous Because o f ( S 4 ) ' ( i ) , B := h2( [A1]~)

Since hl i s an i n c l u s i o n , A1 = hl(A1)

and t h e r e f o r e , because of ( S 5 ) , a l s o [A1]v. i s S3-autonomous.

T

On t h e o t h e r hand, i t f o l l o w s from ( S 6 ) t h a t v l A 1 i s a congru-

ence p a r t i t i o n o f S21A1 = S1 which corresponds t o t h e r e s t r i c t i o n Since (57) i m p l i e s t h a t Sl/(nlA1) fl :=

h2

f20fl.

and ( S 2 1 [A1]T)/(Tl

[A,]n)

b2

o f hp t o A1.

= S31B a r e isomorphic,

and f 2 := i n c i 3 c o n s t i t u t e t h e claimed f a c t o r i z a t i o n o f f . m

Thus the n o t i o n o f homomorphism induced by t h e s u b s t i t u t i o n o p e r a t i o n f u l f i l s the f a c t o r i z a t i o n condition (Ml).

Furthermore, t h e epimorphisms and monomorphisms

are (up t o b i j e c t i o n s ) e x a c t l y t h e s p e c i a l s u r j e c t i v e mappings and i n c l u s i o n s associated w i t h t h e s u b s t i t u t i o n operation.

Hence, a l s o (M4) and (M5) a r e

s a t i s f i e d (up t o the i d e n t i f i c a t i o n o f r e l a t i o n s w i t h d i f f e r e n t r e f l e x i v e t u p l e s and Boolean f u n c t i o n s w i t h complemented v a r i a b l e s o r f u n c t i o n a l value i n t h e sense o f Section I.2,1.4).

O f course, S-autonomous s e t s and congruence p a r t i t i o n s

i n t h e a l g e b r a i c model correspond e x a c t l y t o t h e i r counterparts f o r t h e s u b s t i t u t i o n operation.

Since (M2) and (M3) a r e f u l f i l l e d t r i v i a l l y , t h e embedding o f

t h e s u b s t i t u t i o n decomposition i n t o t h e general model i s completed. This approach may seem r a t h e r l a b o r i o u s , i n p a r t i c u l a r w . r . t .

the v e r i f i c a t i o n o f

(Ml). However t h e obtained i n t e r p r e t a t i o n o f t h e s u b s t i t u t i o n decomposition i n terms o f an a l g e b r a i c homomorphism theory permits t o t h e a p p l i c a t i o n o f methods and concepts from u n i v e r s a l algebra.

From t h i s p o i n t o f view, (Ml) t u r n s o u t t o

be a powerful c o n d i t i o n and e s s e n t i a l l y e q u i v a l e n t t o c o n d i t i o n s ( S 5 ) , (S6) and

( S 7 ) introduced a t t h e beginning o f Section I ( c f . Theorem 3.1.3).

Furthermore,

there are examples o f the general model which cannot be obtained from a s u b s t i t u t i o n operation, i . e . exanples i n which i t i s n o t p o s s i b l e t o "uniquely r e c o n s t r u c t " a s t r u c t u r e S from a q u o t i e n t S / n and the substructures S I B , B E T ( c f . [IOZJ).

Substitution decomposition for discrete structures

313

The second group o f c o n d i t i o n s o f t h e a l g e b r a i c model r e f l e c t s p o s s i b i l i t i e s f o r c o n s t r u c t i n g new congruence p a r t i t i o n s f r o m a l r e a d y known ones. (M6)

'TI,U

(M7)

(i)

E:

V ( S ) and 'TI

V(S)

e

'TI

# 4.

4 u => Epi(S/*,S/u)

=>

B EA(S)

for all B

E

'TI.

n

If

(ii)

= I L 1,...,Ln,Iu31a

'TI

Li E A ( S ) , i = 1 (M8)

'TI

= EBi

=>

I

,..., n,

i E I}E V ( S ) ,

u = IBi,C.

J

I ic

E

A\U

t h e n 'TIBV(S).

[ j

= ICj

T

i s a p a r t i t i o n o f AS w i t h

Li}

i=l

E

J}

E

V(S

I Bi

) f o r some Bi 0

E

'TI

0

I\{i0}, j e J} C V ( S ) .

I n t h e s u b s t i t u t i o n decomposition, t h e s e c o n d i t i o n s correspond t o ( S 4 ) ( i ) , ( S 2 ) and (S3), c f . S e c t i o n I. (M6) i s known as t h e "Theorem o f Induced Homomorphism" i n u n i v e r s a l a l g e b r a . I t i m p l i e s by s t a n d a r d arguments f r o m u n i v e r s a l a l g e b r a ( c f . [66,

p.611) t o g e t h e r w i t h Lemma 3.1.1

t h a t V ( S / ' T I ) i s c o m p l e t e l y determined

by V ( S ) ( c f . a l s o Example 3.2.9):

L e t S be a s t r u c t u r e on A and

Theorem 3.1.3: {U E

V(S)

I

'TI

[a]lT(u/lr)[B]'TI

: 2".

by induction on the l e n a t h V(S) i s finite. :i

# " 2 cover

iil

2a)

Gf

The i e s t f o l l o w s t r i v i a l l y .

a f i n i t e maximal chain t h a t each i n t e r v a l

This i s obvious i f

T~

covers

i n a maximal, f i n i t e chain i n

nl. [T,,IT,].

We show [T.,T,]

I n t h e i n d u c t i v e step, l e t By the i n d u c t i v e

of

Substitution decomposition for discrete stnictures

assumption,

317

i s f i n i t e and, because o f Theorem 3.1.3 isomorphic t o

[IT.IT,]

we may assume t h a t n1 = IT'. Lemma 3.2.4 t h e n y i e l d s [ n ' , ~ ~ ~ / n ~ ] .So w.l.o.g., t h a t 71 has only one n o n - s i n g l e t o n c l a s s , say 6. So, because o f Lemma 3.2.1 ITAU E V ( S ) and ITVO E V ( S ) f o r a l l u E V ( S ) . L e t K be a maximal c h a i n i n [r1,n2].

If there exists u

E

K such t h a t avn < n2,

then we can conclude f r o m t h e i n d u c t i v e h y p o t h e s i s a p p l i e d t o [n1,uv~] and

K i s finite.

that

[uvn,n2]

-

the i d e n t i t i e s

= r2,

ITVU

T h i s i m p l i e s t h a t [r1,n2] as

[~T",IT,]

[To,Tz],

=

and

I f t h e r e i s no such u , one o b t a i n s [ K I \< 3 because of

ITAU

=

= nl f o r a l l a E K \ { ~ T ~ , I T ~ I .

IT^,^^]

i s o f f i n i t e l e n g t h and may t h u s be w r t t e n

{ITOIUlJ[ui,r2], where ui, i E I, a r e t h e I i s f i niQ i i e because of Lemma 3.2.7 and 3.2.4.

p r o o f , s i n c e each

atoms o f V ( S ) i n T h i s concludes t h e

i s f i n i t e by t h e i n d u c t i v e h y p o t h e s i s .

[ u ~ , I T ~

I

and v.

L i s called

upper semimodular, i f avb covers a and b whenever a and b cover anb.

Upper semi-

L e t L be a l a t t i c e o f f i n i t e l e n g t h w i t h l a t t i c e o p e r a t i o n s

4

modular l a t t i c e s have t h e i m p o r t a n t p r o p e r t y t h a t t h e y f u l f i l t h e Jordan-Dedekind c h a i n c o n d i t i o n [13],

which s t a t e s t h a t any two maximal c h a i n s between any two

elements of L ( t h u s i n p a r t i c u l a r a l l maximal c h a i n s i n L ) have t h e same l e n g t h . For V ( S ) , we o b t a i n :

Theorem 3.2.5:

I f S f u l f i l s one o f t h e c o n d i t i o n s o f Theorem 3.2.3,

then V ( S ) i s

an upper semimodular s u b l a t t i c e o f Z(A).

