Linear and combinatorial optimization in ordered algebraic structures, Volume 10 (Annals of Discrete Mathematics)

ANNALS OF DISCRETE MATHEMATICS

mathematics Monaging Editor Peter L. HAMMER, University of Waterloo, Ont., Canada

Advisory Editors C. BERGE, UniversitC de Paris, France M.A. HARRISON, University of California, Berkeley, CA, U.S.A. V. KLEE, University of Washington, Seattle, WA, U.S.A. J.H. VAN LINT, California Institute of Technology, Pasadena, CA, U.S.A. (3.4’. ROTA, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.

ANNALS OF DISCRETE MATHEMATICS

10

LINEAR AND COMBINATORIAL OPT1MIZATION IN ORDERED A113 EBRAIC STRUCTURES

U. ZIMMERMANN Mathematisches Institut Universitat zu Koln

8 NORTHHOLLAND

PUBLISHING COMPANY - 198 1

All Rights mservui. No p ~ r of t this publimtion may b mprodurrd. s t o d in a mtrievalsystem. or transmitted, in any form or by any mtuns, electronic, mechanical. photmpying. recording or otherwise, without the prior permision of the copyright owner. Submission to this journal of a paper entails the author's irrevocableand exclusive authorization of the publisher to collect any sums or considerotions for copying or mpmduction payable by third parties (as mentioned in article I7pamgroph 2 of the Dutch Copyright Act of 1912 and in the Royal k r e e of June 20, 1974 (S.351) pumant to article 166 ofthe Dutch Copyright Act of 1912) and/or to act in or out of Court in connection themwith.

PREFACE

The object of this book is to provide an account of results and methods for linear and combinatorial optimization problems over ordered algebraic structures. In linear optimization the set of feasible solutions is described by a system of linear constraints; to a large extent such linear characterizations are known for the set of feasible solutions in combinatorial optimization, too. Minimization of a linear objective function subject to linear constraints is a classical example which belongs to the class of problems considered. In the last thirty years several optimization problems have been discussed which appear to be quite similar. The difference between these problems and classical linear or combinatorial optimization problems lies in a replacement of linear functions over real (integer) numbers by functions which are linear over certain ordered algebraic structures. This interpretation was not apparent from the beginning and many authors discussed and solved such problems without using the inherent similarity to linear and combinatorial optimization problems. Therefore results and methods which are wellknown in the theory of linear and combinatorial optimization have been reinvented from time to time in different ordered algebraic structures. In this book we describe algebraic formulations of such problems which make the relationship of similar problems more transparent. Then results and methods in different algebraic settings appear to be instances of general results and methods. Further specializing of general results and methods often leads to new results and methods for a particular problem. We do not intend to cover all optimization problems which can be treated from this point of view. We prefer to select classes of problems for which such an algebraic approach turns out to be quite useful and for which a common theory has been developed. Therefore nonlinear problems over ordered algebraic structures are not discussed; at the time being very few results on such problems are known. We assume that the reader of this book is familiar with concepts from classical linear and combinatorial optimization. On the other hand we develop the foundations of the theory of ordered algebraic structures covering all results which are used in the algebraic treatment of linear and combinatorial optimization problems. This first part of the book is self-contained; only some representation theorems are mentioned without explicit proof. These representation theorems are not directly applied in the investigation of the optimization problems considered, but provide a general view of V

vi

Preface

the scope of the underlying algebraic structure. In the second part of the book we develop theoretical concepts and solution methods for linear and combinatorial optimization problems over such ordered algebraic structures. Results from part one are explicitly and implicitly used throughout the discussion of these problems. The following is an outline of the content of the chapters in both parts. In chapter one we introduce basic concepts from the theory of ordered sets and lattices. Further we review basic definitions from graph theory and matroid theory. Chapter two covers basic material on ordered commutative semigroups and their relationship with ordered commutative groups. Lattice-ordered commutative groups are discussed in chapter three. Without proof we expose the embedding theorems of CONRAD, HARVEY and HOLLAND [ 19631 and of HAHN [ 19071 . Chapter four contains detailed characterizations of linearly ordered commutative divisor semigroups. In particular, we prove the decomposition theorems of CLIFFORD ( [1954] , 119581, [ 19591 ) and LUCOWSKI [ 1964 ] . The discussion of the weakly cancellative case is of particular importance for the applications in part two of the book. Many examples complete this chapter. In chapter five we introduce ordered semimodules generalizing ordered rings, fields, modules and vectorspaces as known from the theory of algebra. Basic definitions of a matrix calculus lead to the formulation of linear functions and linear constraints over ordered semimodules. Chapter six contains a discussion of linearly ordered semimodules over real numbers. We provide the necessary results for the development of a duality theory for algebraic linear programs. Chapter seven is an introduction to part two. Chapter eight covers algebraic path problems. We develop results on the stability of matrices and methods for the solution of matrix equations generalizing procedures from linear algebra. Several classes of problems are considered; the shortest path problem is covered as well as the determination of all path values. Chapter nine contains algebraic eigenvalue problems which are closely related to algebraic path problems. In chapter ten cxtremal linear programs are considered. A weak duality theorem in ordered semimodules leads to explicit solutions in the case of extremal inequalities. The case of equality constraints can be treated using a threshold method. In chapter eleven we develop a duality theory for algebraic linear programs. Such problems are solved by generalizations of the simplex method of linear programming. We discuss primal as well as primal dual procedures. As in classical linear optimization duality theory is a basic tool for solving combinatorial optimization problems over ordered algebraic structures. Chapter twelve covers solution methods for algebraic flow problems. We develop a generalization of the primal dual solution method of FORD and FULKERSON. Clearly primal methods can be derived from the simplex method in chapter twelve. We discuss several methods for the solution of algebraic transportation and assignment problems. Chapter thirteen contains solution methods for algebraic optimization problems in independence systems; in particular, we investigate algebraic matroid and 2-matroid intersection problems. Algebraic matching problems can be solved similarly; both

Preface

vii

classes of problems are mainly differing by their combinatorial structure whereas the generalization of the algebraic structure leads to the same difficulties. In writing this book I have benefited from the help of many people. In particular, I want to express my gratitude to Professor Dr. R.E. Burkard, who encouraged my work throughout many years. A great deal of the material on algebraic combinatorial optimization problems originated from collaborative work with him in the Mathematical Institute at the University of Cologne. He made it possible for me to visit the Department of Operations Research at Stanford University. Valuable discussions in Stanford initiated investigations of related lattice-ordered structures. I am much indebted to the Deutsche Forschungsgemeinschaft, Federal Republic of Germany, for the sponsorship of my stay in Stanford. The manuscript was beautifully typed by Mrs. E. Lorenz.

U.Z. Cologne

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TABLE OF CONTENTS

PREFACE

V

PART I : ORDERED ALGEBRAIC STRUCTURES 1. Ordered Sets, Lattices and Matroids 2. Ordered Commutative Semigroups 3. Lattice-Ordered Commutative Groups 4. Linearly Ordered Commutative Divisor Semigroups 5. Ordered Semimodules 6. Linearly Ordered Semimodules over Real Numbers

1

30

41 51 85 97

PART I1 : LINEAR ALGEBRAIC OPTIMIZATION 7. Linear Algebraic Problems 8. Algebraic Path Problems 9. Eigenvalue Problems 10. Extremal Linear Programs 1 1. Algebraic Linear Programs 12. Algebraic Flow Problems 13. Algebraic Independent Set Problems

111 116 168 188 212 253 30 1

CONCLUSIONS

337

BIBLIOGRAPHY

339

AUTHOR INDEX

369

SUBJECT INDEX

373

ix

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PART I

1.

ORDERED ALGEBRAIC STRUCTURES

O r d e r e d S e t s , L a t t i c e s and M a t r o i d s

In this chapter we introduce basic concepts from the theory of ordered sets and we consider certain types of lattices. Further we review basic definitions from graph theory and matroid theory as given in BERGE [ 1 9 7 3 1 and WELSH [ 1 9 7 6 ] . This chapter covers the necessary order-theoretic background for the -discussion o f the optimization problems in part 11. In particular, some representation theorems from lattice theory give a general view o f such structures. For further results and for some proofs in lattice theory we refer to GRATZER [19781.

The basic properties of the usual order relation numbers IR

5

of the real

are described by the following four axioms:

(RefZesivitp)

a z a a z b ,

b c a

*

a = b

( A n ti s y m m e t r y )

a'b,

c ' b

*

a c c

(Trans i t i o i t y )

or

b c a

(1.1)

a c b

(Linearity)

for all a , b , c E IR. In general we consider a binary relation 5 o n a nonempty set H satisfying some o f these axioms. We always assume that the binary relation is transitive. Such a binary relation is called an o r d e r i n g . If an ordering is reflexive then it is called a

p r e o r d e r i n g or q u a s i o r d e r i n g ; if an ordering is antisymmetric then it is called a p s e u d o o r d e r i n g . 1

A

reflexive and anti-

Ordered Algebraic Smrcritrcs

symmetric orderinq is called a p n r t i a ? , o r d e r i n g ; if a part.ia1 ordering is linear then it is called a l i n e a r orderinq.

(or total)

system (H,f) with such an ordering o n H is

A

called a (pre-, quasi-, pseudo-, partially, linearly) ordered set. A

-

?reorder

on

14

satisfyinq

I S

called an e q u i v a 7 e n c e r e 7 n t * ’ o n . Then R : = { b € HI

LS

called the e q u i z a z e n c e c ? n s S of a. The set { R

a ~ c r f * ‘ + i oof~ H ,

i.e.

n Rb +

Ra

@

-

ard H is the union of all R

t

-

a}

a E H 1 is

Ra = Rb :

Vice versa a partition of H defines an equivalence relation cn H with a

-

b if and only if a and b are elements of the

same element ( b l o c k ) of the partitior..

T w 3 elements a , b in an ordered s e t H are called if

neither a

5

b nor b

5

a.

a

5

b but a

+

ixcomparable

b is denoted by a < b .

T h e ? r , t ; e r s e o r d u a l o r c i e r f n g of 5 is the ordering

a > -b

: -

L

defined by

b 5 a

fcr all a , b E H . A linearly ordered subset A of H is called

?ox?:r?nZ if there is no linearly ordered subset B o f H such that

A

*

R.

In order to obtain a general view o f ordered structures it is v e r y useful to introduce the concept o f isomorphism.

Ordered Sets, Luttices and Matroids

Let (HI()

V: H

+

and (H

, z ' ) denote

3

two ordered sets. A mapping

H' is cal ed an o r d e r i s o m o r p h i s m if

v

is bijective

and a c b for a

a,b E H. Then I? an

H' are called o r d e r - i s o m o r p h i c .

If (9 is an order isomorphism hetween H and V ( H ) but not necessarily surjective on H',then

UY

is called an O r d e r e m b e d d i n g

of H into HI. We say that H can be o r d e r - e m h e d d e d

into

H I .

For

example, each finite linearly ordered set with n elements is order-Isomorphic to the set {lI2,...,~n~linearly ordered by the usual order relation of the natural numbers and each finite linearly ordered set with less than n elements can be order-embedded in every finite linearly ordered set with at least n elements.

Ordered sets are often visualized by graphic representations. We assume that the reader is familiar with the concepts of a g r a p h and a directed graph ( d i g r a p h ) . We denote a graph G by a pair ( V , E ) where V = V ( G ) is the nonempty vertex set and E = E ( G ) is the set of e d g e s ; a digraph G is denoted similarly by ( V I A ) where V = V ( G ) I s the nonempty vertex set and A = A ( G ) the'set of a r c s . Arcs and edges are denoted as pairs (a,b) of vertices; then a,b are called e n d p o i n t s of (a,b); in particular, for an arc (a,b) the vertices a and b are called i n i t i a l and t e r m i n a l

e n d p o i n t s of (a,b).

...,en )

sequence of arcs (edges) (el,e2,

A finite

such that ei has one

common endpoint with ei-l and the other endpoint in common with ei+l for all i = 2,3,...,n-l

is called a c h a i n . The length

Ordered Algebmic Structures

4

of a chain is the number of its edges (arcs). V e r t i c e s o n a chain are called c o n n e c t e d

( b y the chain). A chain with arcs

e . = (ai,ai+l) for i = 1,2,. ..,n is called a p a t h from a to a n + l . a l and a

n+ 1

1

are called i n i t i a l and t e r m i n a l e n d p o i n t

of the path. The e n d p o i n t s o f a chain are defined similarly. A

chain (path) is called s i m p l e if its edges (arcs) are

mutually distinct.

A

simple chain (path) with coinciding end-

points is called a c y c l e

(circuit).

A

chain (path) not using

any vertex twice is called e l e r e n t a r y . A

s u b g r a p h o f G g e n e r a t e d by V'

5

V is the graph (V',E') con-

taining all edges (arcs) o f G which have both endpoints in V'. A g r a p h G is called c o n n e c t e d ted. The binary relation

if all vertices in G are connec-

-, defined o n V by a

-

b iff a and b

are connected, is an equivalence relation o n V. The elements

of the induced partition V 1 , V 2,... ,Vk of V generate the

c o n n e c t e d c o r p o n e n t s of

G.

A

graph is called s t r o n g l y c o n n e c t e d

if for all a , b E V there exist paths from a t o b and from b to a. The strongly connected subgraphs o f G are called s t r o n g l y

c o n n e c t e d c o m p o n e n t s and define another partition of the vertex set V.

F i g u r e 1.

Directed and undirected graph

An ordered set ( H , z ) is represented b y the (directed) graph with vertex set H and (arc set

A)

edge set E = {(a,b)( a 5 b).

Ordered Sets. Lattices and Matmids

5

In the case of a graph a < b is expressed by placing a below b in a drawing of the graph. Clearly, the above examples of graphs represent the same ordered'set. Let A denote a subset of an ordered set E and let b E H . Then b is called an U p p e r ( l o w e r ) bound of A if a

5 b (b 5 a) for

all a E A . An upper (lower) bound b of A i s called a l e a s t

u p p e r ( g r e a t e s t l o w e r ) bound or supremum ( i n f i m u m ) of A if b

5 c

(c

5 b) for all upper (lower) bounds

c of A.

An element a E A is called a maximum ( m i n i m u m ) o f A if a'

5 a

(a 5 a') for all a' E A . An element a E A is called maximal

( m i n i m a l ) if a 5 a' (a' 5 a) implies a' 5 a (a 5 a') for all a' E A . The corresponding sets are denoted by U(A) , L(A)

(upper, lower bounds),

sup A , inf A

(suprema, infima),

max A, min

A

(maxima, minima),

Max A, Min

A

(maximal, minimal elements).

The following example illustrates these denotations. On the

0

set o f the vertices of the graph G = (V,A) in figure 2

3

Figure 2.

Digraph of an ordered set

an ordering i s defined by a

5 b iff (a,b) E A . This ordering

is only transitive. For A = 1 3 1 we find sup A = {l,Z], and Max A = A. For B = { 1 , 2 }

max A = @

we find sup B = max B = Max B = B.

Ordered Algebraic Structures

6

Suprema (infima) and maxima But if Max

is a finite subset of an ordered set H , then

A

A =

(minima) do not always exist.

a.

ordered set is called b o u n d e d if it has a maximum

AI:

tion: 1 or -1

and a minimum

(denotation: 0 o r - - I .

we find 1 E Sup H = max H = Max H

and

O E

(denotaClearly,

inf H = min H = Min H.

If H is pseudoordered then these sets contain exactly one element. A

partially ordered set

Let

called a r o o t s y s t e m if the

a 5 b} is linearly ordered for all a € H.

set { b E H I

(1.2)

(H,c) is

Proposition

(H,Z)

be a partially ordered set. Then H is a root system

if and only if n o pair of incomparable elements in H has a

common lower bound.

Proof. The set { d E H I c 5 d} for cEL({a,b))

is not linearly

ordered if a and b are incomparable. This immediately shows the proposed characterization of root systems.

A partially ordered set

and inf{a,b}

(H,L)

is called a l a t t i c e if sup{a,b}

exist for all a , b E H . For convenience we use the

denotations a

A

b:= inf{a,b},

in lattices.

A

a

sublattice

V

b:=

(H',z)

sup{a,b) of a lattice (H,z)

subset H' o f H which is closed with respect to m e e t j C i n v taken in H , i.e.

is a A

a , b E H' imply a v b , a h b E H ' .

not sufficient that (H',I) is a lattice.

A

and It is

lattice H is called

Ordered Sets. Lattices and Matroids

C o m p l e t e if sup that sup

A

for all B

7

and inf

A

exist for all

exists for all

A

C_ H if and only if inf B exists

A

A

H. We remark

5 a;

Two lattices H and A' are called l a t t i c e - i s o m o r p h i c iff there exists a bijection Ip: H + H' such that a

5 b iff Ip(a) 5 Ip(b)

for all a , b E H . This condition is equivalent to Ip(aAb) = Ip(a) A

Ip(b)

Ip(a v b ) = Ip(a) vIp(b) for all a , b E H . Then Ip is

and

called a lattice isomorphism. If Ip: H morphism then

@

+

cr)(H) is a lattice iso-

is called a l a t t i c e e m b e d d i n g of H in H'. We

say that H can be l a t t i c e - e m b e d d e d

into H ' .

In particular,

q(H) is a sublattice of HI.

Now let A be a root system and let H A be the linearly ordered set lR of real numbers for all A E A .

Let V = V(A) be the

following subset of the Cartesian product

x AEAHA: x =

belongs to V if and only if the set Sx = { A E A 1 x A

(.

* O }

. .xx.. . ) contains

no infinite ascending sequences. Further let M

XIY

:= { h E h l

xA * y A

for x , y E V . Then we define x hEM

XtY

.

As

and

x

lJ

= y p for all A -c

5 y if and only if x A 5 y A for all

in this definition of the binary relation

5 on V

we will often use the same symbol for different order relations; from the context it will always be clear which of the relations considered is meant specifically.

( 1 . 3 ) Proposition Let A be a roqt system. Then V = V(A) is a lattice. Proof. At first we show that V is a partially ordered set with respect to the above defined binary relation. Reflexivity

a

Ordered Algebraic Structures

follows from M

5

let x

5

If x

implies x A

z

and x

MX

A

for all 1-1 > A. y

= @.

= M implies antisymmetry. Now IY YrX y, y < z and let A E M x , z . Then x z A and x = z

xrx

= z6 =

yA

A

5

1-1

=

y p for all 1-1 > A

u u then x 5. y and

Otherwise let A < y with x

zA.

y6 for all 6 > y .

5

Then x

y and y

5

z

Y - z Y *yY

leads to

= x < yy. Secondly, for a , b E V we Y Y b, a v b in the following way.

the contradiction y define a

= z

1-1

*

< z

Y

Let M be a maximal linearly ordered subset of A. Then the p r o jeCtiOn V

M

vM : =

of V onto M, defined by ( v ~ v lv

=

A

o

for all A

M is linearly ordered. For v E M we call vM with vA:= v A for A E M and vM = 0 otherwise the p r o j e c t i o n Of

V.

A

5

if aM

b

M

For A E M we define

,

(a A b) A : = if aM > bM i f a

M

L b

(a v bIA : = ifa' 0 ) endowed with multiplication and ordered by divisibility. Secondly let (G,+) denote the additive group of all real functions f: [ 0 , 1 1 + I R f(x)

with (f +g)(x):=

5 g(x) for all x E

f(x) + g ( x ) and f

[0,1]. Then

(G,+,()

5

g

iff

is a lattice ordered

commutative group. The third example does not look so familiar. For k E N let V denote the set of all subsets A of lNU { O ? with IAI trary B

zk. For

arbi-

{O} let k - m i n ( B ) denote the set of the k smallest

elements of B (in particular, for IBi < k let k - m i n ( B ) =

€3).

Now for A , B E V let A i B

k-min(AUB) =

:-b

A

.

This binary relation defines a lattice on V. Meet i s given by A A B =

k

-

min(AUB)

,

and join is (under consideration of its commutativity) given by

Ordered Commutative Semigroups

A v B = (AnB) U

(2.10)

31

i'

if IAI, IBl C k ,

{bEBl b>a)

if IAl =k, IS1 Ck,

{bEBl b>a}U{B+l, . . . , B +

r)

if IAI, IS1 = k ; a 5 8

with a = max A, 8 = max B and r = k - I ( A n B ) U { b E B I

b>a}l.

The

meet formula is obvious and the join formula is proved in the following proposition. Beforehand we define

* B:=

A

k

-

min(A

+ B)

with A + B : = { a + b l a E A , b E B ) for all A , B E V .

(2.11) (V,*,L)

Lemma is a lattice-ordered commutative monoid with neutral

element {O), ment

least element

2 =

{O,l,

...,k-1)

and greatest ele-

a.

Proof. The properties o f the neutral element, the least element and the greatest element are easily verified. The same holds for the properties of the partial order, the meet and for commutativity, associativity and monotonicity of the internal composition. The remaining part is the verification of the join-formula in ( 2 . 1 0 ) . At first we remark that A B E A Further let

C:=

or

C

d E (AnB)\

C

to A n B

5

C

B is equivalent with

(IAI = k and b E B \ A

implies b > m a x A).

A V B , a = max A, B = max B, y = max

Secondly we prove A n B C_ B we find

5

C.

2

(max

@=-).

As A n B is an upper bound of A and

5 A n B . Assume A n B we find a , B

C

$

C.

Then I C I = k and for each

d > y . The assumption

and thus to the contradiction A n B =

C C.

c_

A n B leads

Therefore

Ordered Akebraic Structures

38

there exists c E C \ ( A n B ) . W.1.o.g. A

let c 6 A . Together with

5 C this implies c > a contrary to a >

y.

Thirdly consider the different cases in formula (2.10). IAI, IBl < k then C

A and C

B shows C

If

A n B . As we know

A n B C_ C this proves C = A n B. In the second case IAl = k and IBl < k we know that c E C \ A

implies c > a and that

C

E B.

Therefore C C ( A n B ) U { b E B I c > a)=:D. Thus D

1. C.

As D is an upper bound of A and B we find D = C.

In the third case IAl = IBl = k we know that c E C \ A c > a and that c E C \ B

implies c > 3 . Therefore ( a

N

N

Then D = k -min(D) fulfills D

implies

5 B)

N

1. C. Further D is an upper bound

N

of A and B. Thus D = C.

The order relation in this example is neither positively ordered nor is ( 2 . 7 )

fulfilled. Therefore ( V , * , l )

may not be

embedded in the positive cone of an ordered commutative group.

An ordered commutative semigroup H is called r e s i d u a t e d if for all a , b E H there exists c E H such that (2.12)

a * x l b

a-~

x l c .

If such an element c exists for a , b E H then it is called a

r e s i d u a l o f b with respect to a. If the order relation in H is antisymmetric then a residual is uniquely determined and denoted by b:a.

If for all a , b E H there exists c E H such that

39

Ordered Commutative Semigroups

(2.12')

a * x z b

Q

x z c

then H is called dually residuated. Such an element c is called a d u a l residual and is denoted by b/a. (2.12)

(2.12')

in the lattice-theoretic sense.

A

is the dual o f

(dual) residual b:a

(b/a) is the greatest (smallest) solution o f the inequality a

*x

5 b

(a * x

2

b). Therefore the (dual) residual exists if

and only if the set { x E H I a * x

5 b)

({xEHI a * x

1. b))

has

a greatest (smallest) element. An important implication of the existence of residuals is the distributivity of the least upper bound.

(2.13) Proposition Let H be a residuated, lattice-ordered commutative semigroup. Then a

*

( b V c ) = (a * b ) V (a * c ) for all a , b , c E H .

* b,a * c 5 a * (b v c). x:= (a * b) V (a * c).

Prdof. Let a,b,c E H. Monotonicity implies a Therefore (a Then a * b

* b)

v (a * c) 1. a

*

(b v c). Let

5 x and a * c 5 x. Then b 5 x:a and c 5 x:a. Thus

< x:a which implies a * ( b v c ) 5 x. bvc -

On the other hand the greatest lower bound is not necessarily distributive.

( 2 . 1 4 ) Proposition Let G be a lattice-ordered group. Then G is residuated and dually residuated are distributive.

and greatest lower and least upper bound

rn

40

Ordered Algebraic Structures

Proof. In a group b:a = b/a = b * a

v follows from

of

(2.13)

-1

.

Therefore distributivity

and distributivity of

follows from

A

(2.13) applied to ( G , * ) ordered with respect to the dual of

the lattice ( G , Z ) .

(2.14)

shows that lattice-ordered commutative groups are

examples for residuated lattice-ordered commutative semigroups. A

more detailed discussion is given in chapter 3 .

Let (H,h,z) denote the lattice-ordered commutative semigroup derived from a bounded lattice (H.5). If (H,h,5) is residuated then from ( 2 . 1 3 )

we know that for all a , b , c E H

a h ( b v c ) = ( a h b ) v (ahc). We have found a stronger result in chapter 1 .

this case that (H,()

(H,L)

(2.12)

means in

is a pseudo-Boolean lattice. Therefore

fulfills a V (b A c) = (a v b)

A

(a v c)

for all a , b , c E H , too. If the lattice ( H , Z ) is Boolean then the residuals of ( H , h , L ) are given.by b:a (see 1.15).

=

bva*

Clearly, for pseudo-Boolean lattices the ordered

commutative semigroup (H,V,()

fulfills ( 2 . 1 2 ' ) ,

a , b E H there exists c E H such that a v x z b

Y

x l c .

i.e.

for all

3.

Lattice-Ordered Commutative Groups

In this chapter we consider lattice-ordered commutative groups (G,*,L). A characterization of such groups can be found in CONRAD, HARVEY and HOLLAND 119631. For a detailed discussion we refer to CONRAD [1970]. We introduce

basic concepts and

properties which allow a precise formulation of this characterization. Due to proposition ( 2 . 1 4 )

we know that a lattice-ordered commu-

tative group is d i s t r i b u t i v e , i.e. least upper and greatest lower bound are distributive over the internal composition.

(3.1) P r o p o s i t i o n (1)

partially ordered commutative group G is lattice-

A

ordered if and only if a v e exists for all a E G . Let G be a lattice-ordered commutative group. Let a,b E G and n E N . Then n

2

(2)

a

bn

(3)

(ave)”

y

a

2

n a ve

=

(G i s

b

isotated)

,

.

Proof. ( 1 ) Necessity is obvious. Otherwise let a : = [(a*b-’) ve] *b. Then a = a v b . Further let (2)

a

n

> -

B:=

-1 (a

b-l)-l

.

Then 8 = a

b.

A

bn is equivalent to (a * b - l ) n > e. Thus it suffices

to show a Let a

n

2

2

e

o

a 2 e .

e. Then monotonicity leads to a

n

2

e. Let now a

n

2

e.

n = 1 i s trivial. Let n > 1. Then distributivity leads to (a A e)” = an

A

an-’

A

... A

e = an-1 41

A

a n-2

A

...

A

e = (a A e) n- 1

.

42

Ordered Algebraic Structures

Cancellation implies a h e = e; i.e.

2

e.

For 0 < k < n we find

(3)

(an-k va -k ) n = a (n-k)nv (2)

Therefore

implies a

(aveIn = a

A

a

n

va

n-k

n-1

v a

. . .va (n-k)k-k (n-k)v . . . v ~- kn

>

e.

2 e , i.e. a n v e 2 a k . Then

-k

v...v~

= a

n

ve.

semigroup (group) (H,*) is called d i v i s i b Z e or r a d i c a b l e if

for all a E H and all nclN there exists b E H such that a = bn.

(3.2) P r o p o s i t i o n A

lattice-ordered commutative group G can be embedded into a

divisible, lattice-ordered commutative group Proof. Let set G

X

denote the set of all equivalence classes of the

{ -n1I

nclN) with respect to the equivalence relation

defined by 1

(a,;) Then

(G, 0

c.

-

1

(b,i)

:

-

am = bn

N

.

is a commutative group with internal composition

)

(a,;) 1

0

(b,;):= 1

(ambn,

L, mn

where elements in

are represented in the usual form. This

definition is independent of the choice of the representing elements as G is cancellative. Let W: G

-

+

G defined by Q(a):=

(a,l). Then

The lattice ordering o n G is given by the positive cone G+ of G. Now

G+:=

1

{(a,;)

1

a E G + j defines a partial order on

e.

This

Lattice-Ordered Commutative Groups

43

definition is independent of the choice of the representing elements as G is isolated. Clearly, (P(G+) = (P(G) fore a

5 b

-

V(a)

cellation arguments. Thus tative group. From (3.1.1)

(E,@,z)is

1 upper bound of (e,l) and (a,;).

2

and can-

a partially ordered commu-

we know that it suffices now to

1 show the existence of (e,l) V (a,;)

xn

There-

1. V(b)

for all a , b E G . Monotonicity follows from (3.1.2)

fore using (3.1.3)

n G+.

e ~a

m

in

Then x

G. 2

Let (x,:) e and xn

m = ( e v a ) , i.e.

denote an

.

2

m a

1

1 z(eva,-). n

(x,;)

There-

Similarly we find that ( e V a , L ) is such an upper bound. Thus n 1 1 (e,l) v (a,-) = ( e v a,-) n n

A

.

special divisible, lattice-ordered commutative group can be

constructed from the root lattice V = V(A) in chapter 1.