Proof

(cf.

[98]

and Lemna 3.2.1, must show t h a t that

~

~

f o r r e l a t i o n s and PO61 f o r c l u t t e r s ) : V ( S ) i s a s u b l a t t i c e o f Z(A). n1

v nz covers r1 and

L e t n1,n2 e V ( S ) cover u .

1

B 2 ) > i f B 1 n B2 = $ and n 1 v n2 =

=

We

IT^. Because o f Theorem 3.1.3 we may assume

a r e, atoms n i~n V ( S ) , i . e . have t h e form n . = {Bi,{a}la

i s p r i m e because o f Lemma 3.2.4.(i

1,Z).

IBIU

Then

e A\Bi}

v n 2 = IBl,B2,{a}lcr

where SIBi

A\(BIU 1 A\(BIU B z ) > i f B 1 n B2 # $. 2a) and (M7) t h a t n1 v IT^ covers IT ?I

E

B2,{alIcre

I n b o t h cases, i t f o l l o w s f r o m Theorem 3.1.4, and n

Because o f Theorem 3.2.3

1

2'

Some f u r t h e r l a t t i c e t h e o r e t i c a l p r o p e r t i e s o f V ( S ) w i l l be g i v e n i n t h e s t r o n g e r case t h a t (M7)*,

(M9) and (M10) h o l d , which i s t h e case f o r r e l a t i o n s and c e r t a i n

c l u t t e r s , c f . Section I V .

We o m i t t h e p r o o f s and i n s t e a d r e f e r t o @05].

p a r t i a l o r d e r s , a r b i t r a r y r e l a t i o n s , and c l u t t e r s , see a l s o [122], [106],

respectively.

For

1981, D51],

R. Ii. Mfikring and F.J. Radermacher

318

We f i r s t c h a r a c t e r i z e complemented congruence p a r t i t i o n l a t t i c e s .

t h a t o n l y s t r u c t u r e s S f o r w h i c h A ( S ) i s degenerate ( i . e .

It turns o u t

A ( S ) = P(A)\{$I,

where

P(A) denotes t h e power s e t o f A ) or l i n e a r ( i . e . t h e r e i s a l i n e a r o r d e r i n g 6 on

A such t h a t A ( S ) = A ( $ ) ) , and which a r e o f importance i n c o n n e c t i o n w i t h (P6), can have complemented congruence p a r t i t i o n l a t t i c e s .

Under t h e assumptions (M7)*,

Theorem 3.2.6:

(M9) and (MlO), V ( S ) i s complemented

i f f one o f t h e f o l l o w i n g c o n d i t i o n s h o l d s : Then V ( S ) i s a 2-element Boolean a l g e b r a .

1.

S i s prime.

2.

S i s l i n e a r and t h e a s s o c i a t e d l i n e a r o r d e r 6 i s l o c a l l y f i n i t e ( i . e .

3.

S i s degenerate.

interval o f 4 is finite).

each

Then V ( S ) i s a Boolean a l g e b r a .

Then U ( S ) i s t h e p a r t i t i o n l a t t i c e Z ( A ) .

Under t h e above assumptions t h e f o l l o w i n g statements a r e e q u i v -

C o r o l l a r y 3.2.7: ilent

1. 2.

V ( S ) i s r e l a t i v e l y complemented

3.

V ( S ) i s a p a r t i t i o n l a t t i c e o r a Boolean a l g e b r a .

V ( S ) i s complemented

Theorem 3 . 2 . 6 , 2 i s a s p e c i a l case o f a theorem o f Hashimoto [70,

Th. 8-41 s t a t i n g

t h a t t h e congruence p a r t i t i o n l a t t i c e o f a d i s t r i b u t i v e l a t t i c e ( c o n s i d e r e d as an t h e j o i n and meet o p e r a t i o n s ) i s a Boolean a l g e b r a i f f t h e l a t t i c e

algebra w . r . t .

i s locally finite.

This f o l l o w s from t h e f a c t t h a t l i n e a r orders a r e l a t t i c e s ,

and t h a t t h e homomorphisms d e f i n e d v i a t h e s u b s t i t u t i o n decomposition c o i n c i d e f o r t h i s Lase w i t h t h e l a t t i c e homomorphisms.

T h i s means t h a t , f o r l i n e a r o r d e r s ,

l a t t i c e congruence p a r t i t i o n s a r e t h e same as t h o s e f o r t h e s u b s t i t l i t i o n decomposition. F o r t h e c h a r a c t e r i z a t i o n of modular and d i s t r i b u t i v e congruence p a r t i t i o n l a t t i c s we need t o s p e c i f y t h e degree t o which a s t r u c t u r e i s degenerate.

Definition:

L e t S be a s t r u c t u r e on A and l e t Deg(S) be t h e s e t o f a l l degeneraie

s t r u c t u r e s which a r e a s u b s t r u c t u r e o f S o r o f some homomorphic image o f S . t h e maximum c a r d i n a l i t y o f a base s e t o f t h e s t r u c t u r e s o f Deg(S) ( o r

-,

Then

i f the

maximum does n o t e x i s t ) i s c a l l e d t h e d e g e n e r a t i o n degree o f S and i s denoted by deg(S).

319

Substitution decomposition for discrete structures

Theorem 3 . 2 . 8 : deg(S) 6 3 (6

Under t h e above assumptions, V ( S ) i s modular ( d i s t r i b u t i v e ) i f f

2).

The p r o o f shows t h a t deg(S) = n i m p l i e s t h a t V ( S ) c o n t a i n s an i n t e r v a l i s o m o r p h i c t o t h e congruence p a r t i t i o n l a t t i c e o f a degenerate s t r u c t u r e on n elements, i . e . T h i s g i v e s t h e easy d i r -

isomorphic t o t h e p a r t i t i o n l a t t i c e Z(n) o f {l,...,n}. ection o f the proof.

I n t h e o t h e r d i r e c t i o n one shows t h a t deg(S) Q 3 ( 2 ) Because o f t h e repre-

i m p l i e s t h a t complements i n V ( S ) a r e incomparable ( u n i q u e ) .

s e n t a t i o n o f any i n t e r v a l [T,u] c V ( S ) as t h e d i r e c t p r o d u c t [n,.]

=

Y

V((S/n)IB),

Bw/n

a l s o r e l a t i v e complements i n V ( S ) a r e t h e n incomparable ( u n i q u e ) .

T h i s proves t h e

theorem by s t a n d a r d c h a r a c t e r i z a t i o n s o f m o d u l a r i t y ( d i s t r i b u t i v i t y ) , c f . p3],

n54-j. For f u r t h e r r e s u l t s on t h e r e p r e s e n t a t i o n o f congruence p a r t i t i o n l a t t i c e s as a s u b d i r e c t e d p r o d u c t o f s p e c i a l l a t t i c e s c f . [151].

1x1

n,&;;G)

0

1 . Besides the t r i v i a l congruence p a r t i t i o n s no and n l , G has t h e following congruence p a r t i t i o n s : n1 = {{l,Zi,{31,{41,t5l), n2 = {{11,{21,{3},{4,511, n 3 = {{1,21,{3},t4,5}}, n 4 = {{1,4,5l,{21,{3lI,

x5 = {{11,{3l,{2,4,51}, {t1,2,4,51,{3}}. V ( G ) i s a l s o given i n Figure 3.1. Note t h a t t h e meet and j o i n i n V ( S ) correspond t o the l a t t i c e operations i n t h e p a r t i t i o n l a t t i c e n6 =

Z({l,.. .,53).

2 . The quotient graphs G/.rri, i = 1,2,3, a r e given i n Figure 3.2. The associated congruence p a r t i t i o n l a t t i c e s V(G/ni) are given a s the dual principal ideal Gi,n’] of V ( G ) according t o Theorem 3.1.3 ( c f . t h e bold nodes and l i n e s in Figure 3.2).

R.H. Mohring and F.J. Radermacher

3 20

4Y75

11

1 4 3

&

Figure 3.2:

Q u o t i e n t graphs and associated dual p r i n c i p a l i d e a l s of V ( G )

None o f t h e l a t t i c e s W(G) and W(G/ni)

3.

follows alsc, from Theorem 3.2.6, o r degenerate.

i = 1,2,3,

s i n c e none

G f

i s complemented.

t h e graphs G,G1,G2,G3

This i s linear

Note f u r t h e r t h a t a l l these l a t t i c e s are modular, and t h a t V(G/nl)

and V(G/n3) are even d i s t r i b u t i v e .

This f o l l o w s a l s o from Theorem 3.2.8,

since

= deg(G/n3) = 2. For i n s t a n c e f o r G (and G/n2) t h i s f o l l o w s from t h e f a c t t h a t t h e 3-node subgraph G/n2)(11,2,451 of deg(G) = deg(G/np) = 3 and deg(G/nl) i s degenerate.

G/n2

111.3

THE JORDAN-HOLDER

THEOREM FOR COMPOSITION

SERIES

An important instrument o f an a l g e b r a i c theory i s t h e f a c t o r i z a t i o n o f a s t r u c t u r e by means o f homomorphisms, where s p e c i a l i n t e r e s t i s p a i d t o t h e successive f a c t o r i z a t i o n i n steps which cannot be r e f i n e d any f u r t h e r ( c h i e f s e r i e s of an algebra i n the sense o f [30]). Definition:

Here, we s h a l l i n t r o d u c e t h e same n o t i o n .

L e t S be a s t r u c t u r e on A.

f i n i t e sequence S = So,Sl,,..,Sn

A composition s e r i e s o f S i s a maximal o f p a i r w i s e non-isomorphic s t r u c t u r e s Si on Ai

w i t h ( A n [ = 1 such t h a t t h e r e e x i s t s an epimorphism hi i = 1, ..., n.

E

Epi(Si-l,Si)

f o r each

Since the sequence i s supposed t o be of maximal length, the congruence p a r t i t i o n V(Si-l)

ri

induced by hi i s an atom i n V(Si-l).

Thus, because o f Lemma 3.2.4,

hi maps e x a c t l y one n o n - t r i v i a l , prime s u b s t r u c t u r e Si-ll element o f Si,

and maps t h e elements n o t i n Bi b i j e c t i v e l y .

SoIB1, S1 JB2, ..., Sn-,IBn

s

=

Bi o f Si-l

so,sl ,.. . ,sn.

=

onto one

The prime s t r u c t u r e s

Sn-l are c a l l e d t h e f a c t o r s o f t h e composition s e r i e s

321

Substitution decomposition for discrete structures

(M6) and Theorem 3.1.3 i m p l y e a s i l y t h a t , g i v e n a c o m p o s i t i o n s e r i e s S = So,S1,

...,Sn,

t h e r e i s a maximal c h a i n no 4

rl 6

... 6

Si V ( S ) induces t h e c o m p o s i t i o n s e r i e s S,S/rl,.

vn =

TI^ i n V ( S ) w i t h

... .s

= TI in n (where, o f course, d i f f e r e n t

S / T ~ and t h a t , conversely, each maximal c h a i n T O 6

. .,S/TI,,

nl 6

maximal c h a i n s may induce isomorphic c o m p o s i t i o n s e r i e s ) .

So c o m p o s i t i o n s e r i e s correspond e s s e n t i a l l y t o t h e maximal c h a i n s i n V ( S ) .

This

g i v e s t h e f o l l o w i n g r e s u l t s , t h e second o f which i s a " c l a s s i c a l " Jordan-

-

H o l d e r - t y p e theorem f o r c o m p o s i t i o n s e r i e s o f s t r u c t u r e s f u l f i l l i n g ( M l )

Theorem 3.3.1:

(EXISTENCE CRITERION):

one o f t h e statements o f Theorem 3.2.3

Theorem 3.3.2:

(M8).

A s t r u c t u r e S has a c o m p o s i t i o n s e r i e s i f f holds.

(JORDAN-HOLDER THEOREM):

Any two c o m p o s i t i o n s e r i e s o f S have t h e

same l e n g t h and t h e same f a c t o r s up t o isomorphism and rearrangement. Proof:

.3.3.1

and t h e i n v a r i a n c e o f t h e l e n g t h i n 3.3.2 f o l l o w f r o m t h e i n t e r p r e -

t a t i o n o f c o m p o s i t i o n s e r i e s as maximal c h a i n s i n V ( S ) and t h e r e s u l t s f r o m 111.2. The i n v a r i a n c e o f t h e f a c t o r s i s t h e n shown by i n d u c t i o n on t h e l e n g t h o f a comp o s i t i o n s e r i e s , which because o f t h e upper s e m i m o d u l a r i t y o f V ( S ) (and t h e Given two atoms

symmetry of t h e s i t u a t i o n ) reduces t o t h e f o l l o w i n g argument: 0

EA\B}

= {B,{a)la

and

T

= { C , { a j l ~ r E A\C> o f V ( S ) , show t h a t S I B i s isomorphic

t o ( S / T ) I C ' , where C ' i s t h e o n l y n o n - t r i v i a l c l a s s i n

UVT/T

E

This i s

V(S/T).

done as f o l l o w s : Since B E A ( S ) , f := nToinc;

Because o f ( M I ) , and Lemma 3.1.1,

E HOm(S[B,S/r).

f has a f a c t o r i z a t i o n f = i n c i l o h , where h r Epi(SIB,S1), and S1 := S/T has t h e base s e t A1.

inc!lcMono(SIID,S1)

One e a s i l y v e r i f i e s t h a t D = h(B) =

rlT(B).

If

Bn C

:

If

Bn C

# 4 , t h e n I B f ) C I = 1 ( o t h e r w i s e S I C would be decomposable because o f

4 , t h e n n T maps a l l elements o f

B bijectively.

( M 7 ) and Theorem 3.1.4) and one a g a i n o b t a i n s t h a t n T maps a l l elements o f B bijectively. Thus h i s t h e r e s t r i c t i o n o f n T t o B and b i j e c t i v e . a unique s t r u c t u r e T w i t h h (M5) t h a t T

=

S1 I h ( B ) .

E

Hom(SJB,T) and h - l

Hence S I B and SIID

E

Because o f (M2) t h e r e e x i s t s Hom(T,SIB).

= S,[h(B)

I t remains t o be shown t h a t S1 [ h ( B ) = S1 I C ' .

I t f o l l o w s from

a r e isomorphic.

L e t g E Epi(S1,S/cm).

Then

R. II: Miihring and F.J. Radennacker

322

gonT = nuvi and t h u s 1gonT(6)l = 1. S i n c e [ r l T ( B ) I > 1, g maps rl,(B) o n t o one element. T h e r e f o r e t h e congruence p a r t i t i o n U V T / T induced b y g has t h e f o r m

{nT(E),{~llG

E

Al\ni(B)l

which p r o v e s n T ( B ) = C ' .

m

Theorems o f t h e Jordan-Holder t y p e a r e by no means s e l f - e v i d e n t ,

f o r i n s t a n c e , l a t t i c e s fi54] o r

hold i n well-structured algebraic theories, cf., semi-automata [ 6 l l .

and do even n o t

So t h e r e have been e x t e n s i v e i n v e s t i g a t i o n s w . r . t .

b r a i c p r o p e r t i e s under which such a theorem holds, c f . f o r example

t h e alge-

"1,

[45],

U s u a l l y t h e " c o m m u t a t i v i t y " o f t h e congruence r e 1 a t i o n s o f t h e [66], [128]. a l g e b r a o r c e r t a i n g e n e r a l i z a t i o n s t h e r e o f [63], [128] a r e b a s i c t o such a theorem. Such a p r o p e r t y does n o t h o l d f o r t h e s t r u c t u r e s c o n s i d e r e d h e r e .