The

components x A for A E A of an element x E V are real numbers. Therefore an internal composition

+

on V can be defined by com-

ponentwise addition of real numbers, i.e. (a + bIA:= a A for all A € A

+

bA

and a , b E V . Then (V,+) is a divisible, commutative

group. V(A) is a subset of the Cartesian product X X E A H A H A = IR for all

A E A.

with

If ( H A , + ) is a subgroup of (IR, + ) then the

same construction leads to the r o o t l a t t i c e

= ? ( H ~ , A ) with

r e e p e c t t o t h e f a m i l y ( H A , A ) and to a commutative group which is not necessarily divisible.

(v,+)

Ordered Algebraic Structures

44

( 3 . 3 ) Proposition

-

V ( V ) is a (divisible) lattice ordered commutative group. Proof. It suffices to show monotonicity. Let a

proof of (1.3) we know that aM ordered subset of A .

5

b. From the

5 b M for each maximal linearly

This means that either L:= { L I E M I a

P

9 b 1 P

is empty or a A 5 b A for A = max L. Now let c E V . Then L = (uEMI

a

l

+c J

U

*blJ+cU) M

and therefore a M + c M < bM+c

.

As this is satisfied for each

maximal linearly ordered subset of A we find a + c

1. b + c .

V is called a H a k n - t y p e group and is considered in CONRAD, HARVEY and HOLLAND [1963]. They prove the much deeper result that each lattice-ordered commutative group can be embedded into a Hahn-type group. A proof of this important theorem is beyond the scope of our book; but in order to understand a more detailed formulation of this result we introduce some basic concepts. In the following G is a divisible, lattice-ordered commutatfve group. Divisibility can be assumed w.1.o.g.

due to proposition

(3.2). A

subset G' of G is called convex if a , b E G ' imB.ligs that

[a,b] C_ G'.

A

convex subgroup G' which is a sublattice of G is

called a l a t t i c e i d e a l of G. The set I of all lattice ideals is partially ordered by set inclusion.

A

lattice ideal G' is

called r e g u l a r if there exists an element g E G such that G' is maximal in I with respect to the property g EG'. Let (GA,h) denote the family of all regular lattice ideals of G.

45

Luttice-Ordered Commutative Groups

CONRAD,

HARVEY a n d HOLLAND [ 1 9 6 3 ] show t h e e x i s t e n c e o f

unique l a t t i c e i d e a l Gh covering G h f o r h E A.

Gx

a

is called

t h e cover of Gh f o r A E A .

A s a s i m p l e example w e c o n s i d e r t h e Hahn-type

with T = { 1 , 2 , 3 } .

T h e root s y s t e m

r

group V ( r )

is o r d e r e d as d e s c r i b e d

i n figure 4.

Figure 4.

T h e n V = lR

and G I = V,

'.

The r o o t s y s t e m

r

We f i n d t h r e e r e g u l a r l a t t i c e i d e a l s

G 2 = G 3 = G1.

Due t o c o m m u t a t i v i t y a * G

Therefore w e d e f i n e A:=

* a-l

r.

= G h f o r a l l a E G h and h E A ~

h

IGx i s c a l l e d a norrnaz subgroup of G 1.

group

x G /GX

T h e r e f o r e t h e factor

d e f i n e d by h G / G ~ : =C a * G A ] h E A )

is w e l l - d e f i n e d

with r e s p e c t t o t h e i n t e r n a l composition

Ordered Algebraic Structures

46

A

G /GX

Due to convexity

is totally ordered by the well-defined

induced ordering given by (a * G x )

5 (b * G x )

5 b

a

91.

.

Using the fact that G h covers G x it can be seen that the factor group

x

is isomorphic to a subgroup o f the additive

G /Gx

x

group of real numbers. In our above example every G / G x is isomorphic to IR.

u if and only if A

For A , p E h let A be seen that

(A,f)

= p or G

x -3 G ' .

It can

is a root system, which has the following

two properties: (1)

V g E G ,

(2)

g@G'

An index

gPe

*

3 )tEA

3 AEA,

:

h > p :

g E G

h

\G

in ( 1 ) is called a v a l u e of g.

A

N

is a root system. Therefore V ( G / G x , A )

subset 'E; of h N

satisfying ( 1 ) and ( 2 ) is called p l e n a r y .

x

A '

x g E G \ G x .

It can be seen that A

is a lattice-ordered

commutative group (cf. proposition 3 . 3 ) .

In the above example A is order-isomorphic to

r.

Properties ( 1 )

and ( 2 ) are easily checked. Clearly, A = 1 is a value of (l,O,O).

x

Further A is the only plenary set and V ( G / G x , A ) to

is isomorphic

v(r).

In general, a group-embedding CP:

V a l u e - p r e s e r v i n g if

G

+

x

N

V(G / G x , A )

E'E; is a value of g

E

G

is called

if and only if q ( g ) x

is a maximal nonvanishing component of q ( g ) and, in this case, rP(g)x = g * G x . in particular

Such an embedding Q(G)

is always a lattice embedding;

i s a sublattice of

v.

47

Lattice-Ordered Commutative Croups

Now we can give a detailed version of the embedding result of CONRAD, HARVEY and HOLLAND 119631. For a proof we refer to CONRAD [1970].

( 3 . 4 ) Theorem Let G be a divisible, lattice-ordered commutative group and let ( G A , A ) be the family of all regular lattice ideals of G. Let

?i

be a plenary subset of A .

Then G can be group-embedded

into V = V(?i). This embedding is value-preserving and therefore the image of G is a sublattice of V.

In general, there may exist many different plenary sets and for each of them we get an embedding. The discussion in CONRAD, HARVEY and HOLLAND 119631 shows on the other hand that if A is finite then A is the only plenary set and G is isomor-

-

A

phic to V(G /GA,A). Thus, for finite A lattice-ordered

-

commu-

tative groups are represented by V(HA,h), where H A is a subgroup o f the additive group of real numbers and A a root system. Other examples with interesting properties are discussed in CONRAD, HARVEY and HOLLAND 119631.

If the root system A is linearly ordered then V(A) is called a H a h n - g r o u p . Then V ( A ) is a linearly ordered commutative group. In linearly ordered groups a lattice ideal is called

i d e a l . Let G be a divisible, linearly ordered commutative group and let ( G A , li € A )

be the family of all regular ideals

of G. Then A is linearly ordered. Theorem ( 3 . 4 ) the theorem of HAHN 119071.

reduces to

Ordered Algebraic Structures.

48

(3.5) T h e o r e m Let

be a divisible, linearly ordered commutative group and

G

let ( G A , A € A ) Then

G

be the family of all regular ideals of G.

can be embedded in the divisible, linearly ordered

commutative group

V(A).

For a particular proof of this theorem we refer to FUCHS 119661. Theorem (3.4) and (3.5) are valuable characterizations of lattice- (linearly) ordered commutative groups and provide a general view of such ordered groups.

Our simple example illustrating the various definitions and denotations above can easily be extended to finite root systems A.

In particular, if A is a linearly ordered set with n ele-

ments then V ( h ) is isomorphic to the additive group Wn of real vectors which is linearly ordered by the l e x i c o g r a p h i c

o r d e r r e l a t i o n of vectors defined by (3.6)

x < y

:-

x =

y

or

xi A 2 > . . . the component x

1

*y.I. 1

>An

for x E V ( 6 ) corresponds to xi for xEIRn. Ai

An infinite example is the additive group of all real functions f: [0,11 + IR with (f + f(x)

5 g(x) for

g)

(x): = f (x) + g ( x ) and f

1. g defined by

a l l x € [0,1].This is a divisible, lattice-

ordered commutative group isomorphic to V ( A ) for the root system A = [O,l1 trivially ordered by x

5 y iff x

=

y.

Finally w e consider a subgroup of the additive group of all

49

Lattice-Ordered Commutative Groups real functions f: A

[ O , l ] +lR w i t h f ( 0 ) = 0 . L e t x E [ 0 , 1 ] .

f i n i t e p a r t i t i o n Px o f t h e i n t e r v a l [ O , x l c a n b e r e p r e s e n -

1 of i t s interval-endpoints; i n

t e d by t h e s e t { a o , a l , . . . , a p a r t i c u l a r 0 = ao C a l < . . . < a v(x,pX):=

r

I

i= 1

= x.

r

-

If(ai)

Now d e f i n e

f(ai-l)I

If V ( b , P ) i s b o u n d e d by a r e a l b

f o r such a r e a l function f .

c o n s t a n t B i n d e p e n d e n t o f Pb t h e n

e x i s t s a n d i s c a l l e d t h e total variation o f f i n [ O , x ] . t h i s c a s e we s a y t h a t f L e t G denote

the

i s o f bounded variation.

set of a l l r e a l functions f :

bounded v a r i a t i o n w i t h f ( 0 ) = 0. C l e a r l y ,

A

function f :

if x

5

[ 0 , 1 ] +IR

of

t h i s is a subgroup

of t h e a d d i t i v e group of a l l r e a l functions f : I t s n e u t r a l element

In

[ 0 , 1 ] + IR.

( f n 0 ) w i l l b e d e n o t e d by 0.

[ 0 , 1 1 +IR

y implies f ( x )

5

is c a l l e d monotonicaZZy non-decreasing f ( y ) . Such a f u n c t i o n w i t h f ( 0 ) = 0

is o f bounded v a r i a t i o n , I n p a r t i c u l a r V(x) = f ( x ) f o r a l l x E [0,1]. order

1. o n

The s e t G+ o f a l l s u c h f u n c t i o n s i n d u c e s a p a r t i a l

,

G a s G + + G + C_ G+

OEG+

a n d G + fl -G+

o r d e r t o show t h a t G i s a l a t t i c e - o r d e r e d t o show t h a t f v 0 e x i s E Z - f o r a l l f

51 ( V ( X )

(3.7)

f * ( x ) :=

f o r f E G.

Clearly f = f+

f o r a l l x,y with x

5

y.

*f -

= (0). I n

g r o u p i t now s u f f i c e s

rc< L e t

~~

_ ~ _

(XI)

f-

.

F u r t h e r f + , f - E G+ as

Therefore f +

2

0,

f+

2

f.

Now assume

50

Ordered Algebraic Structures

that g is an arbitrary upper bound of f and 0. We will show g

f+

.

As

g

-

f+ = ! ( g 2

ces to prove g - V

2

-

f)

0. Let x

+

1 ~

( -gV ) and g

1. y.

Then g

-

plies

-

If(y)

Now for a partition P Px:= (Py

n

f(x) I f S(Y )

-

f

-

f

2

1. 0 0, g

it suffi-

2

0 im-

.

g(x)

we define a corresponding partition

Y

[O,xl) U { X Iof [O,x]. Then the above inequality

used for points in (P f l [x,y])U{x) Y V(Y,PY) and therefore g - V

2

implies

5 V(X.PX) + g(y) -

g(x)

0. Summarizing we have proved the follow-

ing proposition.

(3.8) P r o p o s i t i o n Let G denote the additive group of all real functions f: [O,ll -+lR of bounded variation with f(0) = 0. Let G,

denote the set

of all monotonically non-decreasing functions of G. Then G is

a divisible, lattice-ordered commutative group with respect to the positive cone G + .

Meet and join are given by

(f+g+V(f- 9 ) ) / 2

,

f v g = (f+g-V(f-g)) / 2

.

f A 9 =

For further examples of lattice-ordered commutative groups we refer to CONRAD 1 1 9 7 0 1 .

4.

Linearly Ordered Commutative Divisor Semigroups

In this chapter we develop a characterization of linearly ordered commutative semigroups (H,*,z) which satisfy the additional axiom (4.1)

a C b

-D

3 cEH:

a * c = b

(Divisor rule)

for all a , b E H . An element a E H is called a d i , v i s o r of b E H if there exists c E H with a * c = b. Thus ( 4 . 1 )

means that an

element a is a divisor of all strictly greater elements. Therefore we call such semigroups d i v i s o r s e m i g r o u p s . For convenience we denote a linearly ordered commutative divisor semigroup shortly as d - s e m i g r o u p . Positively ordered d-semigroups have been characterized by CLIFFORD ( 1 1 9 5 4 1 ,

[1958]

and 1 1 9 5 9 1 ) .

D-semigroups which are

not positively ordered were characterized by LUGOWSKI 1 1 9 6 4 1 in a similar manner extending the results of CLIFFORD. The positively ordered case is also covered by the discussion of positively totally ordered semigroups i n the monograph of SATYANARAYANA 1 1 9 7 9 1 . From proposition ( 2 . 9 ) we know that we can assume w.1.o.g.

the

existence of a neutral element in a positively ordered d-semigroup. Later'on we will see that a d-semigroup containing elements which are not positive is always a d-monoid. Therefore the discussion of d-monoids and d-semigroups leads to the same results. A positive element a E H satisfies b 5 a * b for all b E H (cf. 2.6). Since d-semigroups are linearly ordered there exists b E H with b > a * b if an element a is not positive. 51

Ordered Algebraic Structures

s2

(4.2)

Proposition

Let H be a d-semigroup and a E H . Then the following two properties of a are equivalent: i s not positive,

(1)

a

(2)

a > a*a.

Further both properties imply

2

13)

b

Proof.

(1)

a*b I ,

for all b E H .

(2). Then b > a * b for some b E H and a * b * c = b

for some c E H . Therefore a * b * c > a (2)

by cancellation of b * c . ( 2 )

(2)

*

*

*

(a * b * c ) which implies

( 1 ) is obvious.

( 3 ) . The case a = b is obvious. L e t b < a. Then a = b * c

for some c E H . Thus b * c > a * b * c . Cancellation of c implies b > a * b . Let a

4

b. Then a * c = b for some c E H . Suppose

*c

b < a * b . Then a

Proposition ( 4 . 2 )

< a

*a*c

implies a < a * a contrary t o ( 2 ) .

shows that an element a is negative i f it

is not positive. In general, if the d-semigroup is not a group, such an element is not strictly negative. Motivated by ( 4 . 2 . 2 ) we call it self-negative.

Let

(A,L) be a nonempty, linearly ordered set and let (HA,*;X,LA)

be a linearly ordered commutative semigroup for each A € A .

The

sets H A are assumed to be mutually disjoint. Let H be the union of all HA. Then we continue internal compositions and order relations of the H A on (4.3)

a < b

for all a E H A

and

,

b E H

€3

by

a*b = b*a = b U

with A
X o

(31

all H X are isomorphic either to the strict

positive cone of G X or to the trivial group G X . Proof. It remains to show (3). From (4.10) we know that H A is positively ordered. If H A contains a neutral element e X then e

< a for all a E H A

h -

.

due to (4.6) we find a

On the other hand e X is idempotent and

5

e X for all a € HA. Then H A is isomor-

phic to the trivial group, i.e. H A = {eXj. Otherwise we adjoin a neutral element to H

A'

Then from ( 4 . 1 1 )

it follows that H A

is isomorphic to the strict positive cone ( G A ) + L e X where e denotes the neutral element of G We may assume w.1.o.g.

X W

A'

that all G X , X E A are mutually disjoint.

and let for X > X

0

either F A : = G X (iff H A is

isomorphic to the trivial group) or F

(GA)+

{ e X l . Then the

Linearly Ordered Commutative Divisor Semigroups

ordinal sum F of F A

,

)r

67

E A is also a weakly cancellative

d-monoid. We will assume later on that a weakly cancellative d-monoid is always given in this form. Clearly H can be embedded in F. In the same manner as we previously adjoined a neutral element, here we adjoin the possibly missing inverse element of the first irreducible subsemigroup. We remark that for the irreducible case a characterization of weakly cancellative d-monoids follows from proposition ( 4 . 2 ) and theorem ( 4 . 5 1 ,

i.e. all cancellative d-monoids can be

embedded into a Hahn-group. CLIFFORD ([19581, 119591) characterizes another class of irreducible d-monoids. A d-monoid H is called c o n d i t i o n a l Z y c o m p l e t e if each subset A which is bounded from above has a least upper bound. Then each A which is bounded from below has a greatest lower bound. CLIFFORD considers conditionally complete d-monoids which are positively ordered. Due to the positive order relation the first subsemigroup in the ordinal decomposition is always the trivial group {el. Therefore we will consider only d-semigroups without neutral element. A positively ordered d-semigroup H is called A r c h i m e d e a n if

3 nEH

:

for all a , b E H , a

*

(4.14)

a

n

L b

e. The following results are due to

CLIFFORD ([1958], [1959I).

(4.15)

Proposition

Let H be an irreducible, conditionally complete positively ordered d-semigroup without neutral element. Then H is Archimedean.

68

Ordered Algebraic Structures

Proof. Let a , b E H . If an = a

n+ 1

for some nElN then an is

idempotent and due to ( 3 . 6 ) the maximum of H. Therefore an

2 b.

n n+ 1 Otherwise an < a for all n E N . Suppose a < b for all n E N . Then c:= sup{anl nElN] exists and an < c for all nElN (an = c n implies a n+l - a 1.

Now a * d = c for some d E H . If d < c then

d < an for some n and therefore a * d i.e.

5

an+l < c. Thus c = d ,

a * c = c.

Let L:= (91 g < cl. For x , y E L we find max(x,y) < a nElN and therefore x * y

5

a

n

for some

2n . implying x * y E L . Further

x * c f an * c = c shows x * c = c. Let U : =

(91 g

2

c).

Clearly

positivity implies that U is a subsemigroup. Let y E U , c < y. Then c

*

z

=

y for some z E H . Suppose z E L . Then c * z = c

contrary to c e y. Therefore z E L . Let x E U . Then x * y = x * c * z = c * z = y. Thus we have found that H is an ordinal sum of L

and U , contrary to our assumption. Therefore H is Archimedean. w

Now Archimedean, positively ordered d-semigroups have been characterized by HOLDER [19011 and CLIFFORD 119541. It should be noted that an Archimedean, linearly ordered divisor semigroup always is commutative (HOLDER [19011, LUGOWSKI 119641). Thus our assumption of commutativity is redundant. The following result is due to CLIFFORD [1954]:

(4.16) P r o p o s i t i o n Let H be an Archimedean, positively ordered d-semigroup. If H is not cancellative then

69

Linearly Ordered Commutative Divisor Semigroups

(1)

H contains a maximum u,

(2)

for all a

(3)

if a * b = a * c

*

e there exists nEaJ such that a

*

n

u,

=

u for a , b , c E H then b = c.

Proof. Due to the assumption there exist a , b , c E H with a * b = a * c =:u

and b

*

c. Now b * x = c for some x

u + x = u. Suppose u < y E H then y

5 xn for some n

we find the contradiction u = u * x

n

1. u

*y

1. y >

(1) and ( 3 ) . Now for a E H (a 9 e) we know a

i.e.

a

n

n

*

e. Thus

E N . Therefore

u . This shows

5 u for some nElN,

= u.

Now the following theorem characterizes Archimedean, positively ordered d-semigroups. The result is due to HOLDER [1901] and CLIFFORD [1954]. Its proof is drawn from FUCHS 119661.

(4,. 1 7 ) T h e o r e m H be an Archimedean, positively ordered d-semigroup. If H is cancellative, then it is isomorphic to a subsemigroup of the positive cone

(JR+,+,z) of

the real numbers.

In particular if H has a minimum and contains no neutral element then H is isomorphic to the additive semigroup of natural numbers (IN

,+,z).

If H has a maximum u and u has no immediate predecessor then H is isomorphic to a subsemigroup of ([O,ll,*,5) with respect to the usual ordering and a *b:- min(a+b,l). In particular, if H has a minimum and contains no neutral element, then H is isomorphic to (k/nl O * k scme n E m .

5 n) for

70

Ordered Algebraic Strucrures

(3)

If H has a maximum u and u has an immediate predecessor then H is isomorphic to a subsemigroup of ( [ O , l ] U { = ) , @ , ~ ) with respect to the usual ordering and a @ b : = a+b

1.

1 and a @ b : =

a + b if

otherwise.

m

Proof. We always discard the neutral element e if it exists in H. At first we assume that H contains a minimum a. Let b E H , b

*

a. Then ak < b 5 a k + l for some k Em and ak

c E H . Now a

k + l = a k * a z ak * c = b L a

fore H is generated by a. If a isomorphic to (IN

n

< a

, + , z ) .Otherwise

*c

= b for some

k + l shows a k + l = b. There-

n+l

for all n E B then H is

...

a < a2
a*b)

neg(a):=

CbEH(

a

a*b)

,

dom(a):=

IbEHI

a = a*b)

,

pos(a):=

IbEHI

a f a*b)

,

:=

{bEHI

a < a*b)

.

P(a)

2

The elements of these sets are called s t r i c t l y n e g a t i v e , n e g a -

t i v e , d o m i n a t e d , p o s i t i v e and s t r i c t l y p o s i t i v e

( w i t h respect

t o a).

( 4 . 1 8 ) Proposition Let A be a d-monoid. Let a , c E H and b:= a * c . Then

Ordered Algebmic Structures

7.1

N(c) E neg(a) ;

(3)

P(c)

(4)

dom(a) and pos(a) are d-monoids;

(5)

neg(a) is a linearly ordered commutative monoid.

Proof.

pos(a),

( 1 ) It suffices to prove the inequalities. Let x E d o m ( a )

and y E P(a). Then a * x = a < a * y implies by cancellation x < y. N(a) < dom(a) is similarly proved.

(2) is an immediate

conclusion from the definitions and ( 1 ) . ( 3 ) Let x E P ( c ) .

Then c * x > c = c * e . Therefore x >

for a E H we get a * x

a, i.e.

xEpos(a).

e.

Then

N(c) C_ neg(a) is

shown in the same way. At first consider dom(a).

(4) =

Let x , y E d o m ( a ) . Then a

*

(x * y )

a * x = a. Now assume x < y and x * c = y for some c E H . Then

a S c = a * x * c = a * y = a shows c E d o m ( a ) . Now let x , y E p o s ( a ) . Then a

5 a

* y

5 a

* x * y .

Assume x < y and x * c = y for some

c E H . Then x < x * c shows c E P ( x ) C_ pos(a) (5)

(cf. ( 3 ) ) .

Similarly a s in ( 4 ) the discussion of pos(a).

We remark that in general neg(a) is not a d-semigroup. This is due to the asymmetry in the divisor rule. For example assume

*

(a

and let a E P - . Then a < e and for a solution c E H with

a *c

=

e we know c = a

P-

-1

commutative semigroups

(4.1) '

a < b

=o

> e. We may consider linearly ordered

(H,*,c)which 3 c E H :

for all a , b E H . Then ( H , * , z ) If (H,*,()

fulfill

a = b * c

is called a d u a l d - s e m i g r o u p .

is a d-semigroup and a dual d-semigroup then (H,*,L)

is a linearly ordered commutative group. All results for d-

semigroups can be translated into a result for the corresponding

15

Linearlv Ordered Commutative Divisor Semigroups

dual d-semigroup which we get by reversing the order relation of H. Now we define for a d-monoid H with ordinal decomposition (HA i h E A )

u if a

the i n d e x h(a) o f an element a E H by h(a) :=

is an element of H

u

.

(4.19) P r o p o s i t i o n Let H be a weakly cancellative d-monoid with ordinal decomposition ( H A I X E A ) .

5 min(X(b),X(c))

If h(a)

then a * b = a * c implies b = c.

< min(A(b),A(c)) If h(a) -

5 d*c

and d f a then a * b

implies b f c. For a < b there exists a unique c E H with a * c = b. Then h(c) = h(b). If h(a) = h(b) and a * b = a then b = e or H If an = b If b

n

(a)

=

{a].

then a = b.

< a for j = 1,2,...,n

j -

and a

5 b:= b l * b 2 * . . .

* bn

then X(a) = X(b). Let a:= a l * a 2

* ... * a n ,

b:= b

1

* b 2 * ... *

bn. If

a * c < b * c then there exists p with a * c < b

’

*(*

a )*c.

j*u

j

Proof. (1) Assume b

and h(c)

*

c. Then a

*b

5 h(a). Then h(a)

=

a = a

* c.

Therefore h(b) 5 h(a)

= A(b) = h(c) yields a contradiction

as the irreducible H A are cancellative. (2) From a * b

5 d

*c

and d

5 a we find a * b 5 a

* c.

Now ( 1 )

implies b = c if equality holds. Otherwise b < c. ( 3 ) We know a * c = b for some c E H . Then h(a),h(c)

f X(b).

Ordered Algebmic Structures

76

Suppose X ( c ) < X(b). Then X (a) = A(b) and therefore a

*c

= a< b

contrary to the choice of c. Thus h ( c ) = h(b). If h(a) < h(b) then c = b. Otherwise h ( a ) = X(b) = X(c) and c is uniquely determined as H (4)

A (a)

is cancellative. If X(a) = X o then b = e. Other-

This follows from (4.13.3).

wise suppose that H

X (a)

is not the trivial group. Then H

A (a)

is isomorphic to the strict positive cone of a group. Hence

> a contrary to the assumption.

a * b

(5)

an = bn implies h(a) = X(b). Now H

X (a)

is cancellative and

thus a = b.

...,

= maxib.1 j = 1 , 2 , n}. Then A(b) = X(b n 7 < X'(a). On the other hand a 5 b implies X(a) 5 h(b).

( 6 ) W.1.o.g.

assume b

(7) Suppose that a * c

Then an

* cn z

b

2 bu

* a n-1 * c n .

If

*c

' " > " holds then

* ( *

a.)

for all

j*!J

*c

cellation in H

< b

*c

X (a*c)

implies X(a

* c)

5 X(b

= 1 , 2 , . ..,n.

cancellation leads

to the contradiction a > b. Otherwise A ( b * c ) other hand a

u

5 X(a *c). On the

* c).

Then can-

implies a * c = b * c contrary to the assump-

tion.

In the remaining part of this chapter we give examples for different classes of d-monoids. The relationship between these classes is visualized in the following diagram (Figure 5 ) .

)

Linearly Ordered Cammutative Divisor Semigroups

77

d-monoids

weakly cancellative d-monoids

conditionally complete d-monoids

d-monoids in Hahn groups

d-monoids in the real additive group

cancellative fundamental monoids

F i g u r e 5.

fundamental monoids with maximum

Relationship of classes of d-monoids

A line joining two classes means that the upper class contains the lower one. More details can be found in the following remarks and examples. The f undar nent at monoids are the monoids explicitly mentioned in (4.17).

( 4 . 2 0 ) 0-monoids i n t h e r e a l a d d i t i v e . group The only conditionally complete subgroups are (IR

, + r z ) ,(iZ,+,z);

positive cones of these are the cancellative fundamental monoids in (4.17). If a d-monoid H contains an element aEIR+\iZ+ H is d e n s e in IR+ y E

then

(i.e. for a , B E I R + with a < B there exists

H such that a < y < B ) .

A typical example i s ( p + f + , z ) , the

positive cone of the additive group of rational numbers. Another example can be derived from a transcendental number a € (0,l). Let R denote all real numbers of the form

-

p, a’

with

7b

Ordered Algebraic Structures

r € l N U 10) and p . € Z for each j. Due to the transcendency of 1 a this representation is unique and R is a well-defined dense subgroup of IR. Two linearly ordered commutative groups isomorphic to ( I R , + , z )are considered in the following. The multiplicative group (IR

+

\ { O ) , = , z ) of all positive real numbers

is isomorphically mapped ontoIR by x

-B

In x. Now let

( - 1 ) 1 /P

p:=

for p €IN. On R : = IR+p U IR+ we define an extension of the usual

order relation of IR+ by W + P zlR+ and a , 6 Em+.

ap

1. 6 p

a

>

6 for

With respect to the internal composition a@b:=

the system

(R,@,c)

D-monoids

(aP+bP l/P

is a linearly ordered commutative group which

is isomorphically mapped onto ( I R , + , L ) by x

(4.21)

iff

+

xp.

i n Hahn-groups

A d-monoid in a group is always cancellative and therefore irreducible. In particular, d-monoids in the real additive group are examples. Further from (4.11) w e know that a cancellative d-monoid can always be embedded in a linearly ordered commutative group. Such groups can be embedded into Hahn-groups. Therefore all cancellative d-monoids are isomorphic to a dmonoid in a Hahn-group.

A

the additive group (IRn , + ,

typical example of a Hahn-group is * )

of real n-vectors ordered with

respect to the lexicographical order relation. If n > 1 then the strict positive cone P of this group is a cancellat ve d-semigroup which is positively ordered and irreducible but not Archimedean. For example, l e t n = 2 .

Then (0,l) 4

but there exists no n €IN such that n(0,l) = (0,111

+

(

I

1

(1,O). Thus

P is not Archimedean and hence not conditionally complete

(cf. 4.15). A more general example is the following.

0

Linearly Ordered Cummutah'veDivisor Semigroups

Let ( A , i )

19

be a nonempty, linearly ordered set and let for

each A E A

H A be a cancellative d-monoid. We consider the

subset H of the Cartesian product X A E A H A w h i c h consists in all elements

(...

is w e l l - o r d e r e d .

xA

... )

such that Sx:= { A E A l x A 9

(Each subset of

*

On H an internal composition

S

e,)

contains a minimum.)

X

is defined componentwise.

Further H is linearly ordered with respect to the lexicoy

if

x = y

or

graphical order relation, i.e.

x.6

x A < y A for A = min{u E A 1 x

yu}. Then ( H , * , d 1 is called

u

9

the l e x i c o g r a p h i c a l p r o d u c t of the family tion:

H =

(HA, A € A)

[Denota-

n A E A H ) , ] .H is a cancellative d-monoid if and only

if for all A > min A

H A is a linearly ordered commutative

group. Further H is a linearly ordered commutative group if and only if for all A

H A is a linearly ordered commutative

group. For example, let H denote the set of all functions f: [O,l] +IR

*

such that Sf:= 1x1 f(x)

01 is well-ordered. Together with

componentwise addition and lexicographic order relation H is a linearly ordered commutative group. We remark that H is the lexicographic product ll

AE[O,

H A = IR for all A E [ 0 , 1 1 .

As

11

H A of real additive groups, i.e

all H A coincide with lR we call H

a r e a l h o m o g e n e o u s l e x i c o g r a p h i c a l product. H is isomorphic to the Hahn-group V([O,ll)

(cf. chapter 3 ) with respect to

the inverse ordered set [0,1]. In this way each Hahn-group V(A)

is "inverse" isomorphic to the real homogeneous lexico-

graphical product

nAEAH A

'

Ordered Algebraic Structures

80

(4.22) Weakly cancellative d - m o n o i d s A l l weakly cancellative d-monoids are ordinal sums of cancella-

tive d-semigroups. Therefore all Irreducible weakly cancellative d-semigroups can be embedded into Hahn groups. As

first example we consider ([O,l],*,z)

with internal compo-

sition defined by a*b:= a + b

-

ab

and by

Both are weakly cancellative, positively ordered d-monoids. The irreducible subsemigroups are { O ) , (0,l) and {l}: in particular the nontrivial subsemigroup can isomorphically be mapped onto ( I R , + , z )by x A

+

-ln(l-x) and by x+-ln((l-x)/(l+x)).

further example is the r e a l bottleneck s e m i g r o u p

with internal composition defined by (a,b)

+

(lR,max,z)

max(a,b) and with

respect to the usual order relation of real numbers. Its ordinal decomposition is ((a);

aEIR), i.e.

all irreducible subsemi-

groups are isomorphic to the trivial group. We may adjoin

-a

as a neutral element. All elements of the bottleneck semigroup are idempotent. This semigroup is not cancellative but conditionally complete. Now we consider (IR x I R + , * , + )

with internal

composition defined by

for all (a,b), (c,d) EIR of the form

{(a,O))

xlR+.

and { a )

Its irreducible subsemigroups are x

( W + \{O))

for aEIR. We define a

81

Linearly Ordered Commutative Divisor Semigroups

suitable index set A : =

{1,2}

x

IR.

Clearly, (1,a) is the index of

{(a,O)}, and (2,a) is the index of {a}

x (IR+x{O}).

A is linear-

ly ordered by - $ . We may adjoin a neutral element (-0.0) or a first irreducible semigroup

Then the new ordinal

( { - a )xIR,+,.$).

sum is called time-cost semigroup. This semigroup is not conditionally complete. The most general example is the following. Let G h be a real homogeneous lexicographic product for each h € A . Let A have a minimum h o . for h > h

. Le.t H

Then H A is either (GA)+ or (Gh)+x{eA}

be the ordinal sum of G h

and all H A . Then H A

0

is called a homogeneous weakly c a n c e l l a t i v e d-monoid. Clearly,

all weakly cancellative d-monoids can be embedded into a homogeneous weakly cancellative d-monoid.

(4.23)

Conditionally complete d-monoids

All fundamental monoids are conditionally complete. In the positively ordered case all irreducible, conditionally complete d-semigroups are isomorphic to the strict positive cone of a fundamental semigroup. A weakly cancellative example, the Our next example is

bottleneck semigroup, IS given in ( 4 . 2 2 ) . not weakly cancellative. We consider (IR

x

[O,l],*,+)

with

internal composition defined by

(a,b)

*

if a > c,

_.

if c > a,

(c,d):= (a,min (b+d,1)

i f a = c,

for all (a,b), (c,d) E l R x [O,lI. Similar to the time-cost-semi-

group in ( 4 . 2 2 ) {a} x (0,1]

the irreducible subsemigroups are {(a,O)]

for aEIR. Again A = { 1 , 2 )

xlR

and

is a suitable index

set. We may adjoin a neutral element ( - 0 , O ) . The reader will

82

Ordered Algebraic Structures

easily construct further examples using fundamental semigroups (without neutral element) from ( 4 . 1 7 )

in the following way.

Let A be a conditionally complete, linearly ordered set and let H x be a fundamental semigroup for each A E A such that the following rules are satisfied (cf. CLIFFORD [ 1 9 5 8 ] ) . If A E A has no immediate successor, but is not the greatest element of A ,

then H A must have a greatest

element. If A € A has no immediate predecessor, but is not the least element of A , If A . u E A ,

C

then H

u and

A

x

must have a least element.

is the immediate predecessor of

u then either H A has a greatest o r H

u has a least ele-

ment. It is easy to see that these rules are necessary for conditionally complete d-monoids. If we do not assume that the considered semigroup is positively ordered then we may consider the ordinal sum of an arbitrarily with negative

irreducible conditionally complete d-semigroup H A 0

elements and an arbitrary conditionally complete positively ordered d-semigroup H without a neutral element. Again we have to fulfill rules ( 1 ) complete d-monoids.

As

(3).

In this way we get a t 2 conditionally

an example consider

which is defined as the ordinal sum o f the group and the subsemigroup (IR

+

x [ O , l ] , * , 4 )

({--}

xlR,+,4)

of the above example.

Linearly Ordered Cbmmutative Divisor Semigroups

(4.24)

83

0-monoids

All examples considered before are d-monoids. An example which is neither weakly cancellative nor conditionally complete can be found from the first example in ( 4 . 2 3 ) subsemigroups of the form {(a,O)) Then

(1R

X

by eliminating all

in the ordinal decomposition.

(0,13,*,4)is no longer conditionally complete as

the rules in (4.23) are not satisfied, but it can normally be embedded into the original one. Here the question arises whether a d-monoid exists which is neither weakly cancellative nor conditionally complete and, furthermore

cannot be normally

embedded into a conditionally complete one. CLIFFORD 119591 gives the following example. Let

be lexicographically ordered, i.e. (0,l)4 (0,2) 4

.. . 4 (l,-l)4 ( 1 , O ) -4 ( 1 , l ) 4 .. .

. . . 4 (2,O) 4 (3,O)4 . . . .

We define an internal composition by

(n,x)

*

(m,y):=

t

(n+m,x+y)

if n+m < 2 ,

(n+m,O)

otherwise.

Then (H,*,4 ) is an irreducible, positively ordered d-semigroup without a neutral element, A s H is not Archimedean it cannot be conditionally complete. Further H we try to embed

H

into

a

i s

not cancellative. Now

normal completion. The only undefined

supremum (infimum) is a:= sup{ (0,n) I n E N } . We consider H U { a ) with the obvious extension of the linear order relation on H. If the extended internal composition

*

satisfies the monotonicity

Ordered Algebraic Structures

a4

condition then it is uniquely defined by a

*

(O,n):= a

,

f o r all nElN and f o r all z E Z .

a *a:=

(2,O)

But now a < (1,O) and there

exists no x E H U { a ) such that a * x = ( 1 , O ) .

is not a d-semigroup. CLIFFORD [ 1 9 5 9 ] a positively ordered d-semigroup

Therefore H U { a }

shows that, in general,

can normally be embedded in-

to a conditionally complete d-semigroup if and only if all its irreducible subsemigroups are Archimedean. Then this normal completion is uniquely determined.

5.

Ordered Semimodules

In this chapter we introduce ordered algebraic structures which generalize rings, fields, modules and vectorspaces as known from the theory of algebra. Additionally these structures will be ordered and will satisfy monotonicity conditions similar to the case of ordered semigroups. Let (R,@) be a commutative monoid with neutral element 0 and let (R,(D) be a monoid with neutral element 1

(0

*

1 ) . If

(5.1)

for all a , B , y E R then (R,@,@) is called a s e m i r i n g w i t h u n i t y 1

and z e r o 0. We will shortly speak of a semiring (5.1.2)

R.

and

(5.1.1)

are the laws of d i s t r i b u t i v i t y . We remark that

@

is not

necessarily commutative. Otherwise we call R a c o m m u t a t i v e

s e m i r i n g . If all elements of R are idempotent with respect to then R is called an i d e m p o t e n t s e m i r i n g . NOW, if ( R , @ , Z ) is an ordered commutative monoid and

for all a,B,y,6 E R with y

2

0, 8

ordered semiring. 85

5

0 , then R is called an

@

86

Ordered Algebraic Structures

(5.3) P r o p o s i t i o n (I)

A

(2)

An idempotent semiring is partially ordered with respect

semiring is idempotent iff 1 B 1 = 1.

to the binary relation defined by a

5 B iff

a B

B

B

=

for all a , B E R . Proof. ( 1 ) The if-part is obvious. Now let 1 B 1 = 1. Then a

6a

(1

1 ) = a B 1 for a E R .

@

Distributivity shows that a is

idempotent. (2)

If a is an idempotent element then a 5 a . Commutativity im-

plies antisymmetry. Associativity leads to transitivity. is partially ordered a s a = 6

B y for all a , B , y E R .

@

B

=

B implies ( a B y )

Clearly 0

5

y

@

for all y E R .

(B

8)

(R,B,f) y)

Then

distributivity implies ( 5 . 2 . 1 ) .

In particular, if ( R , @ ) is a commutative group then R is called a ring. (5.4)

A

semiring R with

a @ B = o

I ,

a = O

or

B = O

for all a , E E R is called a semiring w i t h o u t ( n o n t r i v i a l ) d i v i s o r s Of

z e r o . A commutative ring without divisors of zero is called

an i n t e g r a l domain. called a f i e l d .

A

A ring R with commutative group ( R \ { O } , B )

well-known theorem from the theory of algebra

is the following.

(5.5) P r o p o s i t i o n A

is

(linearly ordered) integral domain R can be embedded in a

(linearly ordered) field.

Ordered Sem'rnodules

87

Proof. We give only a sketch of the proof which is similar to the proof of theorem ( 2 . 8 ) . An equivalence relation is defined by ( a , B )

R x (R\{O))

-

(y,6)

-

on

iff a O 6 = 8 O y .

We denote the equivalence class containing ( a , B ) by

a/B

and

we identify a E R with a / l . Then the internal composition may be extended by

for all a / B , y / 6 .

It can be seen that then the set F of all

equivalence classes is a well-defined field. With respect to the linear order relation defined by

for all a / B , y / 6

this field is linearly ordered.

The following result for the Archimedean case is similar to (4.17.1).

(5.8) Proposition An Archimedean linearly ordered field ( F , @ , O , Z ) i s isomorphic to a subfield of the linearly ordered field ( I R , + , * , zof ) real

numbers. proof. We only indicate how this result follows from the proof of ( 4 . 1 7 ) . Let

6

The details are left to the reader.

resp.

that F':=

1

denote zero resp. unity of F. It suffices to prove

{aEFI a >

of a subfield of IR

6 , is

isomorphic to the strict positive cone

(IR' : = {aEIR[ a > 0)). F' does not contain a

minimum. Therefore using the proof of ( 4 . 1 7 )

we can choose an

Ordered Algebmic Structures

88

isomorphism rp with a ( ? ) = 1 such that

l ! l

maps (F',B,Z) isomor-

phically onto a dense subsemigroup of ( W ' , + , ( I . cp maps =

$:=

frp

-1

We find that

(9)I q E Q , q > 0 ) onto Q such that rp(a Q 5 )

(P(a)rp(B) for all a , B E Q ' . Then

for all a E F ' leads to cp(a

Q

5 ) = cp(a)cp(5)

for all a , B E F ' .

Now let (H,*) be a commutative monoid with neutral element e and let (R,tB,@) be a semiring with unity 1 and zero 0. If n:

R X H + H

is an e x t e r n a l c o m p o s i t i o n such that

(a 8 6) o a = a n (6

(5.9)

(a B 5) n a =

(a

=

(a

an(a*b)

a) 0

,

a)

*

(5

a) * ( a

0

a)

,

b)

,

O O a = e , l O a = a ,

for all a , B E R and for all a , b E H then ( R , @ , Q ; U ; H , * )

is called

a s e m i m o d u l e o v e r R. We remark that the adjective 'external' does not exclude the case R ' = H .

In our definition we only

consider a left external composition

0:

R XH

one may consider a r i g h t external composition

+

H.

0:

Similarly, H x R + H.

If

we want to distinguish between such semimodules then we will use the adjective "left" or "right". Usually we will only treat left semimodules and omit the adjective "left". A

direct consequence of (5.9) is the relation a o e = e for all

a E R . This follows from e = O n e = (a

(D

0) n e = a n ( O n e ) = a n e .

Ordered Semimodules

89

In particular, if R is a ring then H is a commutative group with a

-1

=

( - 1 ) O a for all a E H (-1 is the additive inverse

of 1 in R ) .

Then H is called a m0dUZe over R .

then H is called a v e c t o r s p a c e over R .

If R is a field

We remark that if R is

a ring and H a commutative group then O o a = e is a redundant assumption in (5.9). N o w let

(H,*,L)

be an ordered commutative monoid with neutral

element e and let ( R , @ , @ , Z ) be an ordered semiring with unity 1 and zero 0. Let

0:

R x H

+

H be an external composition satis-

fying (5.9),

for all a , B E R , (5.10.2)

c , d E H with d 5 e 5 c and

a z b

*

yoa(yob

for all a , b E H , y E R + : = { E E R I Then (R,@,@,Z;O;H,*,5)

E

and

2

6 0 a 2 6 0 b

0 ) and 6 E R - : =

{EERI

E

5

0).

is called an o r d e r e d semimodute over R .

We remark that we will use the same denotation different order relations in R and H .

5 for the

From the context it will

always be clear which one is meant specifically. If, in particular, ( R , @ , @ , Z ) is the positive cone of a field then H is called an o r d e r e d s e m i v e c t o r s p a c e over R .

Some direct consequences should be mentioned:

(5.11) P r o p o s i t i o n Let H be an ordered semimodule over R . (1 1

If H is a group then a ( 6

*

aoc(B0c

Ordered AIgebmic Structures

90

for all a , B E R and c E H + implies (5.10.1). If R is a ring then

(2)

*

a z b

yoa(y0b

for all a , b E H and y E R + implies (5.10.2). N

(3)

If R is the positive cone of a linearly ordered ring R N

and H is the positive cone of a linearly ordered group H then the external composition can be continued in a N

W

N

unique way on R x H such that H is a linearly ordered N

modul over R.

5 d Is equivalent to c

Proof. ( 1 ) In an ordered group c

This follows from composition of c

5

d with c

-1

*d

-1

.

2 d -1

-1

NOW let

a , B E R with a

5 8 and let d E H-. Then d-' E H+ a n d , due to our

assumption, a

0

d

-1

(B

= e shows that a o d

-1

Od-l. Now = (a od)

(2) In particular, ( H , * ) and let 6 E R - .

(-6) O a

1.

(a

-1

.

od

-1

)

*

(-6) o b . Now [ (-6)

(Rfl'ii- = ( 0 ) ) and

;=

Ex;

by

on

0

-1

*

a]

.

( 6 Oa) = "-6)

Therefore 6 O a

HUZ-

a ob:=

Bob:= for all aER\{O), N

61 o a

=

e

2 6 Ob. = RUE-

I

- 1 -1 (aob )

8 oa:= ( ( - B )

@

( H n z - = {el). Therefore we may

OOb:= e (5.12)

Bod.

Then -6 E R + a n d , due to our assumption,

( 3 ) A s R and H are linearly ordered we know

0

2

*d)

is a group. Now let a , b E H with a f b

shows that (-6) o a = (6 n a )

continue

-1

( a o d ) = a 0 (d

Therefore a o d

.

oa)

(-8) O b

-1

-1

, , I

B E z i R , a E H , bE'ii\H.

Then

is a module

over R. In a module the equations used in (5.12) are necessarily

Ordered Semimodules

fulfilled. Therefore the extension of

0

91

is uniquely determined.

As B is an ordered semimodule over R we know that a

5 5 im5 b im-

c 5 5 O c for all a , B E R and c E H and that a

plies a

0

plies y

0a

1. y

0b

for all a r b€ H and y E R .

It suffices to show

that the first implication holds for all a , 5 E'ii and that the N

second implication holds for all a , b E H . Then (1) and ( 2 ) N

is a linearly ordered module over R .

imply that

At first

we remark that for a 5 B only the additional cases a E R N

and a , 5 (-a)

0

c

€s

occur. Then a

(-B)

0

c 5 00 c = e 5 5

O c lead to a O c

0

c

, 5ER

resp.

5 5 o c . The second implication

follows in a similar way.

Proposition ( 5 . 1 1 )

shows that for special semimodules some of

the monotonicity conditions are redundant. Next we consider some examples.

(5.13) P a r t i a l l y o r d e r e d s e m i r i n g s Let ( R , @ , @ , ( )

be a partially ordered semiring. Then (R,@,z)

is a partially ordered semimodule over ( R , C e , @ , L ) to the "external" composition defined by a a,BER.

In particular, ( 5 . 9 )

follows from ( 5 . 2 ) .

0

5:=

follows from ( 5 . 1 )

with respect a @

5 for all

and ( 5 . 1 0 )

As an example, (Z+,+, 5 ) is a linearly

ordered commutative semimodule over the linearly ordered commutative semiring ( Z Z + , + , ~ , 5A) s. in ( 5 . 1 1 . 3 )

this semi-

module can be extended to the linearly ordered commutative

,+,z) over

module (Z

the linearly ordered commutative ring

(22 , + , * , 5 ) .Further the additive group of a subfield of the

real numbers is a vectorspace over this subfield. The positive

92

Ordered Algebmic Structures

cone of such an additive group is a semivectorspace over the positive cone of such a field. Finally we remark that ( R " , @ , Z ) with respect to componentwise composition and componentwise order relation is again a partially ordered monoid. Therefore (R

n

,@,i)is

a (right as well as left) partially ordered semi-

module over ( R , @ , O , z ) .

In particular 23

n

n (Q

and

ordered commutative modules (vectorspaces) over

n IR ) are latticeZ(Q

and IR)

.

( 5 . 1 4 ) Semirings Let ( R , @ , 8 ) denote a semiring. Then a i b

:-

a @ b = b

defines a pseudo-ordering on R with minimum 0. In general ( R , @ , Z ) is not a pseudo-ordered monoid but satisfies

(5.14.2)

a z B

for all a , B , y , 6 E R .

and

y i 8

*

a @ y z B

Further (5.2) is satisfied. If R is an

idempotent semiring then R is a partially ordered semiring. From (5.13) we conclude that ( R , @ , L ) is a partially ordered semimodule ( R , @ , @ , z ) .

In this partial order the least upper

bound is well-defined, i.e. sup(a,b) = a @ b but the greatest lower bound may not exist, in general.

( 5 . 1 5 ) R e s i d u a t e d l a t t i c e - o r d e r e d commutative monoids Let ( R , O , < )

be a residuated lattice-ordered commutative monoid

with neutral element 1 . At the end of chapter 2 we have

Ordered Semimodules

93

mentioned lattice-ordered commutative groups, pseudo-Boolean and, in particular, Boolean lattices as examples for this ordered structure. If not present we adjoin a minimum 0 and a maximum

m.

The internal composition is extended by

0 = 0 8 a = a 8 0 for all a E R ' : =

= a 8

03

for all a E R U { m } .

R

U

and

{O,m)

Let a @ B : =

=

-

8 a

a V 8 for all a , B E R ' .

Then (5.1) is satisfied by all a , B , y E R ' .

In particular

distributivity follows from (2.13). Therefore ( R ' , @ , @ , Z ) is a lattice-ordered, idempotent, commutative semiring with unity I and zero 0. From ( 5 . 1 4 )

we conclude that ( R ' , @ , L ) is a lattice-

ordered commutative semimodule over R'.

(5.16) L i n e a r l y ordered s e m i m o d u l e s o v e r real n u m w Let R + be the positive cone of a subring of the linearly ordered field of real numbers with {O,l)

5

Let ( H , * , z )

R+.

be a linearly ordered commutative monoid and let

0 :

R+xH + H

be an external composition such that ( R + , + , = , L ; O ; H , * , ( ) linearly ordered commutative semimodhle over R +

is a

.

In particular we consider the d i s c r e t e semimodule H over i Z + with external composition defined by z Oa:= az for all z EZ+ and for all a E H .

As

we discuss only semimodules over semi-

rings containing distinct

zero and unity such a discrete

semimodule is contained in each semimodule. Linearly ordered semimodules over real numbers are discussed in chapter 6 .

Now we introduce matrices for semirings R and for semimodules H over R . Matrices with entries in R and H will be denoted by A =

(aij), B,

C

and

X =

(xij), Y ,

2.

For matrices of suitable

Ordered Algebmic Structures

94

size we define certain compositions. Let A , B and X , Y be two m x n matrices over R and H. Then

C = A

B and Z = X * Y

(5.17)

yij:= a

ij

for all i = I r 2 , . . . , m

are defined by

,

e Bij

and j = I , 2 ,

...,n.

Let A be an m x n matrix

over R, let B be an n x q matrix over R and let X be an n x q matrix over H. Then C = A 0 B and

Z

= A O X

are defined by

n (5.19)

yik:=

(5.20)

n z. - = ik’ j=l

j=1

*

for all i = lr2,...,m

(a 0 Bjk) ij

(a

xjk)

ij

I

’

and k = 1 , 2 , ...,q.

Properties of these

compositions correspond to properties of the underlying semiring and semimodule. If the semiring or semimodule considered is ordered then, for matrices of the same size, an order rela-

tion is defined componentwise. A matrix containing only zeroentries is called z e r o - m a t r i x and is denoted by [ O ] . Similarly the n e u t r a l m a t r i x [el contains only neutral elements. A matrix containing unities on the main diagonal and zeros otherwise is called u n i t y m a t r i x and is denoted by [I]. considered for any size m

(5.21)

x

Such matrices are

n.

Proposition

Let H be an ordered semimodule over R. Let A , B , C , D , E over R with D 2 [ O ] and E H with U

[el and V

1.

[O].

Let X , Y , Z , U , V

be matrices

be matrices over

5 [el. The following properties hold for

95

Ordered Semimodules

m a t r i c e s o f s u i t a b l e size. (1)

( A @ B ) @ C = A @ ( B $ C )

I

A @ B = B @ A ( A 8 B )

8 C = A 8

(2)

8

I

I

( A 8 C )

@

( B 8 C )

,

( A @ B ) =

(CtBA)

@

( C 8 B )

,

101

[O]

A ( B A

( B O C )

=

( A @ B ) 8 C

I

B

A L B

C

8 A =

= A 8 101

;

-D

A @ C L B @ C

*

A 8 D C B 8 D

and

D @ A ( D Q B ,

I .

A O E ) B @ E

and

E 8 A Z E t B B ;

I

the set of all m x m matrices over R i s an ordered semiring with unity (4)

(X*Y)

x

111

* z

= X * ( Y * Z )

*Y

= Y

and zero [ O ] ;

I

*x

I

( A 8 B ) 0 2 = A n (BOZ) ( A a B ) 0 2 =

101 (5)

=

*

(BOZ)

[el

I

;

X

i

Y

-b

X * Z L Y * Z

A

I

B

-D

A O U Z B U U

-D

D O X ~ D O Y and

X Z Y

(6)

O X

( A n Z )

I

I

and

A O V L B O V , E n X L E O Y

;

the set of all m x m matrices over H is an ordered semim o d u l e o v e r t h e s e t of a l l m x m m a t r i c e s o v e r R.

Proof. These properties are immediate consequences o f the definitions of the composition considered. A s an example we prove (A 0 B) 0 2 = A 0 ( B O Z )

Ordered Algebraic Structures

96

for m x n and n Z

X

q matrices

over H. An entry x

is

A

and B over R and an q x r matrix

of the left-hand side of the equation

is given by

Using (5.1) and (5.9) we find

which is an entry of the right-hand side of the equation considered. 8

For special semimodules a theory similar to the classical theory of linear algebra has been developed. We refer to m o dule theory (cf. RLYTH [1977]) and to min

- max

algebra (cf.

CUNINGHAME-GREEN 119791). A linear algebra theory in the general case is not known. Further properties will be discussed in the chapters on optimization problems. Here we have described the necessary basic concept for the formulation of equations, inequations and linear mappings over ordered semimodules. Duality principles will be discussed in chapter 10 and chapter 11.

6.

L i n e a r l y O r d e r e d S e m i m o d u l e s O v e r Real N u m b e r s

Let (R,+,=,Z) denote an arbitrary subring of the linearly ordered field of real numbers with zero O E R and unity 1 E R . Let (H,*,z) be a linearly ordered commutative monoid. In this chapter (R+,+,.,l;O;H,*,() dered) semimodule.

As

will always be a (linearly or-

multiplication is commutative the semi-

module is commutative, too. Further (R+,+,z) is a d-monoid as for u,B E R + with (4.17)

a

5 B we know B

- a €

R+. From the proof of

we conclude that R+ is either equal to Z+ or R + is a

dense subset of IR+. In any case, z+c_R+ as { 0 , 1 1

c_

R+.

Therefore the properties of the d i s c r e t e semimodule (Z+ ,+,

,f;O;H,

*,z) play

an Important role.

We do not assume in the following that the semimodule considered I s ordered. Therefore the monotonicity conditions (cf. 5.10) may be invalid. Nevertheless, for discrete semimodules we find: (6.1)

e C a O a

and

a o b c e

f o r all a E Z Z + x f 0 ) and all a , b E H with e

(6.2)

a l e

*

a o a f B o a ,

and

C

a, b

C

e,

uob,Bob

--1

for all a , B E Z+ and all a,b E H with e 5 a , b 5 e l and

for all a E Z +

and all a , b E H . We remark that (6.2) and (6.3)

for all a , B E R + and all a,b E H are equivalent to (5.10). We will show how these properties are related and that they hold In all r a t i o n a l semimodules (R+ 91

g+).

96

Ordered Algebraic Stntctures

(6.4) Proposition L e t H be a l i n e a r l y o r d e r e d commutative monoid which i s a

semimodule o v e r R+.

*

a ( b

(11

If

a o a z a o b

VaER+

V a,bEH

then t h e following p r o p e r t i e s are s a t i s f i e d :

v

< a o a

(2)

e

(2')

a o a < e

(3)

e

(3')

a o a

(4)

a

5 B

9

a o a

5

(4.')

a

5 8

*

aOa

2

5

aER+L{O)

V a E H

with e C a,

V aER+\{O}

V aEH

with a < e ,

v

aER+

V aEH

with e

5

a,

V aER+

V a € H

with a

5

e,

Boa

V a,BER+

V aEH

with e

5

a,

Boa

V a,BER+

V aEH

with a

5 e.

a o a

5

e

Furthermore

(2),

( 3 ) and

as w e l l as

(4)

(Z'),

(3')

and

(4')

are

equivalent. Proof.

(1)

*

( 3 ) . L e t a = e . Then a

same way b = e s h o w s of

(21,

( 3 ) and

(4).

(1)

*

(3'1.

0

a = e leads t o

(3). In the

We p r o v e o n l y t h e e q u i v a l e n c e

The e q u i v a l e n c e o f

(Z'),

( 3 ' ) and

(4')

follows similarly. (2)

*

( 3 ) i s obvious.

(3)

*

( 2 ) . L e t a E R + L { O j and l e t e < a E H .

Now 1 5 n a

Suppose e = a O a .

f o r some n E N . T h e n

contrary t o e < a. (4)

*

( 3 ) . Let a = 0 i n

(3)

*

(4).

e

5 (B

- a)

This leads t o

(4).

Let a,BER+ with a 0

a. T h e r e f o r e

(a

0

5 B a)

*e

and l e t e

5

(a

0

a)

(3).

5

*

aEH. [ (8

-

a)

Then 0

a]. m

Linearly Ordered Semimodules over Real Numbers

99

(6.5) T h e o r e m Let H be a linearly ordered commutative monoid which is a semimodule over R + .

If R + C_ Q + then H is a linearly ordered semi-

module. Proof. Due tc proposition ( 6 . 4 )

it suffices to prove ( 6 . 4 . 1 ) .

Let a E R + and let a , b E H with a

1. b.

a

Ua

a =

5

a

If a = b or a = 0 then

O b . Now assume a > 0 and a < b. A s R + C _ , Q +

we find

n/m f o r some m , n E W with greatest common divisor 1. There-

fore there exist p , q E Z with p n + q m = 1. This implies that

Let a' = (l/m) O a and b' = (l/m) O a . Now a < b implies a' Cb'.. The latter yields (at)mn 5 (b')mn, i.e.

a O a f aob. rn

In the general case R + may contain irrational or, in particular, transcendental numbers. Then even in the case (H,*,z) rn

( m , +-, < )

the monotonicity conditions may be invalid. This is shown by the following example.

(6.6) S e m i m o d u l e c o n t a i n i n g t r a n s c e n d e n t a l n u m b e r Let a € (0,l) be a transcendental number and consider the subring R generated by

ZL

U{a).