T h i s becomes,

f o r i n s t a n c e , apparent b y t h e f a c t t h a t t h e congruence p a r t i t i o n l a t t i c e s a r e ( i n t h e f i n i t e case) o n l y upper semimodular, whereas f o r a l g e b r a s w i t h commuting congruence r e l a t i o n 5 t h e y a r e modular [13].

I n t h i s r e s p e c t , t h e s t r u c t u r e s con-

s i d e r e d h e r e c o n s t i t u t e a s e p a r a t e case, i n which a Jordan-Holder Theorem h o l d s under c o n d i t i o n s d i f f e r e n t f r o m t h e u s u a l ones.

A f u r t h e r i n d i c a t i o n as t o t h e d i f f e r e n t n a t u r e of t h i s r e s u l t i s an a d d i t i o n a l i n v a r i a n c e ( a Church-Rosser p r o p e r t y ) f o r c o m p o s i t i o n s e r i e s which i s known f o r r e l a t i o n s and c l u t t e r s , and which does n o t h o l d f o r a l g e b r a s , i n g e n e r a l : Any two c o m p o s i t i o n s e r i e s of S have t h e same l a s t f a c t o r up t o isomorphisni.

I n t h e general model, t h i s i n v a r i a n c e does n o t f o l l o w f r o m (Ml)

-

(M8), b u t

r e q u i r e s an a d d i t i o n a l assumption, w h i c h h o l d s i n t h e s p e c i a l cases o f 1.2 - 1.4. (M12)

L e t SIB and S I C be p r i m e s u b s t r u c t u r e s o f S such t h a t B fl C # $. Then S I B and S I C a r e i s o m o r p h i c .

I f t h e c a t e g o r y K f u l f i l l s ( M l ) - (M8) and (M12), t h e n any two c o m p o s i t i o n s e r i e s o f a s t r u c t u r e S have t h e same l a s t f a c t o r up t o isomorphism.

Theorem 3.3.3:

Proof:

I f , i n t h e p r o o f o f Theorem 3.3.2,

UVT

#

nl,

t h e n t h e l a s t f a c t o r i n any

c o m p o s i t i o n s e r i e s o f S i s i s o m o r p h i c t o t h e l a s t f a c t o r o f S / O V T by t h e i n d u c t i v e a s s u m t i on. If

OVT

= nl,

then

Bn C

#

$

and S I B = S I C because o f (M12).

Since

two c o m p o s i t i o n s e r i e s c o n s i d e r e d i n t h e p r o o f o f Theorem 3.3.2 and S, S / T , S/svr,

i.e.,

S / u and

S/T

are the l a s t factors.

t h a t SIB and S/T as w e l l as S I C and S/u a r e i s o m o r p h i c . a l l f o u r f a c t o r s S I B , SIC, S/o,

S/T are isomorphic. m

UVT

= nl, t h e

a r e S, S / u , S/UVT,

Theorem 3.3.2

shows

So, because o f (M12),

323

Substitution decomposition for discrete structures

S i m i l a r t o t h e examples i n 111.1, t h e r e e x i s t subcategories o f KO i n which (1:12) holds b u t n o t

Example 3.3.4:

(M11), and v i ce-versa.

L e t 0 and G be t h e p a r t i a l o r d e r and i t s a s s o c i a t e d c o m p a r a b i l i t y

graph o f F i g u r e 3.3.

A composition series o f

0 ( i n activity-on-arc

representa-

t i o n ) and a corresponding one f o r G a r e g i v e n i n F i g u r e 3.3 t o g e t h e r w i t h t h e composition series

composition s e r i e s

factors

factor.

0

*

I

0

in

c .

F i g . 3.3 Composition

W

Series for a P a r t i a l Order and f o r i t s Comparabi 1it y

0

0

Q

9-

m

.

.

Graph.

R.H. Mdhring and F.J. Radermacher

324

Note t h a t each composition s e r i e s f o r 0 induces one f o r G

associated f a c t o r s .

(because of Theorem 1.5.1),

b u t n o t vice-versa.

S i m i l a r l y , t h e g i v e n composition

s e r i e s of 0 a l s o induces composition s e r i e s f o r the c l u t t e r s C := C(G) o f c - m a x i ma1 cliques, arc] o f c-maximal independent sets, and b[C]

o f 5-minimal c l i q u e

separating sets, and the associated monotonic Boolean f u n c t i o n s ( c f . Example

1 . 1 . 1 ) v i a the connections given i n 1.5.

This a l s o holds f o r t h e r e s p e c t i v e

factors.

111.4

THE COMPOSITION TREE

We s h a l l show t h a t the r e p r e s e n t a t i o n o f t h e decomposition p o s s i b i l i t i e s i n a tree, which i s known f o r c l u t t e r s n38],

[2q,

p81 and

graphs [32],

p a r t i a l orders

i s a l s o v a l i d i n t h e general theory, i f , f o r A ( S ) , t h e a d d i t i o n a l c o n d i t i o n

(M9) i s imposed, which s t a t e s t h a t i f C1,C2

E

A ( S ) overlap, then C1\C2

E

A(S).

This c o n d i t i o n w i l l be assumed throughout 111.4 and 111.5 and i s v a l i d for t h e s t r u c t u r e s considered i n Section I .

Related r e s u l t s f o r t h e f i n i t e case based

o n l y on the p r o p e r t i e s o f A ( S ) a r e a l s o contained i n c49]. The c o n s t r u c t i o n o f the t r e e i s based on two decomposition p r i n c i p l e s , the f i r s t of wh'ich i s the "maximal d i s j o i n t decomposition".

Definition:

Let S be a s t r u c t u r e on A .

a partition

a* o f

A maximal d i s j o i n t decomposition o f S i s A i n t o c-maximal S-autonomous s e t s B # A.

The f o l l o w i n g decomposition p r i n c i p l e i s then obvious.

Theorem 3.4.1: I f S admits a maximal d i s j o i n t decomposition a*, then each Sautonomous s e t o t h e r than A i s SIB-autonomous f o r some B E u*. I f , furthermore, a* V(S)\{a?J,

E

V ( S ) , then u* i s the coarsest congruence p a r t i t i o n i n

and S/O* i s prime.

The existence o f a maximal d i s j o i n t decomposition o f S depends t o some degree on the non-existence o f special q u o t i e n t s o f S.

Definition:

A s t r u c t u r e S on A i s c a l l e d s e m i - l i n e a r i f t h e r e i s a l i n e a r order-

i n g 6 on A such t h a t A ( + ) c A ( S ) , i . e . i f A ( S ) contains a l l on the b a s i s o f 4 as usual: a a R - 6 4 a r~

6 andaP B

B -a
-

MAX-SEPARABLE OPTIMIZATION PROBLEMS Max-separable o p t i m i z a t i o n problems are d e f i n e d as o p t i m i z a t i o n problems over

E~

o f t h e f o l l o w i n g form: f ( X ) :max f j ( x j ) d min o r max jaN subject t o

where S1

u S2 = S

5

C1,

max r .1J . ( x .J) jeN

a b

V

iES1

max r i j ( x j ) jrN

\< bi

W

i E S 2

N = 11 ,..., n l , bi,

..., ml,

k . , K . a r e g i v e n elements o f J J

E.

We s h a l l now formulate general c o n d i t i o n s under which e x p l i c i t formulae f o r optimal s o l u t i o n s of (P) can be found.

This extends t h e r e s u l t s g i v e n i n [3]

We s h a l l confine ourselves t o m i n i m i z a t i o n problems o f t h e form (P).

[4].

w.., L~ and M as f o l l o w s :

L e t us d e f i n e t h e s e t s Vij, E

I

kj 6 xj

€ E

1

kj

Vij

= 1X. t

w l. J.

= {x.

J

L.

1

1

= 1j E N

M = fx

Lma 1

E

E

n

1J

6

r . . xj) 1J

Kj,

x j < Kj, rij

3

i E S1, j r N,

bi}

x . ) 4 bil J

w i E S2,

j E N,

V . . # 01 W i € S 1 , 1J

I

max r i j ( x . ) J j EN

%

bi w i E S , ,

[ 3 i 0 t S1 such t h a t Li

0

=

0J

-

k . 4 x j 6 K . W j e N}. J J

M =

0,

and

On ma-separable optimization problems

Proof

0, i o e S1 and l e t X Therefore

L e t Li

=

f o r a l l j E N?