Then the elements of R have the

form (cf. 4 . 2 0 ) r j 1 P, a j=o with p tion

j

0:

€ 2 3 , j = O,l,...,r.

R x l R +lR

by

Now we define an external composi-

100

Ordered Algebraic Structures

for all a E I R . Then lR is a module over R. B u t now

shows that ( 6 . 4 . 3 )

is invalid. Therefore the monotonicity con-

ditions (5.10) are invalid. On the other hand we will show that for an important class of monoids all monotonicity properties hold.

( 6 . 7 ) Proposition Let H be a d-monoid which is a semimodule over R+. Then ( 6 . 4 . 1 1 , (6.4.21,

(6.4.31,

and ( 6 . 4 . 4 )

and ( 6 . 4 . 4 ' ) .

(6.4.3'1,

Proof. Due to ( 6 . 4 ) (6.4.1).

are equivalent and imply ( 6 . 4 . 2 ' 1 ,

it suffices to show that (6.4.2) implies

Let a , b E H , u E R + and assume ( 6 4 . 2 ) .

If a = b or a = O

then a O a = a a b . Otherwise a * c = b for some c E H with c > e. Then e
0.

(a, a r )

We identify a

t)

(a,l) if l~ > X N

*he order relation on H

v

and a

c--)

(a,O) if p = X

0

.

is the usual one with respect to the N

second component. In this way H

p

replaces H

p

in the ordinal

decomposition. If this is done for all trivial semigroups the new semimodule

is called e x t e n d e d . This is again a linearly

ordered semimodule over R+. We remark that therefore an extended semimodule is always linearly ordered by definition.

linear-

A

ly ordered semimodule without trivial semigroups in its ordinal N

decomposition is clearly extended. In particular, H is a weakly cancellative d-monoid. The ordinal decomposition of an extended semimodule H has w.1.o.g.

(cf. remarks after 4 . 1 3 )

the following form. H A

=: G I

0

0

is a nontrivial linearly ordered commutative group with neutral element e which is also the neutral element of H. For X > X o we find that H A is the strict positive cone of a nontrivial

Ordered Algebraic Structures

106

linearly ordered commutative group G A . We may identify the neutral element of G A with e. Then G The order relation on G tive cone H A u {el

A = {a-ll a E H A l U {el U H A .

is completely determined by the posi-

(cf. remarks after 2.7). Using (5.12) to

continue the external composition on R x G A

+

G A for A > A.

find that G A is a module over the ring R for all A € A .

we

We

remark that it is not useful to consider the ordinal sum of the groups

GA.

> 1 then such an ordinal sum is not an or-

If I h l

dered semigroup. Several cancellation properties in extended semimodules will be helpful in part 11.

(6.17) Proposition Let H be a weakly cancellative d-monoid the ordinal decomposition of which contains no trivial subsemigroups. Then

-

for all a , b E H . If H is an extended semimodule over R+ then (4 1

a z B

(5)

a ( B

c,

a a a ~ B o a , a o b z B o b ,

for all a , 6 E R + and all a , b E H with b < e < a and (6)

a z b

for all a E R + , then

c.

u o a z a o b

a > 0 and all a , b E H .

If a < 6 or A(a)

5 A(b)

107

Linearly Ordered Semimodules over Real Numbers

for all a , b , c E H and all a , B E R + with a Clearly, a < a

Proof. ( 1 )

*b

5 8.

implies e < b and X (a)

the reverse implication assume e < b and A ( a ) A(a)

5

A(b).

For

5 A(b). If

< X(b) then a < a * b follows from the definition of ordi-

nal sums. If X(a) = X ( b ) then a < a * b follows from ( 4 . 1 9 . 4 ) applied to an extended semimodule. (2)

and ( 3 ) are proved similarly.

( 4 ) As

(5.10)

implies a a

0

a =

B

0

is satisfied it suffices to prove that a n a

5 B if e

C

a. Suppose

5 B Ua

a. If a o a < B O a then a < 8 . Now assume

B
X(b) and a < 6 .

Then h(a) > X o and therefore a > e . In

an extended semimodule this implies a o a

C

B O a . Again

(6.13)

shows (7). 8

From ( 1 )

-

(3)

in proposition ( 6 . 1 7 ) we conclude the following

corollary which describes the sets corresponding to the different definitions of positivity and negativity considered in the discussion of d-monoids in chapter 4 .

(6.18)

Corollary

Let H be a weakly cancellative d-monoid the ordinal decomposition of which contains no trivial subsemigroups. (1)

The set H+ of all positive elements is equal to {a1 e

5 a);

strictly positive elements exist only if A has a maximum

Ordered Algebraic Structures

and then the set of all strictly positive elements is (a1 e < a , i(a) = max A } .

The set H- of all negative

elements is nonempty only if l A l = l ; then H-

=

{a1 a z e )

and the set of all strictly negative elements is equal to H - \ {el. The set P -

of all self-negative elements is equal to H'H+;

the only idempotent is e and the set { a / a < a * a ]

self-positive

of all

elements is equal to H + \{el.

Let a E H . The partition N(a) < dom(a) < P(a) of H in strictly negative, dominated and strictly positive elements with respect to a is given by {el
O ,

a

0

a:= otherwise.

Then we find the linear algebraic function z(x)

= minIc. I x

I

j

> 01

f o r x E I R y with coefficients c i E l R U

{-I,

i E N.

I n particular, we may consider the semimodule (R,@) over the semiring ( R , @ , @ ) .

Then the linear algebraic function has the

form z(x) = x

T

Q

c =

cl)

(XI @

@

(x,

Q

c,)

@

...

@

(Xn

0 Cn)

for x E R n with coefficients c . E R , i E N . Further, a commutative semigroup ( H , * ) is a discrete semimodule over the semiring ( Z +, +

Then the linear algebraic function has the form

, a ) .

X

z(x) = x T n c = c for x E 22;

1

X

X

* c ,*...*c 2

with coefficients c

i

EH,

n

iEN.

NOW we consider a subset S o f Rn. The elements of S are called

f e a s i b t e o r f e a s i b Z e solutions. In ordered semimodules H over R we define the l i n e a r a l g e b r a i c o p t i m i z a t i o n p r o b l e m b y (7.2)

N

z:=

inf XES

xToc

Linear Algebraic Problems

113

with c o s t c o e f f i c i e n t s c i E H, i E N . In general, existence of an optimal value Y ' E H is not known; even if is possible that

exists then it

is not attained for any feasible solution.

Such questions are discussed for the respective problems in the following chapters, For most of these problems we can give a positive answer. As for classical linear programs and combinatorial optimization problems a characterization of the set of all feasible solutions by linear equations and inequations is very useful. The necessary basic definitions for matrix compositions over ordered semimodules are introduced at the end of chapter 5 .

L i n e a r a l g e b r a i c e q u a t i o n s l i n e q u a t i o n s ) appear in various forms similarly to linear algebraic functions; for example A B x L b for an m

x

n matrix

A

over R and b E Rm. Then {x E RnI

A @ x

5

b)

is the corresponding set of its solutions. In the discussion of duality principles we find linear algebraic equations of the form A for an m

x

T

n u l c

n matrix A over R and c E Hn.

In view of the discussion of ordered algebraic structures in part I it is highly improbable that one can develop a reasonable solution method for the general problem (7.2). Nevertheless, it is possible to solve (7.2) for several classes of problemei the simpler the underlying combinatorial structure of the problem the more general the algebraic structure in which a solution method can be developed. By analyzing a

Linear Algebraic Optimization

114

g i v e n c l a s s i c a l o p t i m i z a t i o n problem and a g i v e n method f o r i t s s o l u t i o n w e may f i n d a n a p p r o p r i a t e a l g e b r a i c s t r u c t u r e

i n which t h e g e n e r a l i z e d method r e m a i n s v a l i d f o r t h e s o l u t i o n o f t h e g e n e r a l i z e d o p t i m i z a t i o n problem.

Using t h i s a l g e -

b r a i c a p p r o a c h we c a n g i v e a u n i f y i n g t r e a t m e n t o f d i f f e r e n t o p t i m i z a t i o n problems and d i f f e r e n t s o l u t i o n methods;

in this

way t h e r e l a t i o n s h i p b e t w e e n v a r i o u s p r o b l e m s a n d m e t h o d s becomes more t r a n s p a r e n t .

F u r t h e r by s p e c i a l i z i n g t h e g e n e r a l

method f o r t h e subsumed c l a s s i c a l p r o b l e m s i t i s o f t e n p o s s i b l e t o f i n d new s o l u t i o n m e t h o d s ;

i n t h i s way we may d e v e l o p m o r e

e f f i c i e n t algorithms.

We d i d n o t t r y t o c o v e r a l l o p t i m i z a t i o n problems which have been d i s c u s s e d o v e r o r d e r e d a l g e b r a i c s t r u c t u r e s .

We selected

two c l a s s e s o f o r d e r e d a l g e b r a i c s t r u c t u r e s which h a v e been d i s c u s s e d by many a u t h o r s d u r i n g t h e l a s t t h r e e d e c a d e s .

As

t h e s e i n v e s t i g a t i o n s w e r e made s e p a r a t e l y many r e s u l t s h a v e been r e i n v e n t e d from t i m e t o t i m e .

We d o n o t c l a i m t h a t w e

succeeded i n f i n d i n g a l l t h e r e l e v a n t

l i t e r a t u r e b u t w e hope

that the resulting bibliography is helpful to the reader.

I n t h e f i r s t c l a s s we c o n s i d e r o p t i m i z a t i o n p r o b l e m s o v e r s e m i r i n g s (R,@,@).

W e d i s c u s s a Z g e b r a i c p a t h prob2ems

(chap-

t e r 8 ) which c a n b e s o l v e d v i a t h e e v a l u a t i o n o f c e r t a i n matrix s e r i e s I @ A @ A

A*:=

2

@

...

o r equivalently v i a t h e determination of a c e r t a i n s o l u t i o n 2

of the matrix equation 2 =

(2 @ A )

@ B .

115

Linear Algebraic Problems

Then we investigate a l g e b r a i c e i g e n v a t u e p r o b l e m s

(chapter 9)

which are closely related to path problems. Finally we consider (7.2) over semirings, i.e. the e x t r e m a t l i n e a r program (chapter 10) M

z = inf{c

T

Q

x( A

Q

x

2

b, xERn}.

In the second class we discuss a l g e b r a i c l i n e a r programs in extended semimodules over real numbers, i.e. N

z = fx

for R

clR.

T

n a ] cx

2

c, XER:)

We develop a duality theory similarly to the classi-

cal case (chapter 1 1 ) which can be applied for a solution of combinatorial optimization problems with linear algebraic objective function. we discuss a l g e b r a i c n e t w o r k flow p r o b l e m s , including solution methods f o r a l g e b r a i c t r a n s p o r t a t i o n and

a s s i g n m e n t p r o b l e m s (chapter 12). Many combinatorial structures can be described by i n d e p e n d e n c e s y s t e m s . The corresponding algebraic optimization problems are considered in chapter 13; in particular, we investigate a l g e b r a i c m a t r o i d and 2 - m a t r o i d

i n t e r s e c t i o n p r o b l e m s . A l g e b r a i c matching p r o b l e m s can be solved by similar techniques, the difference lies only in the combinatorial structure. All discussed algebraic optimization problems involve only linear algebraic constraints and linear algebraic objective functions. There is no difficulty in formulating nonlinear algebraic problems in the same way. Only very few results for nonlinear algebraic problems are known. At the time being a theory for such problems comparable to the linear case has not been developed to our knowledge.

8.

A l g e b r a i c P a t h Problems

I n t h i s c h a p t e r we c o n s i d e r p a t h s i n a d i r e c t e d g r a p h G = ( N , E ) w i t h v e r t e x s e t N and arc s e t E to the arcs (R,@,(D)

N x N. W e a s s i g n w e i g h t s a , . 17

( i , J ) € E . These w e i g h t s a r e e l e m e n t s o f a s e m i r i n g

w i t h u n i t y 1 a n d z e r o 0. Such a g r a p h i s c a l l e d a

n e t w o r k . The l e n g t h I f p ) o f a p a t h ( e l , e 2 , of

its arcs, i.e.

...,e m ) i s

t h e number

l ( p ) = IE; t h e w e i g h t w ( p ) i s d e f i n e d b y

w(p):= a

8 a

el

B

2

...

B ae

m

.

L e t P d e n o t e t h e s e t of a l l p a t h s i n G and l e t P . denote the lj

s e t o f a l l p a t h s i n G f r o m v e r t e x i t o v e r t e x j. The d e t e r m i n a t i o n of t h e v a l u e

for Some i , j E N a path P E P

i j

( i f it e x i s t s )

( o r f o r a l l i , j E N ) and t h e d e t e r m i n a t i o n of

with w(p) = a*

i j

( i f i t e x i s t s ) f o r some i , j E N

( o r f o r a l l i , j E N ) i s c a l l e d t h e a l g e b r a i c path problem.

For c o n v e n i e n c e w e i n t r o d u c e t h e w e i g h t f u n c t i o n

0:

2’

+

R

d e f i n e d by i f S = @ ,

0

(8.2)

a ( s ) :=

i f S + @ .

This function i s n o t always well-defined

a s P is often infinite.

L a t e r on w e d i s c u s s a s s u m p t i o n s which imply t h a t a t l e a s t o ( P

i s well-defined.

Then a *

i j

= 0(Pij).

116

i j

)

117

Algebraic Path Problems

At first we consider the classical shortest path problem. In this case the weights are elements of (IR U {-),min,+)

with

zero = and unity 0. We may as well consider the linearly ordered commutative monoid

(W

u

{ - ) , + , z )with

absorbing maximum

= and neutral element 0. In any case, the special resulting path problem a* : = min ij PEPij

(8.3)

z

, Ci,j E N )

a

VV

(V,V)EP

is called the s h o r t e s t p a t h p r o b l e m . Then p E P

ij

with a* = w ( p ) ij

is called a s h o r t e s t p a t h from i to j. The first discussion

of this problem seems to be due to BORfVKA 119261. He admits only nonnegative weights. In this case shortest paths from a given vertex i to all remaining vertices j

$

i in G can simul-

taneously be determined using an elegant method proposed by DIJKSTRA [ 1 9 5 9 ] and DANTZIG [19601.

Let N = 11.2 ,...,n).

The weight ui of a shortest path from

vertex 1 to vertex 1 = 2,3,...,n procedure. W.1.o.g.

we assume a

is computed in the following

ii

= 0 for i E N and a

ij

:=

0

for (i,j) 6 E.

(8.4)

S h o r t e s t paths i n networks w i t h r e a l nonnegative weights a

for j = 1 , 2 , . . .,n; T:= N \ { 1 }

Step 1.

u

Step 2 .

Determine k E T with

j

:=

= min{u I j ET); k j T:= T\{k).

u let

11

118

Linear Algebraic Optimizarion

Step 3 .

If T =

then stop. min{uj, u k + a 1 for all j E T ; kj

Otherwise u := j go to step 2.

In particular u . =

at the end of the procedure indicates

3

that there exists no path from vertex 1 to vertex j. We will give an inductive proof of the validity of this method.

we claim that at each stage of the algorithm two properties ( Q : = N'T

hold

for the current T ) :

for all jE Q

(8.5)

uj is the weight of a shortest path

connecting 1 and j ,

for all jE T

(8.6)

a, is the weight of a shortest path

connecting 1 and j subject to the condition that each node in the path belongs to Q U {j). Clearly

( 8 . 5 ) and

(8.6) are satisfied after step 1. Now assume

that the current Q,T,u fulfill ( 8 . 5 ) ,

( 8 . 6 ) before step 2 .

Let k E T be determined in step 2 and suppose the weight of a shortest path p connecting 1 and k is v -c uk. Then p contains a first node j b Q u Ik).

Therefore v

2

u k contrary to our

assumption. Thus (8.5) is satisfied by Q U {k),

too. In parti-

cular, if T = {k} then the weights of all shortest paths are determined. Otherwise let v denote the weight of a shortest path connecting 1 and j ET\{k) for Q u [k,j).

If j

*

subject to the condition in (8.6)

Let i denote the predecessor of j in such a path.

k then v = u

j'

Otherwise v = u

is satisfied for Q U {k}, T\{k] in step 3 .

+ akj. Therefore (8.6)

and the revised weights u . 1

119

AZgebraic Path h b l e m s I

The performance of this method needs O(n ) comparisons and O(n

2

additions.

Next we discuss the generalization of this method for weights from o r d e r e d c o m m u t a t i v e m o n o i d s

(H,*,Z)

with neutral element

,+,-,~) e. W e r e m a r k t h a t H is an o r d e r e d s e m i m o d u l e o v e r (Z+

with respect to the trivial external composition

0:

Z+ X H + H

d e f i n e d by (8.7)

z

a:= az

for all a E H , z E Z +

0

( i n p a r t i c u l a r a : = e). A p a t h p in t h e

d i r e c t e d g r a p h G m a y b e r e p r e e e n t e d b y its i n c i d e n c e v e c t o r x E {Oil IE d e f i n e d by

T h e s e t o f a l l i n c i d e n c e v e c t o r s o f p a t h s in P by S i j .

ij

ie d e n o t e d

Then we may rewrite the shortest path problem for the

c a s e of s u c h a s e m i m o d u l e a s (8.9)

a * := min ij XES ij

with coefficients of a

ij

weight

xToa

(weights) a

VV

I

(i,j E N )

E H . If we assume the existence

and the existence o f a path connecting i and j with

*

aij

(an a l g e b r a i c s h o r t e s t p a t h ) then A has to be

t o t a l l y o r d e r e d , in general. A n a l g e b r a i c v e r s i o n o f (8.4) can b e g i v e n by s i m p l y r e p l a c i n g t h e e q u a t i o n in step 3 by (8.10)

A g l a n c e t h r o u g h o u r p r e v i o u s p r o o f o f t h e v a l i d i t y o f (8.4) s h o w s t h a t t h e a l g e b r a i c v e r s i o n is v a l i d if B i s a p o s i t i v e l y

120

Linear Algebraic Optimization

linearly ordered commutative monoid. Commutativity is not necessary for a proof of the validity; but without commutativity H is not a semimodule over 22 +

.

If we try to avoid the

assumption that H is linearly ordered then we cannot be sure that we can perform the steps in the algorithm: in step 2 we

I j E T ) does not exist, in general. All j weights have to be nonnegative (i.e. e 5 a 1 ; otherwise ij even in the classical case counterexamples can easily be may find that min{u

given. In terms of a semiring the algebraic version has been discussed by GONDRAN [1975c], MINOUX 119761 and CARRE 119791. GONDRAN mentions the following examples of positively linearly ordered commutative monoids. F o r an application of the algebraic version of ( 8 . 4 )

let w.1.o.g.

a

ij

:=

(maximum of H)

00

for (i,j) a E and aii = e (neutral element of H) for all i E N .

(8.11)

Examples (lR+U ( - ) , + , L ) with neutral element 0 and maximum Here we get the c l a s s i c a l s h o r t e s t p a t h ([0,11,*,2)

00.

problem.

with neutral element 1 and maximum 0.

Here w(p) is the product of arc-weights; this is the socalled r e l i a b i l i t y problem. (fO,l),-,z) with neutral element 1 and maximum 0.

*

Here a , . = 1 iff a path connecting i and j exists 11

( c o n n e c t i v i t y problem). (JR+U{-),min,))

with neutral element

00

and maximum 0.

* the Here w(p) is the capacity of a path p and a ij maximum c a p a c i t y of a path connecting i and j.

Algebraic Path Problems

121

F o r a d d i t i o n a l examples o f s u c h an o r d e r e d a l g e b r a i c s t r u c t u r e we r e f e r t o chapter 4.

In p a r t i c u l a r , t h e solution of

(8.11.4)

i s r e l a t e d t o maximum w e i g h t e d t r e e s i n u n d i r e c t e d g r a p h s ( c f . KALABA [19601).

For t h e remaining p a r t of t h i s c h a p t e r w e drop t h e assumption t h a t t h e u n d e r l y i n g a l g e b r a i c s t r u c t u r e is p o s i t i v e l y l i n e a r l y o r d e r e d . The s e m i r i n g (R,$,@) and,

is only pseudoordered

a path with weight

i n general,

* a

ij

(cf.

w i l l not exist.

5.14)

* a

i j

is

t h e supremum o v e r t h e w e i g h t s o f a l l p a t h s f r o m i t o j ; c l e a r l y , t h i s supremum w i l l n o t a l w a y s e x i s t ,

as P

i j

may b e i n f i n i t e .

F o r c o n v e n i e n c e w e i n t r o d u c e some d e n o t a t i o n s f o r s e t s o f p a t h s which w i l l a p p e a r i n t h e f o l l o w i n g d i s c u s s i o n . k E N U {O}. Then

b

i j

P[kl i j

'

a n d Pk

i j

0

L e t i , j E N and

denotes t h e set of a l l elementary paths i n

i j

d e n o t e s t h e s e t o f all p a t h s p E P

denotes t h e set of a l l paths p

E

P

i j

i j

with l ( p )

1. k

with l ( p ) = k . For

t e c h n i c a l r e a s o n s we a s s u m e t h e e x i s t e n c e o f a c i r c u i t p o f l e n g t h 0 and u n i t y w e i g h t c o n t a i n i n g o n l y v e r t e x i b u t no a r c . S u c h a c i r c u i t i s c a l l e d t h e n u l l c i r c u i t o f i. Then u ( P L y l ) = 1 f o r e a c h i E N a n d U ( P [ ~ ] ) = 0 f o r e a c h ' ( i ,j ) w i t h i i j

As

+

j.

an i n t r o d u c t o r y example we, c o n s i d e r t h e , c l a s s i c a l s h o r t e s t

p a t h problem with p o s s i b l e n e g a t i v e weights i n t h e semiring

(IR U { - ) , m i n , + )

with zero

of n e g a t i v e w e i g h t be u n b o u n d e d .

If P

0

and u n i t y 0; I f G c o n t a i n s c i r c u i t s

( d e n o t a t i o n : n e g a t i v e circuits) t h e n i j

+

( 8 . 3 ) may

@ and i f G d o e s n o t c o n t a i n a n e g a t i v e

c i r c u i t t h e n t h e r e e x i s t s a s h o r t e s t p a t h from i t o j which is elementary.

122

Linear Algebraic Optimization

We a s s u m e a

= 0 for all i E N and a

ii

f o r u ( P [ ~ ’ ) , k = 0,1,2,... ij

-

f o r k = 1,2,. =

UCP;;])

.. .

for i

if (i,j) d E . Then

=

we find t h e following recursion:

F o r k = 0 w e k n o w u ( P [ O ’ ) = 0 f o r i E N and ii

*

j. A n i n d u c t i v e p r o o f o f t h i s r e c u r s i o n

is g i v e n in t h e f o l l o w i n g .

F o r k = 1 v a l i d i t y of (8.12) i s o b v i o u s . W.1.o.g. t h a t G is a c o m p l e t e d i g r a p h . T h e n P [ k l i, and let p E P I k 1 be a path with weight i j

+ 0

we assume

for k

[kl) U(Pi,

.

2

1.

Let k

2 :

If 1 ( p ) c k

u ( P i[, k - l l ) + a .

O t h e r w i s e 1 ( p ) = k. T h e n l e t n 1,In d e n o t e t h e p r e d e c e s s o r of j i n p. N o w w ( p ) = u(P:;’) +a hj then

w(p)

=

.

b o t h c a s e s (8.12) is s a t i s f i e d . A s G contains no negative circuits we can derive the weights

o f t h e s h o r t e s t p a t h s f r o m (8.12). (8.13)

j a i j = 0 ( P iIn-1

We find

I) L e t U [kl

s i n c e a n e l e m e n t a r y p a t h h a s at m o s t l e n g t h n - 1 . d e n o t e t h e m a t r i x w i t h entrie.s u(P [kl 1 . ij

Then

(8.12) c a n b e

i n t e r p r e t e d a s m a t r i x - c o m p o s i t i o n o v e r t h e s e m i r i n g (IR U min,+) (8.14)

A

*

= U In-’]

=

(ai,) w e f i n d

( ( ( V [ O 1 + A ) +A)...

U I O 1 i s t h e i d e n t i t y o f t h e s e m i r i n g of n

As

IR U

-

An 1 s

(cf. c h a p t e r 5). F o r A:=

{-I

t h i s l e a d s t o A * = An-’

.

) +A. x

2 n-1 w e m a y d e t e r m i n e A

n matrices over

Therefore a computation o f

solves the shortest path problem. As A S = n- 1

{-I,

by t h e s e q u e n c e

-

An 1

for all

123

Algebraic Path Robkms

A,A

2

,A

4

,...,A Ik

;

therefore O ( l o g 2 n) matrix multiplications are necessary, each 3

3

of which needs O(n ) comparisons and O(n ) additions. In particular, we can determine the weights of all shortest paths from node 1 to node j for all j

*

I from the first row

A:

of A f

1.e.

this recursion is the classical B E L L M A N - F O R D method. It needs

3 3 O(n ) additions and O(n ) comparisons. Improvements of this method can be found in Y E N [19751. In the general case, i.e. for weights in arbitrary semirings, a solution of (8.11, if it exists, is determined by computation of Ak and A[k1:=

(8.16)

I

Q A Q A’

Q

...

Q Ak

where I is the unity matrix in the semiring of all n x n matrices over the underlying semiring ( R , Q , b ) .

Problems which can

be reduced to (8.1) and methods for the computation of these matrices, called matrix-methods, have been considered by many authors during the last thirty years. We mention only those which explicitly uae some algebraic structures. The following list will hardly be complete. LUNTS

[I9501 considers applications to relay contact networks,

SHIMBEL KLEENE

([1951], [19531 and [1954]) to communication nets. [1956] bases a theorem in the algebra of regular events

on a Gaussian elimination scheme which is separately developed

for the transitive closure of a graph by ROY [19591 and

Linear Algebraic Optimization

174

WARSHALL [1962].

MOISIL [1960] discusses certain shortest

path problems, YOELI [1961] introduces Q-semirings and PANDIT

I19611 considers a matrix calculus. Further results and generalizations are due to GIFFLER ([19631, [19681). CRUON and HERVE [1965], PETEANU ([1967a],

[1967b],

[19691, [19701,

[1972]), TOMESCU [1968], ROBERT and FERLAND [19681, BENZAKEN

I19681 and DERNIAME and PAIR [19711. Methods based on fast -atrix multiplication are discussed by ARLAZAROV et al. [19701, FURMAN [1970], FISCHER and MCYER [19711 and MUNRO [19711. CARRE 119711 develops a systematic treatment of several linear algebra procedures. This approach is continued by BRUCKER

([19721,

[1974]), SHIER ([19731, [19761), MINIEKA and SHIER [19731, GONDRAN (119751, [1975a],

[1975bl, [1975c1, [1975d],

[19791

and [1980]), AHO, HOPCROFT and ULLMAN [19741, MARTELLI [1975]),

“19741,

ROY [1975], BACKHOUSE and CARRE [19751, TARJAN ([1975],

[1976]), FRATTA and MONTANARI

119751, WONGSEELASHOTE

[1979]), LEHMANN [19771, MAHR

(119791, (1980aI and [1980bl),

FLETCHER [1980], and ZIMMERMANN, U.

([1976],

[19811. GONDRAN and MINOUX

[1979b] a s well a s CARRE [1979] summarize many of these results.

The following proposition shows the different character of A k and A[k1.

T h e proof is drawn from WONGSEELASHOTE [1979].

(8.17) P r o p o s i t i o n L e t A be an n x n matrix over a semiring

for all i,] E N and all m E l N U { O ) .

(R,@,@).

Then

Algebraic Path Problem

Proof. For paths P E P

125

let p o q denote the path jk in Pik traversing at first p and then q. For Q C_ Pij and

S C_ P

jk

let

Q oT:=

o(Q

(3)

0

and q E P

ij

{PO q

T) =

O ( Q

p E Q , q E T 1 . Then 8 o(T)

r

provided that the weight function is well-defined for Q and T. Now for m

2

1

Pm = ij

(4)

u

1 CPir. 1

Pr

0

1 2

Therefore (1) is satisfied for m

... 2

0

P

jl

m-1

pairwlse different rlf...,r m

1. For m = 0 we find

Ao = A 101 = I. As Po contains the null-circuit with weight ii equal to the identity 1 of the semiring ( 3 ) and ( 4 ) show (1). AS

(2)

is Implied by ( 1 ) .

In our introductory example (IR U {-),min,+) A [sl = A [n-tl for all s

2

Am = A[m1 and

n-1. Such results do not hold in

the general case. If there exists r E N U ( 0 ) such that

then A is called s t a b l e . The minimum of the integers satisfying (8.18) I s called the s t a b i l i t y i n d e x of A. For stable matrices the values a* in (8.1) exist and from (8.17) we find ij

for a l l i , j E N . We remark that ( 8 . 1 8 ) all s

2

implies A[']

= A [rl

for

r. Sufficient conditions for a matrix to be stable

are conveniently formulated in terms of the circuits of the

I.

126

Linear Algebraic Optimization

network G. Let C denote the s e t of a l l e l e m e n t a r y non-null

c i r c u i t s and W:= {w(p) I p E

C ) .