E E ~ .

359

Then r . . ( x . ) < bi and k . 6 x . 1oJ J 0 J J




i cS1 3 j(i)

N such t h a t x j f i )

E.

Vij(.)

M, t h e n

E

max r i j ( k j ) jeN

so t h a t W i e S 1 3 j ( i )

E

= r lj(i) .

(

~ 3 (bi ~w i ~E S1, ,

~

N such t h a t xj(.)

E Vij(i).

b e f u l f i l l e d and l e t i be an

L e t t h e r i g h t hand s i d e o f t h e - > - r e l a t i o n a r b i t r a r y i n d e x o f S1. rij(i)(xj(i)

E

j e N.

Then xj(i)

S .

Vij(i)

>

bi and thus max r ( x . ) jaN i j J

>/

f o r some j(i) rij(i)(xj(i))

>, bi

N so t h a t and X

E

M, Q.E.D.

THE EXPLICIT SOLUTION OF MAX-SEPARABLE OPTIMIZATION PROBLEMS We s h a l l assume now t h a t problem (P) s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s .

(1)

A l l nonempty Vij,

I. J

(2)

=

{x. J

1

k. J

W . . are subintervals o f 1J

4 x. &

J

KjI;

F o r f i x e d j t h e s e t s Vij

form a chain ( w . r . t .

set inclusion).

( 3 ) The f u n c t i o n s f . a t t a i n t h e i r minimum on any s u b i n t e r v a l o f I

j.

J

If S 2 # 0 and ( 1 ) i s s a t i s f i e d , t h e n f o r a l l j c N: R.

J

5

ix. J

4 E

1

r..(x.) 1J J

6 bi,

x . 6 K., W i & S 2 } = (l W . . J J ieS2 'J

k. J

and t h u s f o r each j e i t h e r R . = 0 and hence t h e s e t o f f e a s i b l e s o l u t i o n s o f (P) J i s empty o r t h e r e e x i s t k ! , K'. e E such t h a t J J

R. = {x. J

J

E

E

1

k'. J

4

X.

J

c K!]. J

Therefore ifR . # 0, W j E N, t h e n t h e o r i g i n a l o p t i m i z a t i o n problem can be J reduced t o t h e problem of t h e form (P) w i t h these new bounds k ' K j and w i t h

j'

K. J

K. Zimmermann

360 S1 = S , S2 =

0.

Therefore we can c o n f i n e ourselves i n the sequel t o problems ( P )

w i t h S1 = S and S 2 = 0.

We s h a l l show t h a t under t h e c o n d i t i o n s ( l ) , ( 2 ) , ( 3 )

e x p l i c i t fonnulae f o r t h e components of an optimal s o l u t i o n o f ( P ) can be given. Theoran 2

L e t us suppose t h a t t h e s e t o f f e a s i b l e s o l u t i o n s o f ( P ) i s nonempty,

S1 = S, S 2 = 0 and c o n d i t i o n s ( 1 )

xj(i)

-

( 3 ) are s a t i s f i e d .

L e t us d e f i n e elements

and sets S ( j ) , H ( j ) as f o l l o w s : f. . J(1)

(X.J ( i ) )

min

min

= j e ~. x. r ~

j

i

L e t X = ( x l,...,xn)

x

Then

fj(xj),

ij

be d e f i n e d according t o t h e f o l l o w i n g r e l a t i o n s :

i s an optimal s o l u t i o n o f t h e m i n i m i z a t i o n Droblem ( P ) .

I t f o l l o w s immediately from Lemma 1 t h a t Li # 0 t/ i t S so t h a t f o r each i C S there e x i s t s j ( i ) e N such t h a t f . . . j = m i n min f j ( y j ) and ~ ( 1 ) ~ ( 1 ) jcLi y.rv

Proof

(x.

i e S(J(i)).

Then according t o ( 2 ) ^x

j(i)

E

J ij H(j(i)) c V.. 1 J ( i ) so t h a t rij(.)(Xj(.))

3 bi and thus . .(X .) 5 r i j ( i ) ( i j ( i ) ) max r lJ

3 bi,

v iE S

jeN

and

X

i s a f e a s i b l e s o l u t i o n of ( P ) .

I t remains t o prove t h a t

f ( X ) 5 f ( X ) = max f . ( i . ) = f ( j j c ~J J P P f o r a l l f e a s i b l e s o l u t i o n s of (P). solution.

If

=

0,

IfS(p) # 0 and f p ( x p ) =

f(X).

f (x )

P

P

L e t X = (x ly...,xn)

fp(ip).

then again f ( X ) = max [ f . ( x . ) ] + j , ~ J J

I t remains t o i n v e s t i g a t e t h e case t h a t c

be an a r b i t r a r y such

we have

f ( x ) = max f j ( x j ) P P jcN

= f(X) holds.

f (x ) P

P

+

f

P

( iP)

# 0 and a t t h e same time

We s h a l l show i n t h i s case t h a t t h e r e

On max-separable optimization problems

e x i s t s an i n d e x

If fp(xp)


/ f ( x ). t t p p

fp(Xp),

then x

e x i s t s an i n d e x i ( p )

min Y$"i

fp(yp) =

(PIP

36 1

P

+

L e t us n o t e t h a t a c c o r d i n g t o ( 2 ) t h e r e

such t h a t H(p) =

E

min min f . ( y . ) . J J jaL i( p) Y j e V i ( PI j

(k

) = and t h u s f %PIP P P Since X i s a f e a s i b l e s o l u t i o n o f

( P ) t h e r e e x i s t s according t o Theorem 1 an i n d e x

e = l(i(D))

such t h a t xt

V. 1(P)t

and we o b t a i n :

so t h a t

I n a l l p o s s i b l e cases we o b t a i n e d f ( X ) > f ( X ) and t h e r e f o r e X i s an o p t i m a l s o l u t i o n o f t h e m i n i m i z a t i o n problem ( P ) , Q.E.D.

0

Remark 1 The " l i n e a r " o p t i m i z a t i o n problem C ' c o n d i t i o n s (A @ X)i r e l a t i o n s 6.

+,

i = 1,

: bi,

= considered i n

[3]

... ,my , [4]

X e

E

X+

i s a s p e c i a l t y p e o f max-separable o p t i -

m i z a t i o n problem w i t h f . ( x . ) = c . @ x . and r . . ( x . )

J

Remark 2

J

J

The c o n d i t i o n s (l), (2),

max o r m i n under t h e

n , where : stands f o r one o f t h e

J

1J

J

= aij

0 xj.

(3) are f u l f i l l e d f o r instance i f

E

i s a subset

o f r e a l numbers, f . a r e c o n t i n u o u s and monotone, and rij a r e c o n t i n u o u s and

J

monotone f o r a l l i E S2, j

E

N and c o n t i n u o u s and s t r i c t l y i n c r e a s i n g f o r a l l T h i s case was

i E S1, j e Li o r s t r i c t l y decreasing f o r a l l i a S1, j r Li. i n v e s t i g a t e d i n a s l i g h t l y more g e n e r a l f o r m i n

[q.

L e t u s remark t h a t i t can

b e shown t h a t ( I ) , ( Z ) , (3) r e m a i n f u l f i l l e d f o r t h e f u n c t i o n s rij(xj) = min ( r i j ( x j ) , a i j ) ,

Fij(xj)

= max ( r . . ( x . ) , a . . ) 1J

J

1J

=

where r . . s a t i s f y t h e assump1J

t i o n s mentioned above and a . . a r e g i v e n numbers. 13

REFERENCES

p]

Cuninghame-Green, R.A., Minimax Algebra, L e c t u r e Notes i n Economics and Mathematical Systems, ( S p r i n g e r Verlag, 1979) 166.