1 a a a a2 a

(8.20)

Let

...

a am

for a E R and m E N U { O } and let (8.21)

a[ml:=

1

a

@ a

1

(a

1

@a2) B . .

.B

(al @ a 2

@...

@arn)

for a E RS, s z m EN U { O } . The network G is called a b s o r p t i v e if

(8.22)

p 1=

a [11

for all a E w

,

for all a E w

,

rn-regular if (8.23)

[ml = a [m+ll

and m - a b s o r p t i u e (8.24)

If

G

then

if

is absorptive then G

for all a E

a[ml = a[m+l] G

wm+l.

is m-absorptive. If G is m-absorptive

is m-regular. Absorptive networks over certain semirings

were introduced by CARRE 119711; the generalizations (8.23) and (8.24)

are due to GONDRAN [1975a] and ROY 119751.

We remark that a s e m i r i n g R is called a b s o r p t i v e

(m-regular,

m - a b s o r p t i v e ) if the respective property is satisfied for a l l elements of R. An absorptive semiring is idempotent.

(8.25) P r o p o s i t i o n If

G

(1)

(2)

is an m-regular network then for all a E w a[m] = a [Sl

a[ml

= a

for all s

2

m+l,

for all s 2 m+l;

@ as

if G is an m-absorptive network then for all a € W s !3)

a[ml

=

ark]

(4)

a[ml

= a[ml

(a1

...

@ as)

for all s

2

k

for all s

2

m+l.

m+l,

127

Algebraic Path Problems

Proof. We show ( 3 ) and ( 4 ) .

A

proof of (1) and (2) is similar.

( 3 ) . F o r s = k = m + l this is obvious. Let s = k > m + l and

assume B[s-21

for all 8 E

= S[s-lI

Then we get

=

1 @ a 1 fd (1 @ a2

= a[e-1]

Therefore a[s-l]

=

(B

...

(B

(a2 8

...

8

a 6-1 1 )

.

a[s]

for all a E Ws and for all s

2

m

+ 1.

This leads to ( 3 ) as a E W s implies (al,aS,...fak) E W k for s In particular

( 3 ) implies

arm] = a[sl = a[s-11 = a[ml

for all a E Wsf s

2

(B

(al 8 a2 8

(B

(a1

8

a2 8

... 8 ... 8

as) as)

m + 1. Thus ( 4 ) is satisfied.

In (5.14) we introduced the pseudoordering a c b

a ( B b = b

:c.,

of the semiring ( R f ( B f 8 ) . G is absorptive i f f

f o r all a E W ,

G i s m-regular iff

m+l < a [ml

0

for all a E w and G is m-absorptive i f f a

1

8 a2 8

...

8

a < a[ml m+l

-

for all a €

wm+l.

k.

12s

Linear Algebraic Optimization

Similarly, of

t h i s pseudoordering.

L e t Q,T b e s u b s e t s o f i f

c a n b e i n t e r p r e t e d i n terms

(8.25.1 - 4 )

properties

Then Q i s c a l l e d

P w i t h Q C_ T.

d e n s e i n 2'

there exists a f i n i t e subset H of

for all pET\Q

Q such

5 o(H).

that w(p)

(8.26) P r o p o s i t i o n L e t Q b e a d e n s e s u b s e t o f T. Further, Proof.

i f

B is f i n i t e ,

Let pET\B.

I f Q C_ B

then u(B)

d e n s e i n T.

If

= o(Q).

1. o ( H ) .

B the set B is

As H

B is f i n i t e t h e n B X Q i s f i n i t e .

a n d some H

Q w e know w ( p )

lead t o w(p)

5

u(Q).

o(B) = o(BwQ) @

The s e t o f

T t h e n B i s d e n s e i n T.

and t h e r e f o r e t h e r e exists a

Then p E T \ Q

f i n i t e s u b s e t H o f Q w i t h w(P)

5

5

U(H).

for pEBwQ

Then 0 < U(QwH) and

(5.14.1)

1.e.

< o(Q),

shows o ( B \ Q )

(5.14.2)

Now

= O(Q).

O(Q)

rn

t r a v e r s e an e l e m e n t a r y nonA m 1 (ml null c i r c u i t a t m o s t m t i m e s i s d e n o t e d b y P ij' S i m i l a r l y P ij a l l paths P E P

i j

which

denotes t h e set of a l l paths P E P

i j

e l e m e n t a r y non-null circuits. L e t p (8.27)

therefore

l(p)

5

n

l(q)

5

n m

$:j

c_

IcI

P i[SI j

m + n

+

n

-

and

-

t r a v e r s e a t most m

which

Egm i j

I

qE9:;).

Then

1 =:s,

1 =: t i

-

Pi,( m ) C_ P iIt1 j

.

( 8 . 2 8 ) Proposition (1)

I f G is a b s o r p t i v e t h e n

(2)

i f G is m - a b s o r p t i v e dense i n Pi,

;

6"i j

is dense i n P

i j

;

and R is commutative then $ ( m ) i s

ij

Algebraic Path Problems

129

i f G is m - r e g u l a r and R i s c o m m u t a t i v e t h e n

(3)

dense in P

(1)

Proof.

-

ij

Let P E P

ij

\$:,.

L e t p' E

$yj

3 m,

lj

is

denote the elementary

p a t h d e r i v e d f r o m p by e l i m i n a t i n g a l l c i r c u i t s i n p. A s p c o n t a i n s a t l e a s t o n e e l e m e n t a r y n o n - n u l l c i r c u i t q and w ( q ) t h e m o n o t o n i c i t y c o n d i t i o n (5.2) (cf. 5.14) Thus (2)

Boij

implies w(p) zw(p').

is d e n s e in P

ij' F r o m (8.25.4) we k n o w t h a t f o r e a c h s - t u p l e q = ( q l l q 2 1 . .

2

E C S with s

..,q,) (8.29)

w(ql)

@

m+l t h e r e l a t i o n

w(q2)

Q

i s s a t i s f i e d . L e t p E P i j \$;;).

.. .

Q

5 q[ml

w(qs)

Then p traverses s elementary

non-null circuits q 11q21...lqs with s

2

m+l.

Successive elimi-

n a t i o n o f q s , q s ~ l l . . . , q l l e a d s t o t h e p a t h s pl,pz,...lps. particular p w(ps):=

In

i s an elementary path (or a null path with weight

1). A s R is c o m m u t a t i v e we f i n d

T h e m o n o t o n i c i t y c o n d i t i o n (5.2) t o g e t h e r w i t h

(8.29) s h o w s

' (m) is dense in P A s H C _ AP ( m ) t h i s s h o w s P ij' ij ij (3)

11

F r o m (8.25.2) w e k n o w

(8.31)

w(qIS

5

w(q) I m l

for all q E C and all s

2

m+l.

Linear Algebraic Optimization

130

Let p

E Pij

. Then

,$:

s times with some s

2

p t r a v e r s e s at l e a s t o n e g E C e x a c t l y m+l. L e t p = p l o q o p 2 0 q o

... ~ q o p , + ~ .

A g a i n t h e r e m a y be s o m e n u l l p a t h s i n t h i s r e p r e s e n t a t i o n w i t h w e i g h t 1.

Prom

Commutativity leads to

(8.31) a n d m o n o t o n i c i t y c o n d i t i o n

5

(8.32)

W(P)

Let

p ( l ) :=

W ( P 1 ) 8 w(p2) 8 Pi* P20

*..

( 2 ) := p l o q o p 2 0

p ( m ) : = PI. q.p2.

(5.2)

... 8

OPS+1

w e get

w(ps+l) 0 w(q)

[ml

.

’

... ~ P s + l ~ ’ ” ’ q....

.

oqopm-lD pmo . . . o p s + l

(8.32) together with commutativity shows w(p)

5

u ( { p ( l ) , p ( 2 )I

. . . ,

p(m)I).

F u r t h e r e a c h p ( i ) i s a s u b p a t h of p w h i c h d o e s t r a v e r s e q at most m times. If p contains further elementary non-null circuits t h e n t h e p r o c e d u r e i s a p p l i e d t o t h e p a t h s i n H = {p“’,p

(2)

,...I.

A s t h e r e is o n l y a f i n i t e n u m b e r o f s u c h c i r c u i t s t h e p r o c e s s t e r m i n a t e s a f t e r a f i n i t e n u m b e r o f steps. lead t o w(p) < in P

ij

u(g)

with finite

-

T h e m a t r i x A is c a l l e d a b s o r p t i v e G is a b s o r p t i v e

$m

Ij

(5.14.1)

. Thus

Bmi j

and

(5.14.2)

i s dense

f m - r e g u l a r , m - a b s o r p t i v e ) if

(m-regular, m-absorptive).

The following theorem

s u m m a r i z e s t h e r e s u l t s of C A R R g [ 1 9 7 1 1 , GONDRAN [ 1 9 7 5 a ] a n d R O Y 119751.

19791.

I n t h i s f o r m it c a n b e f o u n d i n W O N G S E E L A S H O T E

131

Algebraic Path Problems

( 8 . 3 3 ) Theorem Let

be a s e m i r i n g a n d l e t A b e an' n x n m a t r i x o v e r R.

(It,$,@)

Then A is s t a b l e w i t h s t a b i l i t y i n d e x r i f o n e o f t h e f o l l o w i n g conditions holds

5 n-1)

(I)

A is absorptive

(then r

(2)

A is m - a b s o r p t i v e

and R commutative

(3)

A

Proof.

is m - r e g u l a r a n d R c o m m u t a t i v e

5

(then r 5 , n m

nm

+

n-lIf

ICI + n

- 1).

F r o m ( 8 . 2 8 ) w e know t h a t

ST,)

(m) ij with szn-1 (;

(then r

is d e n s e i n P ( 8

z n m + n - l ,

( 1 ) ( ( Z ) , (3)) i m p l y t h a t $o ij [sl Therefore t h e f i n i t e set P i j

i,' s znmICl+n-l)

is d e n s e i n P

ij

a n d f o r a l l s u c h s w e f i n d ( c f . 8 . 2 6 ) t h a t U ~ P ' ~is ~ e )q u a l to

if

Then

( a ( $ ( m ) ) fu($y,)). ij

u(;y,)

(8.17)

shows t h a t A i s s t a b l e

and h a s t h e r e s p e c t i v e s t a b i l i t y index.

Theorem

(8.33) g i v e s s u f f i c i e n t c o n d i t i o n s f o r a m a t r i x t o b e

stable.

For such m a t r i c e s t h e a l g e b r a i c p a t h problem

(8.1)

can be solved using t h e equations A* = A[r1

(8.34) for all

8

2

...

=

E

= A 181

r r t h e s t a b i l i t y index o f A.

--

...

By v i r t u e o f

(8.16)

t h e following r e c u r s i v e formula is v a l i d

(8.35) with A

A

Ik']

As A

Ikl

(8.37)

A)

@

a

I

,

k = 1,Zf...

= I where I d e n o t e s t h e n x n u n i t y m a t r i x o v e r t h e

s e m i r i n g (It,$,@).

(8.36)

= ( A Lk-l'

A*

From

I

= (A* 8 A )

8 A = A Q AIkl A*

(8.34) a n d (8.35) w e c o n c l u d e

.

for all k

= (A @ A*)

@

I

.

Em

U {O} w e f i n d

132

Linear Algebraic Optimization

Equations (8.36) and (8.37) show that

A*

certain matrix equations over R. Let Y:=

is the solution of A*

0

B and

Z:=

B 0

A*

for a matrix B over R. Then (8.36) and (8.37) lead to Y = (A 0 Y )

(8.38)

B,

Z = ( Z O A ) $ B .

In particular, if B is the unit matrix then

is a solution

A*

of (8.3811 if B i s a unit vector, then a column resp. a row of A * are solutions of (8.38). The semiring of all n x n matrices over R is pseudoordered by

The solutions

A*

0 B resp. B 69 A * are minimum solutions of

(8.38) with respect to this ordering if R is idempotent. This

follows from

for all s = 0 , 1 , 2 , . . . , s

2

if Y is a solution of (8.38). Then for

r we find

(8.40)

Y

(B

(A*

0 B) =

(As

Thus, if R is idempotent, then

Y)

@

s

5 r

=

0 B

A*

graph contains no circuits then

As

n-1 and (8.39) shows Y =

(A*

(B

A*

0 B) @

1. Y.

(A*

0 B)

.

If the underlying

is the zero-matrix for 0 B is the unique solution

of (8.38). The same remarks hold for the second equation in (8.38).

The solution of such equations can be determined by methods similar to usual methods in linear algebra.

A

systematical

treatment o f such an approach was at first developed by CARRE

119711 for certain semirings. Then well-known methods

from network theory appear to be particular cases of such

Algebraic Path Problems methods.

Several authors

133

( c f . remarks following 8.48) e x t e n -

ded t h i s approach t o o t h e r a l g e b r a i c s t r u c t u r e s o r d e r i v e d methods f o r n e t w o r k s o f s p e c i a l s t r u c t u r e .

BACKHOUSE a n d

CARRE [ 1 9 7 5 1 d i s c u s s e d t h e s i m i l a r i t y t o t h e a l g e b r a o f

regu-

l a r l a n g u a g e s , f o r which t h e m a t r i x A* a l w a y s e x i s t s due t o a s u i t a b l e axiomatic system. N e x t w e c o n s i d e r two s o l u t i o n m e t h o d s f o r

(8.1) which can be

i n t e r p r e t e d as c e r t a i n l i n e a r a l g e b r a p r o c e d u r e s .

A straight-

f o r w a r d f o r m u l a t i o n o f a g e n e r a t i z e d o r a t g e b r a i c JAGOBI-method f o r t h e second equation i n (8.41)

Z(k'l):=

with Z ( O ) : =

(Z(k)

( 8 . 3 8 ) is 8 A)

@ B

,

k = O,l,Z

,...

B.

Proposition L e t A be a n xn-matrix

i n d e x r . Then

(8.41) i s a f i n i t e recursion f o r t h e determina-

t i o n o f B @ A*; (k)

for a l l k Proof. k

2

over the semiring (R,@,@) with s t a b i l i t y

we find = B @ A *

r.

A n i n d u c t i v e a r g u m e n t shows Z(k) = B Q A I k l

= 0,1,2,.

for

.. .

W e remark t h a t f o r t h e c l a s s i c a l s h o r t e s t p a t h problem o v e r

the semiring

(IR U { - ) , m i n , + )

a n d f o r B = (0-... - 1

t h i s method

r e d u c e s t o t h e m e t h o d o f BELLMAN [ 1 9 5 8 ] f o r t h e c a l c u l a t i o n of t h e w e i g h t s of a l l s h o r t e s t p a t h s from node 1 t o a l l o t h e r nodes.

I n t h e g e n e r a l c a s e with B = I a complete s o l u t i o n of

Linear Algebraic Optimzarion

134

(8.1) is d e t e r m i n e d ; t h e v a l u e s a* a r e n o t n e c e e s a r i l y ij

a t t a i n e d as weights of certain paths P E P O(r)

This method needs

ij’

2 i t e r a t i o n s and e a c h i t e r a t i o n r e q u i r e s O ( n 1 8 - c o m p o s i 3

t i o n s a n d ~ ( )n %-compositions.

r = nm I C l

+ n-1. T h e n u m b e r

of e l e m e n t a r y n o n - n u l l c i r c u i t s

ICl is not b o u n d e d by a p o l y n o m i a l i n n. I f

A

is m-absorptive

t h e n r = O(n). In t h i s case t h e method r e q u i r e s O ( n s i t i o n s and O ( n

4

(m*O)

~f A is o n l y m-regular

4

8-compo-

@-compositions.

)

T h e JACOBI-method is a so-called i t e r a t i v e method in l i n e a r algebra. T h e s e c o n d c l a s s of s o l u t i o n t e c h n i q u e s in l i n e a r a l g e b r a c o n t a i n s d i r e c t o r e l i m i n a t i o n m e t h o d s . We w i l l consider t h e g e n e r a l i z e d o r a l g e b r a i c GAUSS-JORDAN method. A(’):=

Let

A a n d d e f i n e r e c u r s i v e l y f o r k = 1,2,...,n

(8.43

for all i , j E N . T h i s r e c u r s i o n is well-defined, ( k - l ) ) * is well-defined. t i o n of (akk

duce t h e set Ti:.’

if t h e computa-

F o r a d i s c u s s i o n we intro-

w h i c h c o n s i s t s i n all p a t h s P E P

ij

s u c h that

p d o e s n o t contain an i n t e r m e d i a t e v e r t e x from t h e s e t

{k+l,k+Z,

...,n ) .

For a c i r c u i t

PET'^) ii

with k < i this means

that p i s a non-null c i r c u i t t r a v e r s i n g i exactly once.

(8.44)

Proposition

(1

bO If A i s a b s o r p t i v e t h e n P kk

(2)

if A is m-regular a n d R is c o m m u t a t i v e t h e n

n

T R(k’ k is d e n s e in T k(k k

,

Akebmic Path Problems

(3)

135

if A is m-absorptive and R is commutative then

n T~~ (k)

i s dense in T (k)

kk

‘

Proof. Follows from the proof of ( 8 . 2 8 ) as in each case the constructed set H for p E T : o

(k) Tkk

.

satisfies H

This proposition is the essential technical tool in the proof of the next theorem.

( 8 . 4 5 ) Theorem Let (R,$,@)

be a semiring and let

A

be an n

x

n matrix over R.

Then A is stable with stability index r if one of the followihg conditions holds: (1)

A

is absorptive

(then r f n-11,

(2)

A

is m-absorptive and R commutative

(3)

A

is m-regular and R commutative

(then r (then r

1. n m + n

- 11,

5 nm I C I + n -

In each case A*

(4)

= A(n)

with respect to the recursive formula ( 8 . 4 3 )

for all k = 1 , Z ’ ,

and

. . . ,n.

Proof. We claim that (8.46)

for all i , j E N and all k E N U { O ) .

As T!o’

13

= { ( i , j ) } we see

that ( 8 . . 4 6 ) holds for k = 0. Now assume that ( 8 . 4 6 ) holds for k-1 with k

2

1 and for all i,j E N .

1).

136

Linew Algebmic Optimization

f i r s t we consider the case i = j = k.

A t

(1).

( 2 ) and

( 3 ) we f i n d

(k) Tkk f o r all s

1

( c f . 8.44) (k) Tkk

is d e n s e i n

r and t h e r e f o r e

In particular,

t h e e x i s t e n c e o f a(TL;))

h a s b e e n shown.

Now

c o n s i s t s i n t h e n u l l c i r c u i t and all c i r c u i t s of t h e

T

-

form p = P 1

u

Is]

'kk

I n e a c h of t h e c a s e s

=

P2

-ps with c i r c u i t s p

- * - .

l , 2 , . ..,s, a n d S E N .

a* = 1

a @ a'

@

e x i s t s as a s = o ( [ p l - pa

and

(k-1)

L e t a:= o(Tkk

(8.47) lead t o

@

E ).

Tki-')

for

Then

...

- -.-

'p

o p S I

for u = 1 , 2 ,

(k-l) Tkk

. . . , sl)

a [rl = a [ r + l l T h u s we h a v e p r o v e d

( 5 ) f o r k and

Secondly w e consider

+

( i ,j )

(8.46) f o r i = j = k .

(k,k).

Then P E T . ( k ) 'Tij 11

v e r t e x k and h a s t h e form p = P I * p g - p 3 w i t h

p 2 E T k( k )

and p

3

E

T(k-l). kj

(8.43) t h a t

A n i n d u c t i v e a r g u m e n t s h o w s now t h a t

a

i j

for all i , j EN,

In particular

= 0fPij)

i.e.

contains

(k-1) Tik

,

Hence

which shows t o g e t h e r w i t h

a l l k = O,l,...,n.

'1

(k-1 f

A(")

= A*

.

(8.46)

is valid for k .

( 8 . 4 6 ) and

(5) hold for

Algebraic Path Problems T h e o r e m (8.45) s h o w s t h a t t h e r e c u r s i o n

137

(8.43) c a n b e u s e d i f

some o f o u r s u f f i c i e n t c o n d i t i o n s f o r s t a b i l i t y o f A a r e fulfilled.

Another proof

i s d u e t o FAUCITANO a n d N I C A U D 1 1 9 7 5 1 a n d c a n b e

f o u n d i n GONDRAN a n d M I N O U X e xis ts then A

( n ) = A*.

[ 1 9 7 9 b ] . T h e y show t h a t i f A ( n )

I t i s claimed t h a t

(k-1) ) ( ak k

diagonal pivoting is incorporated i n recursion

*

exists i f

(8.43). Our

p r o o f shows t h a t p i v o t i n g is n o t n e c e s s a r y i f o n e o f t h e assumptions

(8.45.11,

case a proof

(8.45.2) o r

(8.45.3) h o l d s .

i s a l s o g i v e n by CARRg

"19711,

For t h e a b s o r p t i v e 119791).

R e c u r s i o n (8.43) c a n b e s i m p l i f i e d i n t h e f o l l o w i n g way.

I f a*

e x i s t s f o r a E R then a*

=

( a * @ a ) @ 1 = 1 IB ( a

i s s a t i s f i e d ( c f . 8.36 a n d 8.37).

o

a*)

Therefore, i n

(8.43) w e f i n d

f o r i * j = k

and s i m i l a r l y f o r j

If

and of

a*

*

i = k

e x i s t s f o r a E R t h e n a* @ a*

= a*

. Together

with

(8.43')

(8.43") t h i s l e a d s t o t h e f o l l o w i n g e f f i c i e n t f o r m u l a t i o n (8.43).

Linear Algebraic Optimization

138

(8.48) G A U S S - J O R D A N E L I M I N A T I O N Step 1.

Define the original matrix A; let k : = 1.

Step 2 .

akk:= (akk)

Step 3 .

a.

:=

a

:=

*.

ik

kj

a ik

@

a

@ a

kk

akk kj

Step 4 .

a. := a

Step 5 .

If k = n stop.

ij

(aik

@

ij

Otherwise k : = k

@

a

kj

)

+ 1 and

for all i

$

k;

for all j

9

k.

for all i,j

$

k.

go to step 2.

The method requires O(n) iterations. Each iteration needs O(n @-compositions, O(n (akk)*,

2

2

)

@-compositions and the computation of

)

i.e. (akk)* = 1

$

akk

@

...

(akklr

.

Here we need O(r) $-compositions and O(r) @-compositions. Again if

A

is only m-regular (m

$

0 ) then this does not yield

a polynomial bound. If A is m-absorptive then r = O(n). If A is absorptive then (akk)* = 1 and the method can be simplified. We may w.1.0.g.

assume that the main diagonal of A con-

tains only unities. Then step 2 and step 3 can be eliminated from the algorithm. If A is m-absorptive then the method requires O(n 3 ) 8-compositions and O(n

3

)

@-compositions. Therefore, in general, the

GAUSS-JORDAN-method is superior to the JACOBI-method. Nevertheless, for special structured networks (as for example in treenetworks, i.e.

iCl = 0 ) the JACOBI-method may turn out to be

better (cf. SHIER [ 1 9 7 3 ] ) .

139

Algebraic Path Roblems

Specific forms of this method have been developed by many authors. ROY [19591 and WARSHALL [1962] applied it to the transitive-closure problem in graphs, FLOYD [19621 to the classical shortest path problem, ROBERT and FERLAND [1968], CARRE 119711, BRUCKER ([19721,

[1974]),

BACKHOUSE and CARRE/

[19751, and GONDRAN [1975a] gave extensions to certain algebraic structures. MULLER-MERBACH 119691 used it for the inversion of Gozinto-matrices, MINIEKA and SHIER [1973], MINIEKA [1974] and SHIER 119761 applied it to the k-shortest path problem. MARTELLI

“19741,

[1976])

enumerates all mini-

mal cut sets by means of this method. Other solution methods, as the GAUSS-SEIDEL iteration method o r the ESCALATOR elimination method have been generalized also for certain semirings. The interested reader is referred to GONDRAN and MINOUX [1979b] and CARRE/ [1979]. We remind that stability of a matrix A implies the existence of the closure matrix A*. Theorem (8.45) shows that certain types o f stability imply the validity of the GAUSS-JORDAN method f o r the determination of A*. In general, it is not known whether the GAUSS-JORDAN method is valid for all stable matrices. Nevertheless, no counterexample is known and we conjecture the validity of the GAUSS-JORDAN method for all stable matrices. Instead of assuming that the matrix A satisfies certain conditions we may impose certain conditions on the underlying semiring. We remind that P

ij

is always a countable set. Thus

A* i s well defined for all matrices A over R if we assume that infinite sums of the form

Linear Algebraic Optimization

I40

are well defined in R for a . E R , i EN. We denote such a series by Z I a i for a countable set I. In arbitrary semirings Z

is well defined only if I is a finite s e t . Besides E

I

I

a

i

a . E R we 1

assume distributivity, i.e. b B (1 a , ) = ZI(b I 1 (Za I i

@

ai) ,

,

B b = Z (a. @ b ) I 1

for a l l b c R and associativity, i.e. Z

I

a

ZJ(ZI a . )

=

i

j

for all (countable) partitions

(I

j'

j E J ) of I. Then R is

called a ( c o u n t a b l y l c o m p l e t e semiring.

In a complete semiring the GAUSS-JORDAN method is well defined for all matrices

A

over R. Its validity is shown in the same

way as in the proof of theorem ( 8 . 4 5 ) by verifying

(8.46).

In

a complete semiring the proof becomes much easier since the occuring countably infinite sums 2

a * = l t 8 a @ a

$...=

i

are well defined.

( 8 . 4 9 ) Theorem Let ( R , C I , @ ) over R. Then

be a complete semiring and let A*

Theorem ( 8 . 4 9 )

A

be an n x n matrix

is well defined and A* = A ( n )

.

is proved by AHO, HOPCROET and ULLMAN [ 1 9 7 4 ]

f o r idempotent complete semirings. FLETCHER [I9801 remarks

141

Algebraic Path Problems

that idempotency i s not necessary if the GAUSS-JORDAN method is formulated in the right way (equivalent to (8.43)). We remind that for stable matrices A* is a solution of the matrix equation (8.50)

X = I e ( A Q X ) In a complete semiring

(cf. 8.37).

A*

=

Ai satisfies

z+

(8.501, too.

Next we discuss a weaker condition which does not necessarily imply the existence of

A*;

on the other side a solution of

(8.50) will exist and can easily be determined using a modifi-

cation of the GAUSS-JORDAN method ( 8 . 4 3 ) . We assume the existence of a unary c l o s u r e operation a

+

a

on the semiring R such that = 1

(8.51)

(a

a)

for all a E R . Then R i s called a c l o s e d S e m i r i n g .

A

complete

semiring is always a closed semiring with respect to the

a:=

.

i a* = 1 a On the other z+ side, if R is a closed semiring and if R is m-absorptive for

closure operation defined by

-

some m E m , then a = a* for all a E R . In general, the closure operation does not necessarily coincide with the '*'-operation and is not uniquely determined. If a closure operation is defined only on a subset R' of R then LEAMANN [ 1 9 7 7 ] proposes to adjoin an element m a R , to add the definitions

and to extend the closure operation by

-

a:=

00

142

Linear Algebraic Opiinrizariun

for all a E ( R U (-1) a

Then R

X R ' .

u

I-1

satisfies all axioms of

closed semiring with the exception of

(a.52)

a B O = O B a = O

which is invalid for a =

m.

for all a E R , a c t o s u r e of R .

Ke call R U { - )

in this way an arbitrary semiring can be closed if we drop the assumption that R contains a zero. In particular, this approach is possible for the closure operation a + a* =

a

z+

i

which is

w e l l defined only on a subset of the underlying semiring.

Therefore we will avoid to use (8.52) in solving the matrix enuation

in closed semirings.

(8.50)

The following variant of the GAUSS-JORDAN method defined in a closed semiring. Let B ( O ) := sively for k = 1,2,

A

( 8 . 4 3 ) is well

and define recur-

..., n:

for all i , j E N . T h e following theorem is implicitly proved in LEHMANN [1977]. Its simple direct proof is drawn from MAHR [ 1980b].

(8.54) Theorem Let

(R,(B,@)

be a closed semiring and let

over R . Then I X = I

(B

(A

(B

A

be an n X n matrix

B (n) is a solution of the matrix equation

B X).

Proof. At first we show

for k = O , l , . . . , n. The case k = 0 is trivial. Let i,j E N and

Akebraic Path Problems

1

5

k

C

143

n. We assume that (1) is satisfied for k - 1 and we

denote B ( k - l ) and B(k) by B and B'. Since 1 =

-

(bkk 8

@

Ekk)

bkk we find b' = b kj kj

@

(bkk8gkk8b kj

=

1;

kk

B

b

kj

=:B

...,

Now b' = b @ (bvk 8 8 ) for all v = 1,2, k-1. Therefore vj vj the right-hand-side of equation (1) is equal to

Applying the inductive assumption to the terms in brackets we find equality with

by (8.53) this term i s equal to b! NOW let B:= B (n), 6ij

= 0

c

,

11

C:=

I

@

.

B , and define 6

ij

:=

1 if i = j and

otherwise. Then

ij

= 6

ij

@ b

ij n

= 6 . . @ a . 11

lj

@

n X(a 8 b ) iv vj v=l

for all i,j E N . Therefore C is a solution of the matrix equation

X

=

I a ( A B X). rn

We remark that theorem (8.54) is proved without using (8.52). For k = n the equations ( 1 ) In the proof show that B(n) is a solution of the matrix equation (8.55)

X = A @ (A 0 X ) .

144

Linear Algebraic Oprimizatioti

In a closed semiring R satisfying the additional equation

for all a E R w e may use the G A U S S - J O R D A N method

for

(8.43)

the determination of I @ B(n) if we replace the ‘*‘-operation in ( 8 . 4 3 )

by the closure operation in R. Let

C ( O ) : = A

and

define recursively f o r k = l , 2 , . ..,n:

for all i,j E N .

(8.58) C o r o l l a r y Let (R,@,@) b e an n

x

b e a closed semiring satisfying ( 8 . 5 6 )

n matrix o v e r R. Then C(n) = I

of the matrix equation Proof.

A

X

= I @

comparison of ( 8 . 5 3 )

(A @

A

B (n) is a solution

@

X).

and ( 8 . 5 7 )

for all i ,j E N and for all k = 0,1,.

and let

shows that

. . ,n.

Therefore C(n) = I

We remark that Corollary (8.58) is proved without using

@

(8.52).

Therefore we may consider the ‘*‘-operation a s a particular well-defined example for a closure operation. Thus A always a solution of the matrix equation

X

=

I

@

(A

( a closure of) the semiring R. On the other side A *

exist, in general

(cf. 8 . 6 5 ) .

is @

B (n)

X ) in

does not

.

145

Algebraic Path Problems

I n t h e remaining p a r t of t h i s c h a p t e r w e d i s c u s s s e v e r a l e x a m p l e s f o r p a t h p r o b l e m s . The u n d e r l y i n g g r a p h i s a l w a y s d e n o t e d by G = ( N , E ) w i t h w e i g h t s a

i j

c h o s e n from t h e r e s p e c -

t i v e semirings.

(8.59)

C o n n e c t i v i t y i n graphs

I f p i s a p a t h i n G f r o m i and j t h e n t h e s e v e r t i c e s a r e c a l l e d c o n n e c t e d . The e x i s t e n c e o f s u c h p a t h s i n

(undirected)

g r a p h s h a s f i r s t b e e n d i s c u s s e d by LUNTS [ 1 9 5 0 ] a n d SHIMBEL [ 1 9 5 1 ] w i t h r e s p e c t t o c e r t a i n app i c a t i o n s . problem t o t h e d e t e r m i n a t i o n of t h e .-Ith

They r e d u c e t h e

power o f t h e

a d j a c e n c y m a t r i x A d e f i n e d by a

ij

:=

{

1

if

(i,j) E E ,

0

otherwise

f o r a l l i,j E N o v e r t h e s e m i r i n g

({O,l),max,min).

T h i s is a

c o m m u t a t i v e , a b s o r p t i v e s e m i r i n g w i t h z e r o 1 and u n i t y 0. A b s o r p t i v e s e m i r i n g s h a v e b e e n d i s c u s s e d by Y O E L I c a l l e d them Q - s e m i r i n g s .

In p a r t i c u l a r ,

1 1 9 6 1 1 . He

YOELI proved t h a t

in

absorptive semirings

-

An 1 = A*

(8.60)

f o r a l l matrices A with A equivalent t o a

ii

(B

I = A.

The c o n d i t i o n A

= 1 ( u n i t y ) f o r all i E N .

i n o u r above example A

n- 1

I = A 'is

T h i s shows t h a t

9 QI. In the ij n- 1 c a s e of u n d i r e c t e d g r a p h s A and A a r e symmetric m a t r i c e s .

i j

= 1 i f

(B

and o n l y i f P

The g r a p h w i t h a d j a c e n c y m a t r i x A * = A n - '

sitive c l o s u r e o f problem.

ROY

G,

is c a l l e d t h e t r a n -

and w e s p e a k o f t h e t r a n s i t i v e c l o s u r e

119591 a n d WARSHALL 119621 d e v e l o p e d t h e

Linear Algebraic Optimizatioh

14tl

GAUSS-JORDAN method

(8.48) for the computation o f An-1. Methods

based on STRASSEN's fast matrix multiplication are proposed by ARLAZAROV et al. [1970], FURMAN [1970], FISCHER and MEYER [I9711 arid MUNRO [1971]. The average behavior of such methods i s discussed by BLONIARZ, FISCHER and MEYER [1976] and by SCHNORR [19781.

(8.61)

Shortest paths

The shortest path problem has at first been investigated by BOR6VKA 119261. SHIMBEL 119541 reduced this problem to the computation of the n-lth power of the real arc-value matrix A over the semiring ( W + U (-l,min,+). absorptive semiring with zero

a

ii

m

This is a commutative

and unity 0. SHIMBEL assumed

= 0 for all i E N . Therefore A @ I = I and

again that A* = A

n- 1

.

If negative weights a

ij

(8.60) implies are involved

then we have to assume that the network i s absorptive, i.e. G does not contain a non-null elementary circuit p with weight w(p) < 0 (with respect to the usual order relation of the extended real numbers). Then the semiring (IR U {m},min,+)

is the weight of a shortest path is considered. Hence Ak ij from i to j of length k and A i k l is the weight of a shortest ij path from i to j o f length not greater than k. In particular A* = A [n-11 (cf. 8 . 3 3 ) . Based on the paper of WARSHALL 119621 FLOYD [1962] proposed the application of the GAUSS-JORDANmethod for the determination o f A*. In particular, if all weiqhts are nonnegative then algorithm ( 8 . 4 ) (cf. DANTZIG [1960], DIJKSTRA [1959]).

can be used

Several other Solution

methods can be found in the standard literature (cf. LAWLER

Algebraic Path Problems

147

[19761). Recent developments can be found in WAGNER [19761, FREDMAN [1976l,,JOHNSON 119771, BLONIARZ [1980] and HANSEN [1980]. An excellent bibliography is given by DEO and PANG 119801.

If the weights a

are chosen in R:=

ij

IRk

u

{ (-,-,,,

then

the weight of a lexicographically shortest path may be determined. Let a

(B

b:= lex min(a,b)

for a,bEIRk with respect to the lexicographical order relation

6 of real vectors. Then (R,@,+) is an idempotent, commutative semiring with zero

A* is ij the weight of a lexicographically shortest path p EPij. A* (-,-I...I-)

and unity (O,O,...,O).

exists if and only if the network is absorptive, i.e.

if G

contains no circuit p with w(p) 4 (O,O,...,O). Such a problem was first considered by BRUCKER ([1972],

119741).

(8.62) M o s t r e l i a b l e p a t h s If the weights a

ij

are chosen from the semiring ([0,11,max,=)

with zero 0, unity 1 and usual multiplication of real numbers then the weight w(p) = a il,l=ai

. ...

2 2

8

of a path p = ( ( i ~ , j ~ ) , ( i 2 , j ~ ) , . . . , ( i SS,'1j) may be interpreted as the reliability of the path p . This semiring is again commutative and absorptive. A : j

is the weight of the most reliable

path from node i to node j. This problem was first considered by KALABA 119601.

148

(8.63)

Linear Algebraic Optimization

Maximum c a p a c i t y p a t h s

If the weights a .

are chosen from the semiring (lR+U{w),max,min)

lj

with zero

and unity 0 then the weight w ( p ) = min{a

.a. ,...,a. I i l j l 12j2 lsjs

o f a path P = ((il,jl),(i2,j2), . . . , (is,js)) may be interpreted

as the c a p a c i t y of the path p. This semiring is again commutative and absorptive. A* is the weight of a maximum capacity ij path from vertex i to vertex j. This problem was first considered by HU [1961].

(8.64)

A b s o r p t i v e semirings and a b s o r p t i v e networks

YOELI [1961] discussed absorptive semirings (Q-semirings) and the relevance of A

n- 1

and A*. The application of the GAUSS-

JORDAN-method in absorptive semirings was proposed independent-

ly by TOMESCU [I9681 and ROBERT and FERLAND 119681. CARRE 119711 considered such algorithms for absorptive network5 over idempotent semirings under the assumption that the composition

0

is cancellative. In particular, distributive lattices are discussed in BACKHOUSE and CARRd [19751. F o r Boolean lattices (Boolean algebras) we refer to HAMMER and RUDEANU [19681.

(8.65)

Inversion o f matrices over f i e l d s

A field (F,+,.) with zero 0 and unity 1 is, in particular, a semiring. If A is a stable matrix over F with stability index

r then

A

Irl

=

matrix f o r all

A [r+ll = A [rl s

+

Ar+'

implies that AS is the zero-

2 r+l. Let p be a circuit of length l(p). Then

149

Algebraic Path Problems

2 r+l for some m E N . Therefore w(p)

m.1 (p)

cuits of

= 0.

Thus all cir-

G have weight zero. If all circuits in G have weight

5 n-1. (8.36) and

zero then A is stable with stability index r (8.37) imply

In particular, if A is stable then I - A is a regular matrix over F. The algorithms for the computation of A * given in this section reduce to the conventional linear algebra methods for the inversion of I

Applications for networks without cir-

-A.

cuits have been given by MULLER-MERBACH 119701 (so-called Gozinto-graphs). For the closure IR U { = )

of the field of real numbers LEHMANN

[1977] introduces the closure operation a

-

if aE{l,=),

m

a:=

otherwise Clearly that

a

= 1

+

(a:)

= 1

a*:=

+

(;a)

= a* = 1 + a + a 2 + . . .

The closure operation a if la1

2 1.

a

+

defined by

.

for all a EIR U (-1.

We remark

for all aEIR with la1 < 1 .

a* may be extended on I R U

+

n

Since 1

+ a + a' +

...

{mf

by

is an oscillating di-

vergent series-for a e - 1 the algebraic path problem does not have a solution in IR U { - ) circuit p = (el,e2,...,e

P

if the underlying graph contains a )

with weight

For the different closure operations algorithm

(8.57) leads

to different matrices C(n) denoted by A- and A, which both solve the matrix equation X = I

+ (AX) in IR

U

I-) (cf.

Linear Algebmic Optimizatiun

150

C o r o l l a r y 8 . 5 8 ) . A, A_.

I f A-

w i l l c o n t a i n more i n f i n i t e e l e m e n t s t h a n

does n o t c o n t a i n an i n f i n i t e element t h e n A _ = ( I - A )

The same r e s u l t h o l d s f o r A * .

we find A-l

an i n f i n i t e e l e m e n t . (I -PA)-

For a given nonsingular matrix A

provided t h a t

(I -A)-

=

(I

- A) -

does not contain

I f P i s a permutation m a t r i x such t h a t

does n o t c o n t a i n an i n f i n i t e element t h e n A

U s i n g w e l l known p i v o t i n g r u l e s A

-1

-1

= (I-PA)-P.

can always be determined

i n t h i s way. [I9751 c o n s i d e r e d t h e n u m b e r o f p a t h s f r o m n o d e i t o

GONDRAN

n o d e j. F o r t h e a d j a c e n c y m a t r i x A w i t h r e s p e c t t o t h e f i e l d k o f r e a l numbers w e f i n d t h a t A . . i s t h e number o f p a t h s f r o m 13

i to j

of

of length k , A[k1 i j

i s t h e number o f p a t h s from i t o j

length not g r e a t e r than k and,

circuit, to j.

t h e n A*

i j

= A In-']

i j

I t i s always assumed

(8.66)

i f G does not contain a

i s t h e number o f a l l p a t h s f r o m i that A

ii

= 0 for

all i E N .

Regul a r a1 g e b r a

BACKHOUSE a n d CARRg c o n s i d e r e d c l o s e d

with a closure operation a +

-

a =

a.

idempotent semirings

satisfying

(1 @ a)

(8.67)

f o r a l l a,B,J,

E R and

f o r a l l a,B,$ E R such t h a t

for all X E R .

-1

They d i s c u s s o n l y e x a m p l e s i n which

coincides

.

Algebraic Path Ptoblems

with the usual closure operation a + a * ,

= 1

a*

a

(B

(B

a'

(B

151

i.e.

... .

The standard example of a regular algebra is the following.

1 b e a f i n i t e n o n e m p t y set. V i s c a l l e d a n

L e t V = {v1,v2,...,v

a l p h a b e t and its elements are called l e t t e r s .

A

word o v e r v i s

a f i n i t e s t r i n g o f z e r o or m o r e l e t t e r s f r o m V . , I n p a r t i c u l a r the empty string

is denoted by

c.

of t w o t a n g u a g e s t = t t

, . ..

i s c a l l e d e m p t y w o r d . T h e s e t of a l l words

E

a , B C_

ii

i s a U 5 . For w o r d s w = w l w2...w

r

and

t s we d e f i n e w o

w o

l a n g u a g e i s a n y s u b s e t o f ?. T h e sum a + 5

A

t:= w l w2...w

t t ,...ts

i

t is c a l l e d t h e c o n c a t e n a t i o n o f w a n d t. T h e c o m p l e x c o n c a -

Fenation of two languages a , B

(V,+,O)

C, 0 is d e n o t e d b y a

5 . Then

i s an i d e m p o t e n t s e m i r i n g w i t h zero # and u n i t y

The closure a*

-

~ € B1 a

B a'

B

{E}.

...

i s w e l l - d e f i n e d ; i n p a r t i c u l a r $* = ( € 1 . W e d e n o t e s i n g l e - l e t t e r l a n g u a g e s {a) s h o r t l y b y a. T h e n a r e g u l a r e x p r e c l e i o n o v e r V is any well-formed formula obtained from t h e elements V U operators

+ , a

and

*,

and the parentheses

# * + (v,

D

(v, + Vl)*'

(

and

).

{ # I t,h e

For example

v. 3 ) *

is a r e g u l a r e x p r e s s i o n . E a c h r e g u l a r e x p r e s s i o n d e n o t e s a l a n g u a g e o v e r V w h i c h is c a l l e d a r e g u l a r l a n g u a g e . T h e s e t of

a l l r e g u l a r l a n g u a g e s is d e n o t e d b y

-

s(v).

(S(V)

,+,a,*)

is a

regular algebra ( a = a * ) . The axiomatic system defining regular

152

Linear Algebmic Optimization

a l g e b r a s w a s p u t f o r w a r d f o r r e g u l a r l a n g u a g e s by SALOMAA [1969].

I n a p r o o f o f a t h e o r e m on l a n g u a g e s i n a u t o m a t a

t h e o r y K L E E N E [ 1 9 5 6 ] u s e s a GAUSS-JORDAN m e t h o d for S ( V ) . I n H c N A U G H T O N a n d YAMADA

[ 1 9 6 0 ] t h e same m e t h o d i s d e s c r i b e d

e x p l i c i t l y . For t h e t h e o r e t i c a l background i n automata t h e o r y we r e f e r t o E I L E N B E R G

([19741,

p.

175).

A n o t h e r e x a m p l e o f a r e g u l a r a l g e b r a is a n a b s o r p t i v e s e m i r i n g (R,$,B)

(cf. 8.64).

-

Then a = a* = 1

a = 1 f o r all a E R .

IB

T o g e t h e r w i t h i d e m p o t e n c y t h i s shows

(8.67). Distributive

l a t t i c e s are included as a particular case.

BACKROUSE a n d CARRE [ 1 9 7 5 ] d i s c u s s s e v e r a l l i n e a r a l g e b r a methods f o r s o l v i n g m a t r i x e q u a t i o n s i n r e g u l a r a l g e b r a . I n p a r t i c u l a r , theorem

(8.54)

t h e GAUSS-JORDAN m e t h o d

shows t h e v a l i d i t y o f a v a r i a n t o f

(cf. 8.53).

I f the considered closure

operation coincides with t h e usual *-operation ( 8 . 5 8 ) shows t h e v a l i d i t y o f t h e G A U S S - J O R D A N

(8.69)

then corollary method

(8.57).

Path and cut set enumeration

I n o r d e r t o d e t e r m i n e t h e set o f a l l p a t h s f r o m n o d e i t o n o d e j w e c o n s i d e r t h e r e g u l a r a l g e b r a S ( E ) where E d e n o t e s t h e s e t of

a l l a r c s of t h e u n d e r l y i n g g r a p h G.

Then t h e w e i g h t a

g i v e n by

(8.70)

if

(i.j)

EE,

otherwise and t h e w e i g h t of a p a t h p = ( e l , e 2 ,

...,e s )

, is

i j

is

Algebraic Path Roblem

w(p) = e l o e20

...

o

e

S

153

.

(Here we denote single-letter languages by ei rather than {el}.) For k E l N we get

for all i,j E N . An actual computation of A* is not possible in the general case, as the cardinality of a* may be infinite. ij If we are only interested in all elementary paths then a suitable modification of the underlying regular algebra is the following. A word w is called an a b b r e v i a t i o n of a word t if w can be obtained from t by deletion of some of the letters of t. For example, METAL is an abbreviation of MATHEMATICAL, or if an elementary path p is obtained from a nonelementary path p' by deletion of the elementary circuits in p' then w(p) is an abbreviation of w(p'

).

For a E S(E) let r ( a ) denote the language

containing all words of a for which no abbreviation exists in a.

Let N

s(E):=

{aES(E)I

a =

r(a)l

and define a @ E:=

r ( a +B )

a dD 8:-

r ( a o 8)

for all a , B Ez(E). with zero we find

0

Then ( z ( E ) , @ , @ I is an absorptive semiring

and unity

{E}.

Further, for A as defined by ( 8 . 7 0 1 , .

154

Linear Algebmic Optimization

for a l l k E N and

A*

= A In- 1

1

,

i.e.

a*

11

=

a In-'] ij

contains the

w e i g h t s o f a l l e l e m e n t a r y p a t h s f r o m i t o j. F o r t h e c o m p u t a t i o n of A * KAUFMANN and MALGRANGE I19631 proposed the sequence M,M w i t h M:=

2

,M

4

I . . .

I b A. T h e n M Z k = A* f o r 2 k

2

n - 1 shows that this

m e t h o d , c a l l e d Latin m u l t i p l i c a t i o n , n e e d s O ( l o g n ) m a t r i x 2

N

m u l t i p l i c a t i o n s o v e r t h e s e m i r i n g S ( E ) . M U R C H L A N D 119651 and B E N Z A K E N 119681 a p p l i e d t h e G A U S S - J O R D A N m e t h o d f o r a s o l u t i o n

of t h i s p r o b l e m .

Improvements can b e found in BACKHOUSE and

C A R R E 119751 a n d F R A T T A a n d M O N T A N A R I 119751.

A related problem is the enumeration o f all minimal cut sets o f t h e u n d e r l y i n g g r a p h G = (N,E). L e t

(S,T) denote a partition

of N. T h e n c = (S x T ) flE i s c a l l e d a cut s e t o f G. I f t h e a r c s of a c u t s e t c a r e r e m o v e d f r o m G t h e n t h e r e e x i s t s n o p a t h in G f r o m a n y i E S t o a n y j E T . T h e r e f o r e w e s a y t h a t c

separates i from j. I f a c u t s e t c d o e s n o t c o n t a i n a p r o p e r s u b s e t w h i c h i s a c u t s e t t h e n c is c a l l e d a minimal cut set. In o r d e r t o d e t e r m i n e t h e s e t of a l l m i n i m a l c u t s e t s s e p a r a t i n g I E N f r o m j E N t h e f o l l o w i n g a l g e b r a is i n t r o d u c e d by M A R T E L L I ([19741, [1976]). L e t P ( E ) d e n o t e t h e s e t o f a l l s u b s e t s of E. F o r a C_ P ( E ) l e t

r(a) denote the subset o f a containing all sets in a f o r which n o p r o p e r s u b s e t e x i s t s i n a.

Let

155

Algebraic Path Problems

and define

a @ B:=

r{aUbl

8 B:=

r(a U B)

Q

aEa, bEa)

f o r a l l a , B E C(E). T h e n ( C ( E ),@,a) i s an absorptive semiring with zero

and unity

a,,:=

0.

1 E C ( E ) where a i s defined by ij ij ( 8 . 6 0 ) . T h e weight w(p) of a p a t h p = ( e l , e 2 e ) with

We as s i g n weights

{a

,...,

r e s p e c t t o t h e semiring C ( E ) is t h e set of its arcs, 1.e. w ( p ) = {e,l

k = 1,2,.

. . ,s}.

( 8 . 3 3 ) we find A* = A

Du e t o t h e orem

In-1 1

. Each

element c E a

* i j

h a s t h e f o rm

w h e r e e ( p ) d e n o t e s an arbitrary arc of p. T h u s t h e r e exists n o p a t h from i t o j in t h e g r a p h G

C

=

(N,E'c).

Let S d e n o t e

t h e s e t of a l l vertices k s u c h t h a t t h e r e exists a p a t h from

i to j in G c

,

and let T:= . S ' N

Then c = ( S

x

T ) n E , 1.e.

c

is a cut set separating i and j in. G. T h u s a* is t h e set o f ij

a l l m i n i m a l cut sets separating i and j. T h e o r e m (8.45) shows that

R*

can be computed using t h e GAUSS-

J O R D A N method. T h i s approach i s discussed in d e t a i l by MARTELLI [ 1 9 7 6 ] . An arc

(p,v)

is called b a e i c with respect t o i , j E E i f { ( p , u ) }

is a m i n i m a l c u t separating i f r o m j.

CARRE 1 1 9 7 9 1

proposes

t h e f o l l o w ing semiring for t h e determination o f all basic a r c s in G. Le t P ( E ) d e n o t e t h e set o f all s u b s e t s o f E, and let UbE.

W e d efine

156

Linear Algebmic Optimization

f o r a l l a , B E P ( E ) a n d a d j o i n {a} a s a z e r o . L e t R = P ( E ) U {o}.

i s an absorptive semiring with zero ( w )

(R,@,@)

Then

and with

6. W e a s s i g n w e i g h t s

unity

A g a i n t h e w e i g h t o f a p a t h i s t h e s e t o f i t s arcs. D u e t o t h e o r e m (8.33)

-*

we find A

basic with respect to i and j provided

-*. a

ij

find

*: i j c o n t a i n s a l l P + 0; o t h e r w i s e ij

= A[n-ll. Then

arcs

= {a). S i n c e w e a s s u m e t h e e x i s t e n c e o f n u l l c i r c u i t s we

z:i

=

6 for all i E N .

(8.71)

Schedule algebra

GIFFLER

([1963]. [ 1 9 6 8 ] ) c o n s i d e r e d t h e d e t e r m i n a t i o n o f a l l

path weights in a network with nonnegative integer weights. H i s a p p r o a c h h a s b e e n e x t e n d e d by W O N G S E E L A S H O T E

Let ( H , + , I )

[1976].

be a totally-ordered commutative group with neutral

e l e m e n t 0. In p a r t i c u l a r , w e a r e i n t e r e s t e d i n t h e a d d i t i v e g r o u p o f r e a l numbers.

n

H =

@.F o r

I l z l l :=

bY

fi

=

x E H } be a copy of H with

convenience we define

we consider z +

for all z E Z .

Let

-

x

= x f o r all

a s a u n i t a r y o p e r a t i o n o n Z = €3

;€HI

i.e.

u i.T h e n

if z E H . if Z E H ,

We continue the internal composition

+

of H on Z

157

Algebraic Path hoblems

-

-

x

+

y:= x

+

y

x

+

y:= x

+

y:=

-

-

I

x+y ,

for all x , y E H and for all ;,?Eli.

Then (Z,+) is a commutative

group with neutral element 0. The inverse elements are denoted by -z for z E 2. A

function a: Z

Z+ is called countable if

-B

is countable and a is called well-ordered if

is well-ordered (each subset of H(A) contains a minimum). Let

U denote the set of a l l countable and well-ordered functions a: z

-B

Z+.On U we define two internal compositions by

for a , b E U and z E Z and by (8.75)

(a o b) (z): =

for a , b E U and

~

€

2

E

x+y=z

a(x)b(y)

The . right-hand sides of ( 8 . 7 4 )

and (8.75)

are defined with respect to the usual addition and multiplication of nonnegative integers. It is easy to show that a + b is a well-defined element of U. Now let a , b E U and ~ € 2 .Then we claim that S = { ( x , y ) E Z x + y = z, a(x)b(y) * O ) is a finite set. Suppose infinite. Then

{XI

S

2

I

is countably

(x,y) E S) contains an infinite sequence

IIxlII < IIx211
0

x

and

> 0.

Proof. If X = 0 then A B x = 0. (9.3) shows a all i,j EN. A 6

(R',b) is a group we get a i j

all i,j E N . A s x E V ( A ) there exists x all j EN,contrary to G

A

I

*

ij

= 0

8 x

or x

0. Then a

for

= 0

j

j

i j

= 0

= 0

for

for

strongly connected. Therefore A > 0.

Now suppose xk = 0 and let M:= { j l x

j

*

0). Then M , N X M 9 $

and

for all i EN'M. contrary to G

Therefore a A

ij

= 0

strongly connected.

for all ( i , j ) E (N'M)

x M,

Linear Algebraic Optimization

174

I n t h e d i s c u s s i o n of t h e e i g e n v a l u e problem i n t h e c a s e t h a t

(R',@)

i s a group w e w i l l only consider solutions of

w i t h A > 0, x > 0.

Such s o l u t i o n s a r e c a l l e d finite

t19791). P r o p o s i t i o n (9.13)

CUNINGHAME-GREEN

c i e n t condition such t h a t every solution of

Now we a p p l y

(9.10) t o

(9.12).

L e t A:= N

h

-1

A

and A-'

(cf.

gives a suffi(9.1)

0 A.

is finite.

Then G

d i f f e r only i n the weights assigned t o the arcs

G-

(9.1)

A

and

(i.e.

a. lj

The c o r r e s p o n d i n g w e i g h t f u n c t i o n a n d o t h e r 11 N N N d e n o t a t i o n s i n v o l v i n g w e i g h t s a r e l a b e l e d by ' I - " (U,W,Uj I . . 0 a. .).

.

1.

(9.1-4) P r o p o s i t i o n L e t R be a s e m i r i n g w i t h g r o u p N

N

L e t k E N.

N

(1)

If

1 tB u k = u k

(2)

if

R i s commutative and 1 @ pk =

Proof.

( 1 ) Here

fore N

A:

;:

then

(R',@).

(9.10.1)

A:EVo(X);

vk

for

N

i s f u l f i l l e d w i t h LI = 1 for A .

i s an e i g e n v e c t o r o f

There-

( 9 . 1 2 ) w i t h e i g e n v a l u e 1 . Hence

E Vo(h).

( 2 ) I n t h e commutative c a s e t h e weights o f a p a t h p w i t h respect t o G

A

a n d GX a r e g i v e n b y N

w(p) = x - l ( p )

Therefore

k

= pk

and

(D

w(p).

(1) is applicable.

These s u f f i c i e n t c o n d i t i o n s should be seen t o g e t h e r w i t h t h e following necessary conditions.

Eigenvalue Problems

175

(9.15) Proposition Let R be a semiring with group ( R ' , B ) and let R be partially ordered by (9.2). Let x E V ( h ) . b e a finite solution. w

(If R is commutative then)

(1)

for all circuits p in

G

w(p)

5 1

(w(p)

5 h 1 (PI)

.

A

'

if R is idempotent then

(3)

if R is linearly ordered (and commutative) then there

exists a circuit p in G Proof. -

is absorptive (hence

'ii* exists);

(2)

with ~ ( p )= 1

A

(1) Let p be a circuit of (A-'

a a,,)

B

xj

(w(p) = ~'(p)).

Then (with respect to 9 . 2 )

GA.

5 xi

for all arcs (i,j) of p. Together these inequalities imply N

W(P)

@

Xk

5

Xk

f o r each vertex k of p. Then xk > 0 shows

W(P) 5 1.

N

(4)

In the commutative case y(p) = w(p) (2)

If R is idempotent then ( 4 ) and (9.4) lead to N

w(p) 13 1 = 1

i.e.

@

N

A

is absorptive and

*;

f o r all circuits p in G

ij

B x.1

3

'

exists (cf. 8 . 3 3 ) .

( 3 ) In the linearly ordered case

maxis

A

A B

x = h

x means

j E N ) = A B xi

f o r all i E N . This maximum is attained for at least one j ( i )

in row i . The sequence k , j ( k ) , j ' ( k )

...

contains a circuit p

Similarly to ( 4 ) we find N

W(Pk) = 1

which in the commutative case means w(pk) = h

1 (P,)

k'

Linear Algebraic Optimization

176

If R is linearly ordered, commutative and finite solution then propositions

contains a

and (9.15) show in

(9.14)

particular that at least some columns

V(A)

-* Ak

are elements of

V0(A).

Now we consider again the more general situation without specific assumption on the internal composition 0.

(9.16) P r o p o s i t i o n Let R be a semiring. If R is idempotent then x E V ( 1 )

(1)

implies

0 x = x and

A*

A + @ J x = x ;

(2)

if R is absorptive and commutative (hence

A*

exists) then

implies A * 0 x = x ;

xEV(X)

if R is absorptive and aii = 1 for all i E N then V ( X ) C _ V ( 1 )

(3)

for all A E R. Proof. ( 1 ) Let x E V ( 1 ) . Then A k

8 x

= x

for all k E N . There-

fore Ark]

0 x = x B

for all k E R . Then Let x E V ( A ) .