[Z]

Zimmermann, K., The e x p l i c i t s o l u t i o n o f max-separable o p t i m i z a t i o n problems, Ekonornicko-matematickj obzor, 4 (1982).

362

K. Zimmermann

131

Zimnermann, U., On some extremal o p t i m i z a t i o n problems, Ekonomickomatematick5 obzor, 4 ( 1 9 7 9 ) .

[4]

Zimmermann, U., L i n e a r and c o m b i n a t o r i a l o D t i m i z a t i o n i n ordered a l g e b r a i c s t r u c t u r e s , Annals o f D i s c r e t e Mathematics, 10 (North-Holland, 1981).

Annals of Discrete Mathematics 19 (1984) 363- 382 0 Elsevier Science Publishers B.V. (North-Holland)

36 3

MINIMIZATION OF COMBINED OBJECTIVE FUNCTIONS ON INTEGRAL SUBMODULAR FLOWS

U. Zimmermann Mathematisches I n s t i t u t U n i v e r s i t a t zu KBln 0-5 K o l n 41 Weyertal 86

We develop a n e g a t i v e c i r c u i t method and an augmenting p a t h method f o r t h e m i n i m i z a t i o n o f o b j e c t i v e f u n c t i o n s on i n t e g r a l submodular f l o w s . The c l a s s o f o b j e c t i v e f u n c t i o n s c o n s i d e r e d admits c e r t a i n combinations o f t h e usual l i n e a r o b j e c t i v e f u n c t i o n and a f i x e d - c o s t o b j e c t i v e f u n c t i o n .

1.

INTRODUCTION

I n sequence of a paper o f EDMONDS and GILES [1976] v e s t i g a t e d by s e v e r a l a u t h o r s . t i e s a r e due t o FUJISHIGE [1978], [1980].

submodular f l o w s have been i n -

R e l a t e d models w i t h s i m i l a r c o m b i n a t o r i a l p r o p e r HASSIN ( n 9 7 8 ] ,

[1981])

F o r i n t e g r a l submodular f l o w s , FRANK [1982]

m i n i m i z i n g (maximizing) l i n e a r o b j e c t i v e f u n c t i o n s .

and LAWLER and MARTEL

developed a n a l g o r i t h m f o r T h i s method i s q u a s i p o l y -

nomial and, i n t h e case o f 0-1-valued submodular f l o w s , p o l y n o m i a l .

A polynomial

method f o r t h e i n t e g r a l case i s supposed t o be p o s s i b l e u s i n g s c a l i n g t e c h n i q u e s . The m i n i m i z a t i o n o f c e r t a i n o b j e c t i v e f u n c t i o n s on R-valued submodular f l o w s , where R i s a t o t a l l y o r d e r e d group ( r i n g ) i s discussed i n ZIMMERMANN p982b-J.

In

p a r t i c u l a r , a n e g a t i v e c i r c u i t method i s based on a theorem showing t h a t t h e d i f f e r e n c e o f two submodular f l o w s i s a c i r c u l a t i o n i n a c e r t a i n a u x i l i a r y graph. I n t h e f o l l o w i n g , we use t h a t ' d i f f e r e n c e ' theorem i n o r d e r t o m i n i m i z e f u r t h e r d i f f e r e n t o b j e c t i v e f u n c t i o n s on i n t e g r a l submodular f l o w s . t i o n s g e n e r a l i z e f u n c t i o n s i n FRIESDORF and HAMACHER ([198la],

These o b j e c t i v e f u n c n981b]).

They

p r o v e t h e v a l i d i t y o f a n e g a t i v e c i r c u i t method as w e l l as o f a s h o r t e s t augmentirg p a t h method f o r m i n i m i z i n g t h e s e f u n c t i o n s on network f l o w s o f maximum f l o w v a l u e . We develop b o t h methods i n t h e g e n e r a l case.

I n p a r t i c u l a r , a s h o r t e s t augmenting

p a t h method f o r m i n i m i z i n g (maximizing) l i n e a r o b j e c t i v e f u n c t i o n s on submodular flows i s given. S e c i i c r , ? c n n t a i n s t h e necessary c o m b i n a t o r i a l r e s u l t s ; i n p a r t i c u l a r , a c o n c i s e statement of t h e ' d i f f e r e n c e ' theorem.

I n s e c t i o n 3, we i n t r o d u c e o b j e c t i v e

f u n c t i o n s combining a l i n e a r o b j e c t i v e and a f i x e d - c o s t o b j e c t i v e on submodular

U.Z i m m e m n n

364 flows.

Some examples a r e given i n a theorem.

The c l a s s o f o b j e c t i v e f u n c t i o n s

considered i n general i s d e f i n e d by s t a t i n g c e r t a i n important p r o p e r t i e s . The negative c i r c u i t method i s developed i n s e c t i o n 4 and t h e augmenting p a t h method i s developed i n s e c t i o n 5.

2.

SUBMODULAR FLOWS

EDMONDS and GILES p976] discuss a r i c h combinatorial s t r u c t u r e i n c l u d i n g network flows, polymatroid i n t e r s e c t i o n s and d i r e c t e d c u t s .

I n a previous paper p982b]

we g e n e r a l i z e t h e i r concept i n a c e r t a i n a l g e b r a i c sense and develop a negative c i r c u i t method f o r t h e d e t e r m i n a t i o n o f submodular f l o w s m i n i m i z i n g c e r t a i n That approach i s based on an a u x i l i a r y

f u n c t i o n s on r m g u a l u e d submodular f l o w s .

graph which FRANK p982) uses f o r t h e development of a polynomial a l g o r i t h m f o r maximizing a r e a l - v a l u e d l i n e a r o b j e c t i v e f u n c t i o n on 0-1-valued submodular f l o w s .

I n t h e f o l l o w i n g we l i s t d e f i n i t i o n s and r e s u l t s on i n t e g e r - v a l u e d submodular flows. denote a digraph w i t h v e r t e x s e t V and a r c s e t E.

L e t G = (V,E)

V

A family F S 2

i s called a crossing family i f [S

n T # 0, SU T #

= [S flT,

SU T E F]

(2.1)

Two members S , T o f F w i t h S 6 T, T SS, S flT # 0 and 9 V a r e c a l l e d c r o s s i n g members o f F. A f u n c t i o n h: F + Z i s c a l l e d submodular (on F) i f (2.2) h(S) + h(T) a h ( S n T) + h ( S U T )

f o r a l l S, T E F. S

T

f o r a l l crossing members of 6(s).

Let

5

:= V\S.

F.

The s e t of a l l a r c s l e a v i n g S gV i s denoted by

Then 6 f S ) c o n t a i n s t h e arcs e n t e r i n g S.

f o r A C E we d e f i n e x ( A ) := Zed\

For x

E ZE and

x(e).

E

A vector x E Z s a t i s f y i n g x ( ~ ( S ) )- x ( & ( S ) ) 4 h(S) i s c a l l e d an ( i n t e g e r - v a l u e d ) submodular f l o w . and upper bounds on t h e arcs, i . e . flow x i s called feasible, i f

e

II C

(Z&Jr-m))E,

(5

E

F)

(2.3)

With r e s p e c t t o given lower and c

E

(m{-l)E, a submodular

6 x .c< c.

L e t x be a f e a s i b l e , submodular f l o w .

A member S E: F i s c a l l e d t i g h t ( w i t h

respect t o x ) i f (2.3) holds w i t h e q u a l i t y .

Our previous n o t i o n ' s t r i c t ' i s

replaced by ' t i g h t ' ( c f . ZIMMERMANN c982b])

s i n c e ' s t r i c t ' seems t o be misunder-

standable when used f o r a n o n - s t r i c t i n e q u a l i t y .

NOW, a : 2'

+

ZZ

, defined

by

Combined objective functions on integral submolar flows

-

o ( S ) := X ( 6 ( S ) )

365

X(6(S))

( w i t h r e s p e c t t o x ) i s a modular f u n c t i o n , i . e . a(s)

f o r a l l S, T

For v

GV.