(2)

Ak

A*

( A @ X ) 8)

8 x

=

B

An-'

x. Similarly

A

+

B

...

8 x

= x

= x.

Then commutativity implies

0 x = Ak

8 x

for all k Em. Due to ( 8 . 3 3 )

..

(A2 0 x )

E R shows

A*

= A

rn-'1.

1 B a = 1. Therefore A *

Now a = A

(B

...

h2

0 x = A[n-~l ~

= (1 @ a ) 0 x = x . ( 3 ) Obviously I =

(1 B A )

0

(B

A = A.

Let x E V ( h ) . Then

A 0 x

=

(I @ A) 0 x

x = 1 0 x = x. rn

177

Eigenvalue Problems

We remark that, if R is absorptive and aii = 1 for all i E N , then A

*

exists and (9.10.3) shows that all A

*

j

are elements of

Vo(l) which, by (9.16.31, contains all eigenvectors of A .

(9.17) T h e o r e m Let R be a linearly ordered semiring. If x E V ( 1 ) then x = A* @ x M

with

A*

V

8 x

V

M

E V ( 1 ) for all v E M

5

N.

Proof. From (9.16.1) we know x = A*

0

x. Thus it suffices to

find a reduced representation by terms A* (9.16.1) we also know

+ A

V

@

Q

x V E V ( l ) . From

x = x . We consider the partial di-

graph G' = ( N , E ' ) with A

with respect to an eigenvalue X

(here A = 1). Then we introduce

an ordering on N by 1 5 j if and only if there exists a path from i to j in G'

A'

On the set of maximal vertices this induces

an equivalence relation. This fo1,lows from the fact that for all i E N there exists at least one j E N such that i A 4D x = A

@

x and R linearly ordered). Let Nk

, k

5

j

(as

= 1,2,...,s

denote the corresponding equivalence classes. Then a similar argument shows that the subgraphs G;

generated by Nk in G A are

strongly connected and contain at least one circuit of G' A' Now let p denote a path from v to l~ in GA. Using the equations aiJ

8 x, = xi along p we find

(9.18)

w(p) 8 x u = x

V

.

178

Linear Algebraic Optirhization

Obviously x of

(B

P

> 0 iff x V > 0. From (9.8) and the idempotency

we get

+

(AV 8

w(p))

(A:

xV) @

A

(B

+

= A

v

+

.

U

Therefore (9.19)

Q

0 x

(A:

u)

= A

+ u

8 x

U

which leads to x = A

for M : =

{v,[

A:

k

+

O

xV

@

x

=

A

+

M 8 xM k = 1.2,

9 0,

k

u k E N k for all k = 1,2,. ..,s. p of G h

As

...,s )

provided that

each v E M lies on some circuit

(9.18) yields

w(p)

.

xu = x

8

Idempotency of R implies (w(p) for u E M .

8 X")

Then (9.10.2) A ; @ x

for v E M . Thus x =

leads to

= A : B x A:

B

=

@ XV

x

V

(B

x u = w(p) B X"

*

AV 8 x

E V o ( l ) for v E M .

Hence

V

x M is a representation with

A*

V

8 xVEV(l)

for all v E M .

This theorem shows that all eigenvectors with eigenvalue 1 are

*.

linear combinations of certain columns of

A

vertices lie on circuits of the graph

If R is furthermore

absorptive, i.e.

GA.

1 is the maximum of R, then

The corresponding

(9.16.3) implies

that we can determine all eigenvectors in this way provided that the main diagonal of

A

contains only neutral elements.

179

Eigenwlue Problems

Next we consider the spec$al case of a linearly ordered, absorptive (commutative) semiring with idempotent monoid

(R,@).

We may as well consider a linearly ordered (commutative) monoid with minimum 0 and maximum 1 which has an idempotent

(R,B,()

internal composition

The eigenvalue problem ( 9 . 1 )

8.

is thus

of the particular form

for i E N .

(9.21) T h e o r e m Let R be a linearly ordered, absorptive semiring with idempotent (R,O). Hence A* exists. Let B := A* 8 u, j j

monoid

J:= {j E N 1 u

j

*

for j E N and

0). Then

is the right semimodule generated by { B . I j E J);

(1)

Vo(l)

(2)

if R is commutative then A 0 V o ( l ) = A 8 V o ( h ) ;

(3)

if R is commutative and a

I

A 8 V0(l)

= V0(A).

Proof. ( 1 ) From ( 9 . 1 1 . 1 ) we know B absorptive semiring

* a

(9.10.1)

1

uk

*

@

p.

EVo(l)

j

for all j E N . I n an

= 1 for all j E N . Thus

jj

B . E V ( ~ ) .Absorptivity and =

= 1 for all I E N then

ii

Due to ( 9 . 1 7 )

-

c A

Q

= A 8

Vo(A).

*

and ( 1 ) imply A 8 V o ( l ) If x E V o ( A )

( A @ x), i.e.

A

which implies A b V o ( A )

*

0 implies

iff

*

x E V ( 1 ) .has a representation x = AM 8 xM

for all k E M which shows x = B M 8 xM (9.11.2)

*j

show that Ak @ p E V o ( l )

with Ak 0 x k E V ( 1 ) for all k E M. Therefore Ak

(2)

U

then A

@ xI E

-

0

Vo(l).

x =

.

Vo(A). A 8 x.

8 U

k 8 xk

Then A 8 V o ( l )

Thus A B x = A b ( h e x )

Therefore A

A 8 Vo(l).

xk = :A

8

@

Vo(A)

Vo(l)

180

Linear Algebraic Optimization

(9.16.3) implies A b x = x for x E V ( A ) . Therefore

(3)

A B Vo(A)

Together with ( 2 ) we get A

= Vo(A).

Q Vo(l)

=Vo(A).

Now we consider the special case of a linearly ordered semiring with group (R',b). We may as well consider a linearly ordered group (R',O) with an adjoined least element 0 acting as a zero. The neutral element of R' is denoted by 1 . CUNINGHAME-GREEN ment

[I9791

additionally adjoins a greatest ele-

but mainly considers finite solutions of the eigenvalue

problem (0 < A
0 and 7

j

Let A , B be two finite eigenvalues of A for finite solutions

(2)

of the eigenvalue problem. Then ( 9 . 1 5 . 2 ) and

-1

i:= B @

X < B . From

A

implies that

are absorptive. Suppose 1

9

x:=

A

-1

@ A

B and w.1.o.g.

we know that there exists a circuit p in

(9.15.3)

N

N

such that w(p) = 1. Then w(p) > w(p) = 1 contrary to A ab-

GA

sorptive. Thus A = B . ( 3 ) Let J denote a maximal set of nonequivalent eigenvertices

of G-. A

Let j E J. Clearly, ( 9 . 1 5 ) N

1 CB

j

= u j and, by

get x * E V ( A ) . j

(9.10.1),

-* A

j

implies Eyo(l)

From the proof of ( 9 . 1 7 )

a representation x =

-* A

M

-

@

j is an eigenvertex of

x.

,

= 1. Therefore

= Vo(h).

As

*: j j

= 1 we

we know that x E y ( 1 ) has

xM such that all vertices j E M lie

on a circuit p of GA and w(p) * x A

j

- x,. -

N

Thus w(p) = 1 , i.e.

The elements of

M

were chosen in such

a way that any two of them do not belong to the same circuit

of

Gi.

in

G i

of

x.

As

any circuit p in G-

A

with weight z(p) = 1 is a circuit

we find that M consists in nonequivalent eigenvertices

182

Linear Algebraic Optimization N

for an arbitrary k E J . Suppose : A

Let K:= J\(k)

(4)

for some y e R

K

.

N

= A;

B y

Then N

N

1 = a*

= a*

kj

kk

yj

QD

N

for some j E K and as a? = 1 ij

2

N

a;k

N

N

Now a*

(a*

ik

kj

.

1 @ y . = yj

1

is the weight of a shortest path p (9) from j to

)

q) = a* Q *: 2 1 and p kj jk Absorptivity implies w ( p - q) = 1. Then k

k ( k to j) in GN and therefore w(p A

is a circuit in

GN.

A

N

D

q

and j are equivalent vertices contrary to the choice of k and j. Thus

S

is a minimal generating set of V o ( A ) . As

N

is an equi-

valence relation on the set of all eigennodes all maximal sets of nonequivalent eigenvertices have the same cardinality. 8

Theorem ( 9 . 2 2 ) Vo(x),

i.e.

shows how we can compute the finite eigenvectors,

if the unique finite eigenvalue N

--1

N

we determine A* for A:= h

C3

x

is known. Then

A and select columns

*; j E

(R'In

N

with a ? . = 1. Let J denote the set of the indices of such finite 33

eigenvectors and choose a maximal set of nonequivalent eigenvertices from J. Such a set can easily be found by comparing the N

columns A* j

, j E J , pairwise as N

A* = 3

*; k

(follows from ( 9 . 1 9 ) x:=

-* A , ) .

Q

j

-

k implies that

*; k j

on the corresponding circuit in G' with A

I

The remaining problem is the determination of

x

(if it exists).

We assume now that R is commutative. Then we may assume w.1.o.g. that ( R l . 8 ) is a divisible group (cf. 3.2). F o r b E R '

let

183

Eigenvalue Problems

x = bnlm denote the solution of the equation xm = bn with n/m E Q. From (9.15.3) we know that if

exists then there

exists a circuit p with

-a -

2

and A

= w(p) l/l(P) = :

;(p)

-

w(p') for all circuits p' in G

...

tary then p = p l e p2. k = 1,2,.. . , s .

p, with elementary circuits p

-

, k

A = W(Pk)

A ( A ) :=

(9.23)

exists then

-

B:=

A

Then

=

1,2,...,s.

max{w(p)

I

= A(A).

In any case the matrix

x

A(A)

i s absorptive. Hence

of

k '

contains at least one circuit then let

GA

If

If p is not elemen-

The cancellation rule in groups implies that

If

A'

-1

p elementary circuit in G

A

I.

8 A

i * exists.

i * contains

A(A)

is the finite eigenvalue

i* >

0 with g* = 1. j jj is a finite eigenvector. Otherwise the eigenvalue

if and only if

;*

a column

j problem has no finite solution. A

direct calculation of

A(A)

from (9.23) by enumeration of all

elementary circuits does not look very promising. For the special case of the semiring (IR

u

{--I,max,+)

LAWLER

1 1 9 6 7 1 pro-

posed a method which has a str.aightforward generalization to semirings derived from linearly ordered, divisible and commutative groups

with

A

k

=

(R',@).

k (a ij A(A)

Let

A'k'

be defined by

for all i,j ,k € N (cf. chapter 8 ) . Then = maxEa jk'l

ii

i,kEN).

184

Linear Algebraic Optimization

The computation of Ak for k E N needs O(n 4 ) @-compositions and O(n

4

)

( 5 comparisons). Then aii 'k'

@-compositions

can be determined by solving O(n 1

5

2

for i , k E N

equations of the form ak = 6 ,

)

5 n. Further O(n 2 ) @-compositions ( 5 comparisons) lead

k

to the value

h ( A ) .

For the same special case another method was proposed by DANTZIG, BLATTNER and RAO 119671. Its generalization is an algebraic linear program over a certain module (cf. chapter 11). Its solution

can be found by a generalization of the usual

simplex method (cf. chapter 11). that

x

It can be shown (cf. 11.62) N

X(A) and, in particular, h =

determination of

h(A)

if

1

exists. After

using methods discussed in chapter 1 1 we

-- 1

compute the matrix ( h

@

A)* which exists since

x

2 X(A). If

-

this matrix contains a finite column with main diagonal entry 1 then h = X(A) and the column is a finite eigenvector. Otherwise the finite eigenvalue problem has no solution. In particular, if

7

> h(A) then all main diagonal elements of (x-'

@

A)*

are strictly less than 1. For the above mentioned special case this method is discussed in CUNINGHAME-GREEN 119791.

We remark that the above results for linearly ordered, commutative groups have some implications for lattice-ordered commutative groups. From theorem ( 3 . 4 )

we know that a lattice-

ordered, commutative group (R',@,z) can be embedded in the lattice-ordered, commutative and divisible group V root system

T.

(x) with

a

Two different finite eigenvalues with finite

eigenvectors are incomparable and therefore the number of such finite eigenvalues is bounded by the number of mutually in-

185

EigenvafueProblems

comparable w(p) f o r elementary circuits p of GA ' This number

-

is bounded by the number of the elementary circuits in the number of maximal linearly ordered subsets in A . lar, if

G

A

and

In particu-

consists in a finite number of mutually disjoint

maximal linearly ordered subsets C

,C g ,. .. ,C

then we can

solve the finite eigenvalue problem by solving the finite eigenvalue problems restricted to V(C ) for k = l,Z,...,s. k

In the remaining part of this chapter we discuss some examples for eigenvalue problems. A s mentioned previously the approach described in this chapter has been developed in view of B-compositions which correspond to linear ( o r , at least partial) order relations. Therefore only such examples will be considered although the eigenvalue problem may be formulated for all the examples given in chapter 8 .

(9.24) P e r i o d i c s c h e d u l e s CUNINGHAME-GREEN [1960] considers the semiring ( I R U (-=),max,+) and proves the existence of a unique eigenvalue ces A over lR

circuit m e a m

.

for matri-

In this example X(A) is the maximum of the

-

w(p) : = W(P)/l.(P) for elementary circuits p in G

A'

An interpretation of the

eigenvalue problem as described by CUNINGHAME-GREEN ([1960], 119791) is the following. Let x ( r ) denote the starting-time j

for machine j in some cyclic process i n the rth cycle and let a

i j

(r) denote the time that this machine j has to work

Linear Algebmic Optimization

186

until machine i can start its r+lst cycle. Then

for all i E N . Hence the fastest possible cyclic process is described by

for r = 1,2,...

.

The finite eigenvalue A is then a constant

time that passes between starting two subsequent cycles, i.e.

In this way the cyclic process moves forward in regular steps, i.e. the cycle time for each machine is constant and minimal with respect to the whole process. The right semimodule Vo(A) contains the finite eigenvectors which define possible starting times for such regular cyclic processes. An example is given in CUNINGHAME-GREEN [1979]. The solution of the equivalent eigenvalue problem in the semiring (IR+,max,.) is developed in VOROBJEV (119631, [1967] and

.

[ 19701)

Methods for the determination of the finite eigenvalue A(A) and further applications have been discussed in DANTZIG, BLATTNER and RAO 119671, and LAWLER 119671.

( 9 . 2 6 ) Maximum c a o a c i t y c i r c u i t s GONDRAN [1977] considers the semiring (IR A

be an n

x

n matrix with a . . = 11

m

U {m],max,min).

Let

for all i E N . Let a . denote 1

Eigenvalue Roblems

187

the maximum capacity of a circuit with node j . Then ( 9 . 2 1 ) Implies that the vector B

j

with components

for i E N is an eigenvector for each A E l R + U ( - 1 {B

j

and that

I j E N ) generates Vo(x). An application to hierarchical

classifications are given in GONDRAN 119771.

(9.27) Connectivity GONDRAN and MINOUX [1977] consider the semiring ({O,l),max,min) with the only possible finite eigenvalue 1. If GA is strongly connected then a

*

i j

= 1 for all i , j E N and the only finite eigen-

vector is x with x i = 1 for i E N . In general, if G into

s

A

decomposes

strongly connected components with vertex sets 11,12,..

..,Is then the eigenvectors with eigenvalue 1 are x 1 ,x2

,...,x s

defined by xi = 1 if j E Ii and xi = 0 otherwise. j

1

More examples are considered by GONDRAN and MINOUX [1977] and can easily be derived from the examples in chapter 8. F o r a discussion of linear independence of vectors over certain

semirings and related topics we refer to GONDRAN and MINOUX 119781 and CUNINGHAME-GREEN [1979]. CUNINGHAME-GREEN [1979] considers further related eigenvalue problems.

10.

Extremal Linear Programs

In this chapter we consider certain linear optimization problems over semirings (R',$,O) with minimum 0 and maximum

m

derived from residuated, lattice-ordered, commutative monoids (R,O,L)

(cf. 5 . 1 5 ) .

AS

in chapter 9 the correspondence is

given by a z b

a @ b = b

es

o r , equivalently, by a

(10.1)

Due to 12.13) tive over

(D.

@

.

b = sup(a,bl

we know that the least upper bound is distribuThe e x t r e m a l linear program is defined by

z:= inf{c

(10.2)

T

8

XI

A 8

2

x

with given coefficient v e c t o r c E R n l m

x

b,

xERn)

n matrix

A

over R and

m-vector b over R. Extremal linear programs are linear algebraic optimization problems (cf. chapter 7 ) .

The adjective

'extremal' is motivated by the linearly ordered case. Then the internal composition a

(B

bE{a,b).

(B

attains only extremal values, i.e.

In general, linear functions of the form c

'I'

8

x = supfc

j

Q

x

j

I

x . > 01 l

are considered. We define a d u a t e x t r e m a l linear program by (10.3) As

t:= sup{y

T

8 bl yT

(D

A

5 cTI

yERm).

we assume that the reader is familiar with the usual linear

programming concepts we may remark that this kind of dualization is obviously motivated by the classical formulation of a 188

189

Extremal Linear Programs

d u a l p r o g r a m . I t is, i n fact, a s p e c i a l c a s e o f t h e a b s t r a c t dual program which HOFFMAN [I3631 considered for partially o r d e r e d s e m i r i n g s . He s h o w e d t h a t t h e w e a k d u a l i t y t h e o r e m o f linear programming can easily be extended t o such programs. On t h e o t h e r h a n d i t i s v e r y d i f f i c u l t t o p r o v e s t r o n g d u a l i t y results. At f i r s t w e w i l l g e n e r a l i z e t h i s w e a k d u a l i t y t h e o r e m t o a r b i t r a r y o r d e r e d s e m i m o d u l e s ( R , ~ , 0 , ~ ; n ; H , * , ~ ) ( c f . c h a p t e r 5). T h e a l g e b r a i c l i n e a r program (10.4)

z

= inf(x

T

OcI A 0 x

2

b, x

2

0, x E R n )

5

c, y

2

e, Y E H ~ I

h a s t h e dual

(10.5)

t = sup{b

T

nyl A

The feasible solutions of

T

oy

(10.4) a n d

(10.5) a r e c a l l e d p r i m a l

and d u a l f e a s i b l e .

( 1 0 . 6 ) Weak d u a l i t y theorem L e t H b e a n o r d e r e d s e m i m o d u l e o v e r R. L e t x b e a p r i m a l feasible solution o f

(10.4) and l e t y b e a d u a l f e a s i b l e s o l u -

t i o n o f (10.5). T h e n bToy




bi

bi

and

x

and

x, >_ bi

for I E M . The

2 bi

0 x

2

,x ) j

(A')

:

I

> bi j -

I

,

. 8

Due to the reduction step ( 1 0 . 1 2 )

and proposition ( 1 0 . 1 3 ) we

may assume in the following w.1.o.g.

that A is a matrix over

{ o , l ] . Further we exclude trivial problems and nonbinding constraints. If bi = 0 then we delete the i-th constraint. Thus w.1.o.g.

let b > 0. Then Ai = ( O , O , . . . , O ) implies that no

feasible solution exists. Therefore we assume in the following

Linear Algebraic Optimization

196

w.1.0.g.

that A contains no zero-rows. If A contains a zero-

= 0 for some j E N then an optimal value of the j corresponding variable is x . = 0. Thus we assume w.1.o.g.

column j or c

1

that A contains no zero-columns and c > 0. Summarizing our reductions we have to consider in the following only problems with 0 - 1 matrix A containing no 0 - c o l u m n or 0 - r o w and with b,c > 0. Such a problem is called r e d u c e d .

(10.14) P r o p o s i t i on Let ( 1 0 . 2 ) be a reduced problem over the bounded linearly ordered set R with minimum 0 and maximum 1 and a

0

b:= min(a,b)

for all a , b E R . Then R(Ai,c)

=

R

for all i E M . proof. Due -

to proposition

R(Ai,c) = {bl 0

5

b

Bi:=

(10.11.3)

we know that for I E M

5 ail with min{cj:aij

ai:= max{a

ij

I

I

cj

j EN)

I

5 Bi ,

j EN}.

F o r a reduced problem this ieads to

Bi

=

maxfc.1 a , . = 1 , I 13

EN}

and therefore to a i

-

Proposition ( 1 0 . 1 4 )

shows that for a reduced problem over a

1.

bounded linearly ordered set

is an optimal solution of (10.2).

We summarize these explicit solutions of ( 1 0 . 2 ) theorem.

in the following

Extremal Linear h g m m

197

(10.15) T h e o r e m (1)

Let (R,@,Z) be a lattice-ordered commutative group. Then the optimal value 0.f (10.2) is

and an optimal solution N

xj:=

2

T

is given by

@ c-l

j

for all j E N . (2)

Let (R,Z) be a Boolean lattice with join (meet) U (n), complement a* of a, and a 0 b = a n b

for all a , b E R .

Then the value

and the solution N

xj:=

2

u

T

is given by

c* j '

for all j E N are optimal for (10.2) provided that

is

primal feasible. Otherwise there exists no feasible solution of ( 1 0 . 2 ) . (3)

Let (R,I) be a bounded linearly ordered set with minimum 0, maximum 1 and a 0 b = min(a,b)

f o r . a l l a , b E R . If (10.2)

is a reduced problem then its optimal value is

z

=

max

iEM

min[bi,min{cjI

and an optimal solution xj:=

N

{

1 2

for all j E N .

i j

=

is given by

< z , j otherwise ,

ifc

a'

111

Linear Algebraic Optimization

198

It should be noted that HOFFMAN 119631 gave the optimal solution in (10.15.1) for the additive group of real numbers ( l R , + , z )and a solution o f (10.2) in Boolean lattices. Theorems

(10.10) and

(10.15) show that such problems can be treated and

solved in a unified way. Further they describe in axiomatic terms a class of optimization problems for which strong duality results hold. On the other hand we have to admit that for general residuated,

lattice-ordered, commutative monoids no solution method is known for (10.2). A s we did not assume that R is conditionally complete it is even unknown whether an optimal value exists. A solution o f (10.16)

z:= inf{cT b

XI

A t9 x 5 b,

x E Rn)

is trivial if we assume O E R , a solution of (10.17)

z:= sup{cT

b

XI

A t9 x

5 b,

x E Rn)

is obviously x:= b:A which is the maximum solution of A b x f b. The solution of (10.16)

z:= sup{c T b

XI

A

(D

x

2

b,

x E Rn)

is trivial if we assume m E R .

If we consider such problems as (10.2) or (10.17) with equality constraints then we can only derive some lower and upper bounds on the optimal value. Let (10.19)

z:= inf{cT

t9

XI

A (D

and assume the existence of z.

x = b,

xERn}

199

Extremal Linear Programs

(10.20) P r o p o s i t i o n Let R be a residuated, lattice-ordered, commutative monoid. If S:= { x E R n I A B x = b) 4 @ then ;:=

b:A is feasible and

a maximum element of S. Proof.

is the greatest solution of A 8 x f b. In particular,

5 x. This implies b N

if A 0 x = b then x

=

A 8 x 5 A

@

N

x. Thus

N

A B x = b .

m Obviously

;yields

an upper bound on z. On the other hand

(10.3) leads to a lower bound. We find

If the greatest lower bound i s distributive over B then these bounds are certain row- and column-infima of matrices with entries bi

8 (bi:a ). j ij As in chapter 9 we have to assume that R is linearly ordered 8

(c.:a. . ) resp. c

I

11

if we want to develop a finite and efficient solution method. Thus in the following R is a residuated linearly ordered commutative monoid. Further we will assume that (10.2) and (10.19) have optimal solutions if a feasible solution exists. In particular, for one-dimensional inequations (equations) this assumption means for all a,b E R : (10.24)

if a

B

x

2

b

(a

0

x = b)

has a feasible solu-

tion x then it has a minimum solution. Such minimum solutions are uniquely determined and will be denoted by b/a

(b//a)

.

and we find b/a = b//a.

If b//a

exists then b/a exists, too,

If b//a

exists for all a,b then R is

200

Linear Akebmic Optimization

a r e s i d u a t e d a n d d u a l l y r e s i d u a t e d d-monoid. In a l i n e a r l y o r d e r e d m o n o i d l i n e a r a l g e b r a i c i n e q u a t i o n s ( e q u a t i o n s ) can b e i n t e r p r e t e d i n t h e f o l l o w i n g way. A B x z b m e a n s t h a t f o r a l l r o w s i E M t h e r e e x i s t s at l e a s t o n e c o l u m n j (i) such t h a t

(10.25)

aij (i)

o x

j(i)

> bi -

.

A B x = b m e a n s t h a t for a l l i E M t h e r e e x i s t s at l e a s t o n e

column j (i) such that (10.26)

aij (i)

B x

j (i)

= bi

and that (10.27)

a

ij

o x

< b i 1 -

f o r a l l i E M , j E N . T h i s i n t e r p r e t a t i o n is e x t e n s i v e l y u s e d in c h a p t e r 9 f o r t h e s o l u t i o n of e i g e n v a l u e p r o b l e m s in c e r t a i n linearly ordered semirings. Let G (GI) denote the set of all (i,j) E M x N s u c h t h a t t h e s e t

ordered pairs

({XI a i j 0 x . = bill i s n o n e m p t y

I

(and a

kj

Q

(XI

a

ij

B

(bi//aij)

x

> bi)

j -

5

bk

f o r a l l k E M ) . T h e n w e m a y r e d u c e t h e s e t of f e a s i b l e s o l u t i o n s t o a set containing only solutions x with (10.28)

x j E {bi/a

ij

I

(1,j) € G I

in t h e c a s e of i n e q u a t i o n s and w i t h (10.28'

x

j

ECbi//aij

I

(i,j) E G ' )

in t h e c a s e o f e q u a t i o n s . T h i s l e a d s t o t h e f o l l o w i n g t h e o r e m s .

20 1

Extremal Linear Pmgnzmr

( 1 0 . 2 9 ) Theorem L e t R b e a r e s i d u a t e d l i n e a r l y o r d e r e d c o m m u t a t i v e monoid s a t i s f y i n g ( 1 0 . 2 4 ) . With r e s p e c t t o t h e e x t r e m a l l i n e a r p r o g r a m (10.21, l e t x ( a ) : = j

max{bi/ai,[

(i,j)EG,

f o r a l l j E N and f o r a l l a E R .

If

a

2

c

(bi/ai,)}

8

j

(10.2) has an optimal s o l u -

t i o n with optimal value z then (1)

x ( z ) is an o p t i m a l s o l u t i o n o f

(2)

z E Z:= ( c , 8 ( b i / a i , )

Proof.

(1)

cT Q x ( a )

5

I

( i , j ) €GI.

a f o r a l l a E R and t h e r e f o r e it s u f f i c e s

t o show t h a t x ( z ) i s f e a s i b l e . let i E M .

fore y

,

-> b i / a i , (2)

( 1 0 . 2 ) and

L e t y b e an o p t i m a l s o l u t i o n and

Then t h e r e e x i s t s j = j ( i ) w i t h a

2

bi/aij

and z

1. c,

which i m p l i e s a

i,

Q y,

2

2

8 x,

c,

bi.

0

i j

2

8 y,

bi.

There-

,

T h u s x (z)

( b i / a i,).

Eence x ( z ) is f e a s i b l e .

F o l l o w s from z E ( c T 8 x ( a ) l a E R 1 .

( 1 0 . 3 0 ) Theorem L e t R b e a r e s i d u a t e d l i n e a r l y o r d e r e d c o m m u t a t i v e monoid s a t i e -

With r e s p e c t t o t h e e x t r e m a l l i n e a r program

fying (10.24).

,

,

( 1 0 . 1 9 ) l e t x ' ( a ) : = max{bi//ai,

I

f o r a l l j E N and f o r a l l a E R .

I f t h e r e e x i s t s an o p t i m a l s o l u -

( i , j ) EG', a

2

c

8 (bi//ai,)}

t i o n with optimal value z then (1)

x ' ( z ) is an o p t i m a l s o l u t i o n o f

(2)

zEZ':=

Proof.

( 1 ) W e g e t cT 0 x ' ( a )

A

Q

x ' ( a ) 5 b.

fie6 A Q x(z)

1

{ c 8 (bi//ai,) j

5

( 1 0 . 1 9 ) and

(i,j) EG').

a a n d f r o m t h e d e f i n i t i o n o f G'

T h e r e f o r e i t s u f f i c e s t o show t h a t x ' ( z ) s a t i s -

2

b.

L e t y b e an o p t i m a l s o l u t i o n and l e t I E M .

202

Linear Algebraic Optimization

Then there exists j = j(i) E N with a . . fore y

> bi//a.,

j -

Further a

17

@

PV

Therefore x l

I

(2)

yv

( 2 )

.

1. b

for all

v

2 bi//aij

.

? . I E

Thus

Follows from z E {cT 8 x(a)I

heo or ems

(10.29)

2

This implies z

13

@

y . = bi and there3

c . @ yj 3

2

cj 8 (bi//ai,)

.

M , V E N shows ( i , j ) E G ' .