+ U(T)

n T)

=

t

o(s u

T)

(2.4)

V, l e t P ( v ) denote t h e i n t e r s e c t i o n o f a l l t i g h t s e t s

E.

S w i t h v E 5.

The a u x i l i a r y graph Gx = (V,Ex)

Ex : = E+U

E-u Eo,

contains three types o f arcs ( w i t h respect t o x),

d e f i n e d by

Et := I u v E- := { v u Eo := {uv

I I I

x ( u v ) < ~ ( u v ) , uv E E }

"forward arcs",

~ ( u v )< ~ ( u v ) , uv E E }

"backward a r c s " ,

v h P(u); u,v

E

V,U #

"red arcs".

V l

The backward a r c corresponding t o e i s denoted by Z , i . e . i f e = uv t h e n E = vu. The n o t i o n i s m a i n l y drawn from network f l o w t h e o r y . those i n FRANK [1982]

The r e d a r c s c o i n c i d e w i t h

We i n t r o d u c e p o s i t i v e , upper

w i t h reversed d i r e c t i o n .

bounds d on Ex, d e f i n e d by d ( v u ) : = ~ ( u v )- ~ ( u v )resp. d ( u v ) = c ( u v )

-

x ( u v ) on

backward r e s p . f o r w a r d arcs, and by d ( u v ) := m i n {h(S) A vector

on r e d a r c s .

EX

AX E

Z+

-

o(S)

I

S

E

F, u

E

S, v 4 Sl

i s called a circulation i f

AX(dx(i))

-

AX(dx(V)) = 0

(V c v )

where 6x(S) denotes t h e s e t o f a l l a r c s f r o m Ex l e a v i n g S c V.

6+ ( S ) and 6 0 ( S ) A nonnegative c i r c u l a t i o n i s c a l l e d ( + ) - f e a s i b l e [feasible] i f i t s a t i s f i e s t h e upper bounds on E,- [Ex]. A c i r c u l a t i o n A X i n GX E d e f i n e s a v e c t o r x ' E Z i n G by

are defined similarly.

x'(UV) := ( X @ AX)(uV) := X(UV)

t

AX(lrV)

-

AX(VU)

f o r a l l uv e E (we i n t e r p r e t Ax(uv) by 0 i f an a r c does n o t o c c u r i n Gx). F o r a g i v e n f e a s i b l e , submodular f l o w x ' , we d e f i n e

-

~'(uv) ~(uv) Ax(uv) :=

x(vu)

-

~'(vu)

[o f o r a l l uv Now, f o r Ax

E

i f ~ ' ( u v )> ~ ( u v ) , uv

E

E+,

i f x ( v u ) > ~ ' ( v u ) ,uv r

E-,

otherwise

Ex.

t

EX

Z+

, we c a l l Ax conformal

if

Ax(UV) AX(VU) = 0

U.Zimmermann

366

f o r a l l p a i r s o f forward/backward arcs which a r e d e r i v e d from t h e same a r c i n E C l e a r l y , Ax i n ( 2 . 5 ) i s conformal.

I n a previous paper we prove t h e f o l l o w i n g theorem 2.4).

theorem i n a more general form (p982b],

Theorem 2.1 L e t x, x ' be f e a s i b l e , submodular f l o w s .

Then t h e r e e x i s t s a conformal, ( ? ) f e a s i b l e c i r c u l a t i o n Ax i n Gx such t h a t x ' = x @ Ax. It i s well-known t h a t such a c i r c u l a t i o n can be decomposed i n p o s i t i v e c i r c u i t

flows, i . e . AX = Xi

where each Ci

Ax(Ci,ni)

i s a ( d i r e c t e d ) c i r c u i t i n Gx and t h e c i r c u l a t i o n A X ( C ~ , ~has ~)

constant value ni > 0 on t h e arcs o f Ci

b u t vanishes on a l l o t h e r arcs.

The

number o f c i r c u i t s i n t h a t decomposition can be bounded by t h e number o f p o s i t i v e valued arcs i n Ax.

E + U E- i s an a r c o f some c i r c u i t i n t h a t decomposiTherefore, t h e decomposition i s c a l l e d conformal, i . e . i f

I f uv

t i o n then A X ( U V )> 0.

Q

t h e forward (backward) a r c e occurs i n some c i r c u i t then t h e corresponding backward (forward) a r c

6

does n o t occur i n any c i r c u i t o f t h a t decomposition.

p a r t i c u l a r , e E. C i m p l i e s t h a t property.

6 6 C.

In

I n t h i s paper we o n l y consider c i r c u i t s w i t h

C l e a r l y , i f x and x ' a r e f e a s i b l e , submodular f l o w s w i t h x ( e ) =

x ' ( e ) f o r some e E E then no c i r c u i t i n t h e conformal decomposition o f t h e ' d i f f e r e n c e ' c i r c u l a t i o n AX can c o n t a i n e o r Z . D i f f e r e n t from network f l o w theory i t may happen t h a t x

8 Ax

i s not a feasible,

submodular flow, even i f Ax i s a f e a s i b l e p o s i t i v e c i r c u i t f l o w i n Gx.

The

f o l l o w i n g p r o p e r t y o f a c i r c u i t excludes such a behaviour. L e t ~ x ( C , r i ) be a f e a s i b l e , p o s i t i v e c i r c u i t f l o w i n Gx. C does n o t c o n t a i n consecutive r e d arcs.

W.1.0.g.

we assume t h a t

Now, we consider another graph Gc

corresponding t o C i n which t h e r e d a r c s a r e t h e v e r t i c e s and i n which two vert i c e s uv and r s are l i n k e d by an a r c (uv,rs)

i f f us i s a r e d a r c i n Gx.

We c a l l

C admissible i f GC does n o t c o n t a i n a d i r e c t e d c i r c u i t .

I n o u r above mentioned paper (p982b],

theorem 2.6) we prove t h e f o l l o w i n g theorem

Theorem -2.2 L e t x be a f e a s i b l e , submodular f l o w .

I f Ax(C,n)

i s a f e a s i b l e c i r c u i t f l o w on an

admissible c i r c u i t C i n Gx, then X @ A X i s a f e a s i b l e , submodular f l o w .

367

Combined objecrive functions on integral submolar flows C l e a r l y , a c i r c u i t f l o w Ax(C,u) capacity, i . e . 0 6

p Q

i s feasible i f

u does n o t exceed t h e minimum a r c

d(C) f o r

I

d(C) = min Ed(uv)

uv c C l .

I f d(C) i s a t t a i n e d on a backward a r c e o f C w i t h p o s i t i v e w e i g h t b ( 5 ) ( c f .

s e c t i o n 3 ) and d(C) > 1 t h e n we d e f i n e a reduced c a p a c i t y d(C) : = d(C) o t h e r w i s e d(C) : = d(C). sections

3.

-

1;

The reduced c a p a c i t y w i l l be used i n t h e f o l l o w i n g

.

COMBINED OBJECTIVE FUNCTIONS

I n a r e c e n t paper FRIESDORF and HAMACHER 1 9 8 l b ] discussed t h e m i n i m i z a t i o n o f

I n t h e f o l l o w i n g we g e n e r a l i z e

c e r t a i n o b j e c t i v e f u n c t i o n s on network f l o w s . these f u n c t i o n s i n some a l g e b r a i c c o n t e x t .

Although t h e o b j e c t i v e f u n c t i o n s

c o n s i d e r e d a r e d i f f e r e n t f r o m t h o s e i n ZIMMERMANN [1982b]

t h e discussion f o l l o w s

quite similar lines. L e t (R,t,c)

be a t o t a l l y ordered, commutative and d i v i s i b l e group w i t h n e u t r a l

element 0.

R i s assumed t o be n o n t r i v i a l , i . e . R # {Ol.

Then, R i s a t o t a l l y

o r d e r e d vectorspace o v e r t h e f i e l d Q o f t h e r a t i o n a l numbers. The e x t e r n a l comn n p o s i t i o n i s denoted i n t h e usual m u l t i p l i c a t i v e form, i . e . (Fii,a) + m a := x, where x i s t h e s o l u t i o n o f t h e e q u a t i o n n.a = m.x and n - a := a t a t . ..+a

( n times),

!E Q and f o r a l l a E R. L e t R, := { a E R I a + 01. F o r a d e t a i l e d for all ! m d i s c u s s i o n o f t h e a l g e b r a i c s t r u c t u r e s appearing h e r e and i n t h e f o l l o w i n g sect i o n s we r e f e r t o ZIMMERMANN c 9 8 1 1 .

We remark t h a t , due t o c o m m u t a t i v i t y ,

d i v i s i b i l i t y o f R can be assumed w i t h o u t l o s s o f g e n e r a l i t y . L e t W := I x

E.

t

72

I

L 6 x 6 u } where .t and u a r e t h e l o w e r and upper bounds o f t h e

u n d e r l y i n g submodular f l o w problem ( c f . s e c t i o n 2 ) . F o r g i v e n w e i g h t v e c t o r s E E a E R and b R, we c o n s i d e r t h e l i n e a r o b j e c t i v e f u n c t i o n f: W + R d e f i n e d by

f(xf

= CerE x(e).a(e)

and t h e f i x - c o s t o b j e c t i v e f u n c t i o n g: W

+

R, d e f i n e d by

c

g(x) =

b(e).

x ( e ) > f i (E 1 With r e s p e c t t o a f e a s i b l e , submodular f l o w x we i n t r o d u c e w e i g h t s i n Gx: a(e)

i f e e E,, i f e r E-, otherwise,

U.Zimmermann

368

b x ( e ) :=

E,

. b(e)

if e

-

b(e)

i f e E E-

o

otherwise

l

E

A

x ( e ) = e(e),

A

x ( e ) = e(e)

.

Then, f o r a conformal, ( + ) - f e a s i b l e , c i r c u l a t i o n

t h e weights i n Gx r e f l e c t the

AX

change o f the o b j e c t i v e f u n c t i o n values i f x i s replaced by x fx(Ax) := ZecEx gx(AX) :=

+ 1,

Ax(e).ax(e)

0 Ax.

Let

,

bx(e).

1

Ax( e)>O

Proposition 3.1 L e t x be a f e a s i b l e , submodular f l o w and l e t Ax be a conformal, c i r c u l a t i o n i n Gx w i t h conformal decomposition

AX^

(1)

f ( x @ Ax) = f ( x )

+

fx(Ax) =

f(X)

(2)

s ( x @AX) 6 g ( x )

+

!iIx(AX)

g ( x ) + ZiieI g x ( A x i ) .

Q

(+)-feasible

:= ~ x ( C ~ , r tf ~ o r) i E I. Then

fx(Axi),

t EieI

I f Ax = Ax(C,p) w i t h 0 4 p 6 i ( C ) then

g ( x 8 Ax) = g ( x ) + SX(AX).

(3)

( 1 ) i s obvious. ( 2 ) and ( 3 ) f o l l o w from the f a c t t h a t x and Ax are i n t e g r a l . With r e s p e c t t o t h e l e f t i n e q u a l i t y , t h e c o n t r i b u t i o n t o t h e change i n

Proof.

t h e value o f g i s c a l c u l a t e d e x a c t l y f o r a l l forward arcs as w e l l as f o r t h e backward arcs E(e)

t

e with

x ( e ) = L(e)

t

I f f o r some backward a r c

1.

e with

x(e)

1 we have x ( e ) = AX(^) then g(x

0 Ax)


0. With respect t o t h e r i g h t i n e q u a l i t y , i f t h e conformal decomposition contains more than one c i r c u i t using the same forward a r c e w i t h x ( e ) = n(e), we have g(X) + gx(AX) < g(X) + Z i e ~ S,(Axi)* Contributions f o r other arcs are calculated exactly. 0c

p 6

I f Ax = Ax(C,p) w i t h

d(C) then s t r i c t i n e q u a l i t i e s can n o t occur since Ax(e) < X(e) f o r a l l

backward arcs i n C w i t h x ( e ) > e(e)

t

m

1.

The l i n e a r and t h e f i x e d - c o s t o b j e c t i v e f u n c t i o n a r e combined t o a s i n g l e object i v e f u n c t i o n F by a f u n c t i o n r: D ordered s e t .

Then F: W

+.

-+

T with D

T i s d e f i n e d by

5;

R2 where (T, r(f(x)+a,g(x)+b),

i = 1,2,...,~,

r ( f ( x ) + a i ,g(x)+Bi

+ r ( f ( x ) ,g(x)) ,

i = s+l,

...,k.

Since 0 i s convex, we f i n d bi

:= ( f ( x )

bi := ( f ( x )

>

Now

+

t

1 $atai),g(x)

1 1 + ;Ia,g(x) + 7B) e

and a : = ( f ( x ) r(bi)

+ y1i , g ( X )

r ( a ) f o r i = 1,2

0.

,... ,s

1

7Ei)

+

i = 1,2,...,s

d D

1 $B+Bi))

eD

..., k

i = Stl,

T h e r e f o r e (3.3) i m p l i e s

and r ( b i )

k

i l S ( b -a) = C (ai-a,Bi-B) i=1 i=1 C

3

t

r ( a ) f o r i = s+l, 1

'

...,k .

k

C (ai,Bi) i=s+l

= 0

t o g e t h e r w i t h p r o p e r t y (3.4) l e a d s t o t h e c o n t r a d i c t i o n r ( a ) > r ( a ) .

m

P r o p o s i t i o n 5.1 shows t h a t augmentation l e a d s t o a new f e a s i b l e s o l u t i o n .

For

c o n s e r v a t i o n o f t h e minimum c o s t - p r o p e r t y i t i s o f t e n necessary t o r e s t r i c t t h e choice o f p t o 1.

Let

i(c)

i f g x ( A x ( c , l ) ) \< 0

d(C) := otherwise f o r an augmenting c i r c u i t C i n Gx.

C l e a r l y , d(C) > 0.

Theorem 5.2 Let x

E

P be o f minimum c o s t .

w e i g h t i n Gx t h e n x @Ax(C,u) a l l integral p , 0 6 Proof.

LI

I f C i s a s h o r t augmenting c i r c u i t o f minimum i s a f e a s i b l e , submodular f l o w o f minimum c o s t f o r

6 d(C).

L e t x ' E Pe(X) w i t h A = x ( e ) t p .

By Theorem 2.1, t h e r e e x i s t s a

U. Zimmemann

378

conformal f l o w Ax i n Gx w i t h x ' = x @ Ax. a conformal decomposition o f

AX

with

L e t Axi

1

1

x

1

1 f o r a l l i. W.1.o.g.

rl. = 1

denote t h e augmenting c i r c u i t s i n t h e d e c o m p o s i t i o n ( u 6 k ) . E . = g ( A X . ) f o r a l l i.

,..., k be C1 ,..., Cu

i = 1,2

= Ax(C.,rl.),

Let

let ai

= fx(axi),

1

Then

+

r(f(x)+cci , 9 ( ~ ) + 6 ~ ) r ( f ( x ) , s ( x ) ) for all i

>

u.

P r o p e r t y (3.4) i m p l i e s r(f(x) +

1 ai,g(x) i> V

+

E

i> u

Bi)

3

r(f(x),g(x)).

The c o n s i d e r e d arguments b e l o n g t o D s i n c e e v e r y s u b s e t o f t h e c o n s i d e r e d c i r c u i t s leads t o a f e a s i b l e f l o w ( n o t n e c e s s a r i l y submodular). we can s h i f t t h e i n e q u a l i t y f r o m ( f ( x ) , g ( x ) )

to (f(x)

By p r o p e r t y 3.4

+ z

ai,g(x) idu

+ .z B i ) . l