A

(D

x'(z)

b.

aER1.

and ( 1 0 . 3 0 ) reduce the extremal linear programs

(10.2) and (10.19) to min(aEZ1

(10.31)

A

x(a)

(D

2 b)

and to (10.32)

min(aEZ'1

A

x'(a) = b].

(D

For the determination of 2

( 2 ' )

minimal solutions of O(nm)

a.

lj

@

B = b i and O(n m

@-compositions.

As

2

2

it is necessary to compute the

inequations a

ij

8

B 2

bi (equations

comparisons) and to perform

and

2'

O(n m )

are linearly ordered an optimal

solution can be determined using a binary search strategy together with the threshold method of EDMONDS and F U L K E R S O N 119701. Here we need O(log

2

(n m ) )

x(a) consists in O(n m ) check consists in O ( n m )

steps in which the determination of

comparisons and in which the feasibility @-compositions and O(n m )

comparisons.

This approach can easily be extended to residuated latticeordered commutative monoids satisfying ( 1 0 . 2 4 ) which are finite products of linearly ordered ones. Thus, in particular, extrema1 linear programs over the lattice-ordered group of real k-vectors (IRk K:=

{1,2,

,+,C),finite

..., k),

Boolean lattices ( 2 K

,fl,z)

for

and of finite products of bounded linearly

E x t r e m l Linear Programs

203

ordered sets (R,min,L) can be solved using a similar reduction. As

for such monoids the resulting sets Z resp. Z' are in gene-

ral not totally ordered it will be necessary to perform enumeration. A complexity bound for the solution of such problems will contain the cardinality of 'the finite product considered (in the above examples: k).

Similar results as given in this chapter for residuated, lattice-ordered, commutative monoids can be given for dually residuated, lattice-ordered, commutative monoids by exchanging the role of sup and inf. These results follow from the given results by replacing (R,@,z) by its lattice-dual ( R , @ J , ~ )We . remark in this context that lattice-ordered commutative groups are residuated and dually residuated.

Many authors have considered extremal linear equations, inequations and extremal linear programs. The solution of linear equations and inequations is closely related to the eigenvalue problem discussed in chapter 9. The solution of such equations and inequations has been considered by VOROBJEV (119631, 119671) in the special case (IR+,.

):,;

in a series of papers

ZIMMERMANN, K. ([1973al, [1973b], [1974a1, 11974131, [1976a] and [1976b])

discussed solution methods for extremal equations, .in-

equations and extremal linear programs in ( I R + , . , z ) .A compact solution of all different possible extremal linear programs in this monoid is given in ZIMMERMANN, U. [1979c] (cf. 1 0 . 3 4 ) . Extremal linear equations and inequations in linearly ordered commutative monoids are discussed by ZIMMERMANN, K. and JUHNKE [ 19791.

204

Linear Algebmic Optimizarion

For lattice-ordered groups and related algebraic structures such equations and inequations are considered in detail in the b o o k o f C U N I N G H A M E - G R E E N [19791. F o r B o o l e a n l a t t i c e s a d i s c u s sion i s given in the book of HAMMER and RUDEANU [1968] and in t h e s u r v e y a r t i c l e o f R U D E A N U [1972]. F o r r e s i d u a t e d l a t t i c e ordered commutative monoids the solution o f extremal linear p r o g r a m s is p r e v i o u s l y d e s c r i b e d i n an e x t e n d e d a b s t r a c t o f Z I M M E R M A N N , U.

[198Oc].

T h e d i s c u s s i o n o f s u c h p r o b l e m s i n (IR+

leads in a natural

way t o a g e n e r a l i z e d c o n v e x i t y c o n c e p t w h i c h e n a b l e s a g e o m e t r i c a l d e s c r i p t i o n . S u c h c o n c e p t s w e r e i n v e s t i g a t e d by Z I M M E R M A N N , K.

(119771, [1979a], [1979b]) and in a more general

c o n t e x t by K O R B U T 1 1 9 6 5 1 , T E S C H K E a n d H E I D E K R t k E R ( [ 1 9 7 6 a ] , [ 1 9 7 6 b ] ) , a n d T E S C H K E a n d Z I M M E R M A N N , K.

( 1 9 7 9 1 . Z I M M E R M A N N , K.

( 1 9 8 0 1 d i s c u s s e s t h e r e l a t i o n s h i p of c e r t a i n s e m i m o d u l e s and j o i n g e o m e t r i e s a s d e f i n e d by J A N T O S C I A K a n d P R E N O W I T Z [1979]. A u n i f i e d t r e a t m e n t of t h e l i n e a r a l g e b r a v i e w p o i n t i s d e v e l o p e d in C U N I N G H A M E - G R E E N

([1976], [1979]) for lattice-ordered

groups and related algebraic structures. Further, CUNINGHAMEG R E E N 119791 d e s c r i b e d s e v e r a l a p p l i c a t i o n s . A s a n e x a m a l e w e consider a machine scheduling problem.

(10.33) Machine Scheduling We assume that a production process i s described as

L a

(9.24

by s t a r t i n g t i m e s x w h i c h l e a d t o f i n i s h i n g t i m e s A 0 x o v e r the monoid

(IR U I - - } , + , : ) .

If we have prescribed finishing

times b then a solution of the system of extremal equations A 8 x = b i s a vector x o f feasible starting times. Assume -L -L

-

-------&aA-

.-.----A..-..

4 -

..arroccmrv + n

ptarf

maphinp

4

Extremal Linear Programs

205

which takes time c

This means that after starting machine j j' the next start of machine j is possible not before time cj + xj. In order to be able to start the process as early as possible a solution of min{c

T

8

XI

A 8 x = b)

is required. If the time constraint is not absolutely binding then situations may occur in which we consider inequations instead of equations. It is also possible that A 8 x = b does not have any feasible solution. Then a reasonable solution is given by minimizing max{bi

-

(A 8

1 E MI

x),l

-

N

subject to A 8 x 5 b. Its solution is x = b:A since we know that A @ x

5 A

for all x with

0

A Q

x

5 b. Here x is given

by N

xj = min{bi-a

for j E N

ij

I

i E M )

(cf. 1 0 . 7 ) .

( 1 0 . 3 4 ) Bounded L i n e a r l y Ordered Commutative Groups In the following we assume that ( R , @ , L ) is a linearly ordered commutative group with I R l > 1. R is residuated and dually residuated; the residual b:a, the dual residual b/a, and the minimum solution b/a

(b//a)

in ( 1 0 . 2 4 )

coincide with b 8 a - l .

We consider several different types of extremal linear programs denoted by

(O,O,A,p)

with

An extremal linear program

O,O,A E

{max,min) and with

(O,O,A,p)

is given by

p€{z,z,=).

Linear Algebraic Uptimizarion

206

(10.35)

o{z(x)

I

x €PI

with objective function defined by Z(X) : = o I c . 7

and with the set P

C -

B X.

1

I

jE

NI

Rn o f feasible solutions defined by

AIa.. B x.1 ] E N ) P b i 1

11

for all i E I f .

For example, ( 1 0 . 1 ) corresponds to (min,max,max,z)

Since (R,@,z) is a linearly ordered group, too, each program has a dual version in the order-theoretic sense which is obtained by exchanging max w min and

5

CI

2 . Thus it suffices to solve

one problem from each of these pairs. For example, the dual version of

10.1)

is (max,min,min,

max

min

m

m

max

rnin

max

0

0

min

0

0

max

min

max

z*

0

(n>l)

min

m

max

z

-

OD

-

z

-

min

z

max

T

-

z

0

(IPI > 1 )

rnin

w

(IPI>l)

Table 1 :

T

min

-

z

O p t i m a l v a l u e of e x t r e m a l l i n e a r programs

-

z

(o,o,A,p).

208

Linear Algebraic Optimization

For each problem of the form f m < n , o , A , z )

no comaonent x

j

(j E N ) of a feasible solution is bounded from below. Clearly,

*

P

0. We admit x

0 as a n optimal solution. Similarly,

-

x = = is considered as an optimal solution of problems of the form ( m a x , o , A , ~ ) . We remind that 0 and

are the adjoined

minimum and maximum.

f?%ax,max,max,g) has the optimal solution x

=

b:A (cf. 10.17),

i.e. =

miniayl I i E M I

=

[maxia

x. 3

lj

ij

I iEn11-'

=

j

(max)

for all j E N . Clearly, the dual of this problem in the ordertheoretic sense is f m i n , r n i n , m i n , ~ ) and has the optimal solution &(min). Similarly, we find that ;(max)

and x(min) are

the optimal solutions of f r n a x , m i n , r n a x , ~ ) and f r n i n , m a x , m i n , ~ ) . For ( m i n , m i n , r n a x , ~ ) the case n = 1 has to be treated separately. Then

a,

=

minib

i

a;:/

@

i E M ) = b:A is the optimal solu-

tion. Otherwise the objective function is not bounded from be ow We admit the optimal solution x defined by x l : =

*

x . : = 0 for all j 7

1.

(n = 1 )

MI

1

-1

max{bi@ail

=

and the optimal solution x with x : =

with x.:= = for all j 3

and by

Then z(x) = 0. F o r the dual version

f m a x , m a x , m i n , ~ ) we find the optimal solution 2 iE

GI ,

*

1

2,

and

1.

f m i n , m a x , r n a x , ~ ) is a special instance of the problem (10.1) which is solved in theorem ( 1 0 . 1 5 . 1 )

for lattice-ordered

groups. Thus, its optimal value is

z

=

max iEM

min ( c . a a-') jEN

1

ij

= z*(min,max),

and an optimal solution is x*(min,max).

Then the dual version

Extremal Linear Rogram

209

f m a x , r n i n , m i n , ~ ) has the optimal value z*(max,min) and the optimal solution x*(max,min). Secondly, we discuss.problems with equality constraints. Then the set P(A) of feasible solutions is defined by AjEN(aij 9 b x 1 = bi j

for all i E M. Proposition (10.20) shows that x(max) i s the maximum feasible solution of P(max) i f and only, if P(max)

9

@.

Similarly, x(min) is the minimum feasible solution of P(min)

* @.In

if and only if P(min) P(A)

*

0. This assumption can easily be checked for a given E P(A).

problem by testing ;(A) A

the following we assume that

maximum (minimum) feasible solution i s optimal for maximiza-

tion (minimization) problems. Therefore, ;(A)

is an optimal

solution for each of the problems fmax,max,max,=) fmax, m i n , mar, =)

that P(A)

*

, ( m i n , m i n , rnin, =) ,

,

and f m i n,max, m i n , =) provided

@.

Next we consider f m i n , m i n , max, =)

*

is optimal. Otherwise, let x

.

If

P(max) I = 1 then x(max)

x(max) be another feasible

so,lution. Since s(max) is the maximum feasible solution there exists some l~ E N with x Then

with

lJ

z j := x j (max)

solution for a'l1 a 5

- . lJ

< x

Thus a

lJ

0

for j * p and with

; (max). lJ

x

< 1 for all i E M .

lJ lJ

: = a i s a feasible

Since a is not bounded from below

in R , the objective function is not bounded from below in I?. N

admit a = 0. Then x i s an optimal solution with optimal value N

z(x) = 0. Similarly, the optimal solution of the dual version

fmaz,max,min,=)

i s x(min),

an optimal solution

with

if IP(min) I = 1 , or there exists j

=

j

(min) for j

*

l~ and with

We

2 10 N

x

P

Linear Algebmic Optimization

=

QD

for some

u

E N .

Obviously, the condition I P ( A )

> 1 can

easily be checked. I f P(max)

@ then the optimal value z o f

4

fmin,max,max,=)

satisfies the inequalities z*(rnin,max) 5 z max min ( c . iEM J E N

-1

a.. ) 11

@

x

(10.301. With

=

5

-1

max rnin ( c . 19 a ) ] E N IEM ij

x(max) we find

otherwise,

J

a E

z

*

z(max), i.e.

~f z*(rnin,max) < i(max) then we apply theorem

(cf. 10.23).

for all

5

5

R. Again, x! ( a ) 1

=

0 expresses the fact that this

component of x' is not bounded from below. Theorem (10.30) implies that an optimal solution can be determined using one of the following threshold methods. W.1.o.g. c

1

e x1

c c2 8 -

X 2 5 ...
b

*

3 cEH:

(4.1) i s r e p l a c e d by a * c = b

f o r a l l a , b E H t h e n a s i m i l a r t h e o r y a s g i v e n in t h i s c h a p t e r f o r (11.1) c a n b e g i v e n f o r (11.3) s i m p l y b y r e v e r s i n g t h e o r d e r r e l a t i o n in H. A s o l u t i o n of

(11.3) in a s e m i m o d u l e a s

c o n s i d e r e d i n ' t h i s c h a p t e r can b e found u s i n g a r e d u c t i o n d i s c u s s e d at t h e end o f t h i s chapter. I n fact, i t w i l l t u r n o u t t h a t i n a c e r t a i n s e n s e (11.3) i s a s i m p l e r p r o b l e m t h a n (11.1). W e r e m a r k t h a t for t h e c l a s s i c a l c a s e s p a c e (IR

,+,z) o v e r

, 1.e. f o r t h e vector-

t h e field I R , a r e c e n t result of

HAEI J A N

119791 s h o w s t h a t (11.1) can b e solved by a method w t h

214

Linear Algebraic Optinuzation

polynomial complexity bound.

(ll.l),

Such a result is not known for

in general. We will not try to generalize the

ellipsoid algorithm described in HAEIJAN [1979] which i s based on results of SHOR ([19701, [19771). We prefer to develop a duality theory which leads to a generalization of the simplex method for (11.1) and which is applicable to combinatorial optimization problems (cf. subsequent chapters).

The first results on dual linear problems due to HOFFMAN [1963] and BURKARD [1975a] have been mentioned in chapter 10 as special cases of the weak duality theorem (10.6) which holds in ordered semimodules.

If H is a linearly ordered commutative group then we assume w.1.o.g.

that H is a module over R (via embedding as described

in proposition 5.11.3).

Then a reasonable dualization of (11.1)

is (11.4)

N

g:= sup

CTOS

sED

with

This is the same dualization as used in the weak duality theorem (10.6). Unfortunately such a dualization fails even for simple examples if H is not a module. Then C T o s is not defined if negative entries occur in C (or c ) .

Such difficulties will

appear in the discussion of equality constraints even if all entries in the original constraints are positive. In order to find a proper dualization it is helpful to split

215

Algebraic Linear Pmgram

m a t r i c e s a n d v e c t o r s i n t o " p o s i t i v e " a n d " n e g a t i v e " parts. F o r a E R let a + : =

-a .

a _ : = a+

a if a

2

For matrices

0 a n d a+ = 0 o t h e r w i s e .

(vectors) A over R we define A+

in the same manner componentwise.

and

A-

NOW

m we call S € H + d u a l f e a s i b l e if

5

c T+ O s

(11.6) If ( H , * , z ) As

Then

a*cTOs

.

i s a m o d u l e t h e n (11.6) is e q u i v a l e n t t o C T D s

5

a.

u s u a l a s o l u t i o n x E P i s c a l l e d p r i m a l f e a s i b l e and a p a i r

( x i s ) o f a f e a s i b l e p r i m a l and a f e a s i b l e d u a l s o l u t i o n i s called a f e a s i b l e p a i r . Now let a(x;s) := :=

(c-xlT

s

(C+X)TOS

r. F u r t h e r w e d e f i n e r e d u c e d c o s t c o e f f i c i e n t s

with

a f :=

e i f e q u a l i t y h o l d s in t h e j-th r o w o f (11.6). C o l l e c -

ting all equations after composition with x (11.7) As

a E 8:

(11.8)

a(x;s)

*

xToa = x

T

o a

*

j

yields

~(x;s).

w e f i n d t h e weak d u a l i t y t h e o r e m a(x;s)

*

x

T

D a

5

B(x;s).

We remark that if C does not contain negative entries then C- = 0 and t h e n a ( x ; s ) = e w h i c h i m p l i e s t h e u a u a l w e a k d u a l i t y

-

-

t h e o r e m , a s x T 0 a > ( C X ) ~ O S> c T 0 s .

I f H is a m o d u l e t h e n (11.7) i s e q u i v a l e n t t o

Linear Algebraic Optimization

216

which yields x T

0

-

a > cT

0

s,and xT

0

a = cT

0

s if and only if

In the general case, we consider the ordinal decomposition of the d-monoid H , i.e.

the family ( H A ; h E A )

as described in

chapter 4 . Such a decomposition leads to a decomposition of the semimodule (cf. chapter 6 ) . Then the i n d e x condition

plays an important role. If a(x;s) = e then ( 1 1 . 1 1 ) fied as X(e) = X o = min A . fied as A = { A o }

.

In a group ( 1 1 . 1 1 )

The importance o f ( 1 1 . 1 1 )

is satis-

is always satiscan be seen in the

following optimality criterion.

(11.12) T h e o r e m

(Dual o p t i m a l i t y c r i t e r i o n )

Let H be a weakly cancellative d-monoid which is a linearly ordered semimodule over R+. Let s € :H a(y):=

i(n(x))

dual feasible and let

( ~ - y ) ~ ~ ~s(,y ) : = ( ~ + y ) ~ n for s all ~ E R ; . If X E P ,

5

and

then x is an optimal solution o f ( 1 1 . 1 )

and

217

Algebmic Linear Rogmms

Proof. L e t Y E P . T h e n

*

a(x)

x

T

o a = B(x)

*

x

T

n a = B(x)

c -

~ ( y )* y T o a = a ( y )

< -

a(x)

*

T

*

yToa

yTna.

5

D u e t o t h e i n d e x c o n d i t i o n (11.11) and A(:) X(y

1. B(y)

T min()r(x O a ) ,

a)), cancellation o f a(x) yields optimality.

follows from

(11.12.3)

and

(11.12.4)

(11.7).

T h e o r e m (11.12) i s a g e n e r a l i z a t i o n of t h e u s u a l c o m p l e m e n t a rity criterion.

(11.12.1)

-

(11.12.3) c o r r e s p o n d t o (11.10) and

t h e i n d e x c o n d i t i o n (11.11) is a d d e d t o a s s u r e t h a t n o " d o m i n a t i n g " e l e m e n t s o c c u r in t h e i n e q u a l i t i e s .

( 1 1 . 1 3 ) Corol 1 a r y Let A be a d-monoid which i s a linearly ordered semimodule over

R,.

L e t C,c h a v e o n l y e n t r i e s i n R,.

(1)

cTos =

C T O S

*

I f x E P and

(Cx-c)Tns

*

xToa

t h e n x is a n o p t i m a l s o l u t i o n and (21

c T 0 s = xToa.

(11.14) C o r o l l a r y L e t H b e a l i n e a r l y o r d e r e d m o d u l e o v e r R. I f x E P and (1)

(cx-c)Tos = xToi = e

t h e n x is a n o p t i m a l s o l u t i o n and (2

1

C T o s = xTOa.

218

Linear Algebraic Optimization

Special cases of theorem ( 1 1 . 1 2 )

and its corollaries are used

in solution methods for combinatorial optimization problems in subsequent chapters.

A

sequence of pairs (x;s) is construc-

ted in such a manner that each pair fulfills some of the primal feasibility conditions, some of the "weak complementarity" conditions ( 1 1 . 1 2 . 1 )

-

(11.12.3)

and the index condition ( 1 1 . 1 1 )

In each step a new dual feasible solution is determined. Therefore it is possible to define reduced cost coefficients

:

(sometimes called "transformed costs"). Then a "transformation" a

+

a

is considered with respect to the respective class of

pairs.

A

sequence of such pairs is determined in order to find

a pair fulfilling all these conditions. The construction of such a sequence is rather involved for combinatorial and linear integer problems. This reflects the complicated combinatorial (integer) structure of the underlying primal problems. On the other hand optimality criteria, i.e. feasibility and "weak complementarity" are comparably easy to check for the problems considered. The most difficult condition is the index condition (11.11).

This difficulty is due to the occurence of the unknown N

optimal value z in ( 1 1 . 1 1 ) .

N

Therefore lower bounds on z play

an important role in these primal-dual or dual methods.

In as much as we can interpret "transformation" methods by certain duality principles, it is possible to use theorem (11.12)

without an explicit knowledge of duality. Let T: H n + H n

denote a "transformation" called admissible with respect to (11.1)

if there exist functions a: P

such that

x

Hn

+

H,

B: P

x

Hn

+

H

219

Algebraic Linear Programs

(11.15)

a(y,a)

*

yToa = yToi

Then x E P is optimal if (11.12.1)

-

~(y,a)

v

YEP.

(11.12.3) and the index

condition in theorem (11.12) are satisfied by a ( y ) : = a(y,a) and B(y):=

B(y,a) for all Y E P . In the same way corollaries

(11.13) and (11.14) can be viewed in terns of a transformation

concept. Such an approach was necessary since in the early discussion of algebraic optimization problems a duality theory as described in this chapter was not known. Nevertheless, the transformations were always based on the solution of certain inequalities which could be derived from the primal linear descriptions of the respective problem. Till now it has not been possible to develop transformations for problems the linear description of whiCh is not known. In an extended semimodule the "weak complementarity" conditions in theorem (11.12) can be simplified. If the index condition (11.11) is fulfilled then (11.7) is equivalent to (11.16)

* xToa = xTni ~ T o s

*

( ~ x - c ) ~ o* sc+T o s

(11.16) follows from the cancellation rule (6.17.7).

.

This is

quite similar to the case that H is a module over R. The diffeT

rence lies only in the splitting of the dual objective c 0 s into positive and negative parts. In order to obtain the equation (11.16) for all x E P we assume the s t r o n g i n d e x c o n d i t i o n (11.17)

max(;X(s

k

)I

k = 1,2

,...,m )

5 A(;).

Clearly, the strong index condition implies x(Q(Y;s)) for all Y E P . We call

s

5 a(;) m E H+ s t r o n g l y d u a l f e a s i b l e i f s is

220

Linear Algebraic Optimization

dual feasible and satisfies the strong index condition. The N

set o f all strongly dual feasible vectors is denoted by D. Then the equation (11 -16) holds for all x E P and s E E.

(11.18) Theorem

(Dual o p t i m a l i t y c r i t e r i o n )

Let H be a weakly cancellative d-monoid which is an extended 4. .

semimodule over R+. Let s E D. If x E P and (1)

CTOS = CTOS

*

(Cx-c)Tos

*

xTna

then x is an optimal solution of (11.1) and (2)

cT0s = xTOa

*

.:as.

Proof. In the same way as in the proof of theorem (11.12) we find cTos

*

xT o a

5 cT0s

*

yToa

for all y E P. Cancellation of cT 0 s is possible due to the strong index condition (11.17). (11.18.2) is a direct consequence of equation (11.18.1) and equation (11.16).

Similar remarks as for theorem (11.12) hold for theorem (11.18). The usual complementarity condition (11.14.1) in corollary (11.14) is replaced by the weak complementarity condition (11.18.1). The complementarity condition is only a sufficient

condition, but the weak complementarity condition is necessary and sufficient for the existence of a pair (xis) fulfilling a "duality theorem" (11.18.2). Till now we have described the use of certain duality principles

for optimality criteria. In order to define a dual program with respect to (11.1) we have to introduce a dual objective. If

22 1

Algebraic Linear Programs

c

*

$

0 then cT 0 s is not defined. Therefore let f: D

-B

H be

defined by (11.19)

*

.To.

f(s):=

for all s EE with cIf 0 s

1. :C

C T O S 0

s or A

*

Otherwise let f(s):= e . As c T O S

(CT 0 S ) =

xT O a

2

A(cT

0

=

s)

Ao.

c T O s for all x E P

and s E 5 , we find N

f:=

(11.20)

sup f(s) 5 i n f x T n a s€8 XEP

. N

Equality in (11.20) can occur only if there are s E D with A(f ( s ) ) =

x:= A (;I.

Therefore it seems to be useful to restrict

consideration to values of the objective function which are elements of Hy. This yields another approach. It is easy to s e e is the optimal value of the index b o t t l e n e c k p r o b l e m

that (11.21)

N

A = inf ~

XEP

(

n a) x

.~

Its denotation is motivated by i ( x T o a ) = max({A(a

J

) I xJ

>

01 u

and was introduced in ZIMMERMANN, U. ([1976],

{ A ~ I )

[1979a]) for the

solution of algebraic matroid intersection problems (cf. chapter 13). A determination of

N

is possdble via a combination of the

threshold method of EDMONDS and FULKERSON [19701 and usual linear programs. In fact, (11.21) is a linear bottleneck program as considered by FRIEZE 119751 and GARFINKEL and RAO 119761 and

-

thus is a particular example of an algebraic linear program. If A is known then the algebraic linear program (11.1) reduced to an equivalent problem in a group. Let M:=

< y}. -

can be

Ijl

A(aj)

For a vector u E H m and X E A let u ( x ) denote the vector with

222

Linear Algebraic Optimization

-

(11.22)

if h(ui)

1,

U(ui:=

,

otherwise

..., m.

for i = 1 , 2 ,

We consider the r e d u c e d a l g e b r a i c l i n e a r

program N

t:= inf y YEPM

(11.23)

"'1

T

N

oa()OM

CMy 2 c}. This is an optimization problem

with P M'. = { y E R +

over the module G3; over R (cf. chapter 6 ) . Thus its dual is N

g:= sup

(11.24)

cTos

SED

with D:= { s c ( G Y ) ; ~ C

T M 0 s

5 a(h)M}. Weak duality shows g 5 t. N

N

N

The relationship of these different algebraic linear programs is given in the following theorem.

(11.25)

Theorem

Let H be a weakly cancellative d-monoid which i s an extended semimodule over R,. (1)

(11.1)

Let P

and ( 1 1 . 2 3 )

$

(a.

N

Hence h exists. N

optimal solution for ( 1 1 . 1 ) for ( 1 1 . 2 3 1 -

N

are equivalent. z = t. If x E P is an then x

is an optimal solution

M

If y i s an optimal solution for ( 1 1 . 2 3 ) then

(y,O) is an optimal solution for ( 1 1 . 1 ) . If the weak complementarity condition ( 1 1 . 1 8 . 1 )

(2)

fied for some x E P , W

s

E

'i; then

N

f = g = cTns(T) = x T o a =

Proof.

(1)

X ( a ,)

I

= h.

N

2.

If y c P M then x defined by xj:= y . for j E M and x 3

otherwise is primal feasible. Therefore N

i s satis-

Thus yT

0

a

(Y) =

x

T

x

j

= O

> 0 for some j with N

0

j

a which leads to t

2

-

z.

If x E P

223

Algebraic Linear hgmm N

then X (xT 0 a) = X implies

is an optimal solution of ( 1 1 . 1 )

xJ

= 0 for all j d M . Thus x

Let x E P. If x NT

N

x E P with h(x Therefore

j

E P

N

M

and ~ , o a ( h ) =~ x T D a .

> 0 for some j d M then xT n a > N

m a ) = A.

7 5 y.

M

Otherwise x M E PM and

X

Together with ( 1 ) we find

an optimal solution of ( 1 1 . 2 3 )

=

zT

0

T

a for some N

M

n a ( h ) M = ~ Tm a .

y.

I f y E PM is

then x as defined in the proof

of ( 1 ) is an optimal solution of ( 1 1 . 1 1 . ( 2 ) Due to theorem ( 1 1 . 1 8 )

f(s)

=

xT

0

N

a = z. Then

s(y)

we know c:Os E D and c

T

0

=

s(7)

xTUa =

*

c T o s . Thus

complete the

proof.

w

Naturally we cannot prove an existence theorem for optimal pairs or show the necessity of the assumptions in theorem ( 1 1 . 1 8 ) , as a duality gap can occur in combinatorial and integer problems even in the classical case. On the other hand, we will now develop a primal-dual simplex method in order to determine such a pair provided that R is a subfield of the field of real numbers. Then

€3

is an extended semivectorspace over R + .

For the development of a solution method we consider the algebraic linear program ( 11 . 2 6 )

N

z:= inf x T n a XEP

with P:= { x E R y I Ax = bl with m

X

n matrix A of rank m over R

and b E R y . Every linear algebraic program of the form ( 1 1 . 1 ) can be transformed to such a linear algebraic program. We have chosen this form for convenience in the description of basic solutions. Another convenient assumption 1s the non-

114

Linear Algebmic Optimization

negativity of the cost coefficients a . I

, i.e. a > e for all

j EJ.

j-

This allows an obvious starting solution. Self-negative cost coefficients will play a role in the solution of ( 1 1 . 2 6 ) only N

if h ( z ) = X o ;

then it suffices to solve a reduced problem

similar to (11.23) in the vectorspace

over the field R.

G xO

Such a problem can be solved by the usual primal-dual

(or other

variants of the) simplex method which can easily be seen to be valid in vectorspaces. In the following method a sequence of strongly feasible vectors u,v E :H Ax

is determined corresponding to the inequality constraints

2 b and

(11.27)

-Ax A

2 -b. From T

+

Ou

*

(11.6) we derive

ATOv

5 a

*

T A-OU

*

T

A+Ov

and the strong index condition (11.17) reads h(ui) ,h(vi)

...,m}.

for all i E I:= { l r 2 ,

5

A(;)

n

Reduced cost coefficients a E H + are

n defined accordingly. A corresponding sequence of vectors x E R + will satisfy x T o a = e. The method terminates when the current vector x is primal feasible. Then the complementarity condition (11.10) is fulfilled which implies (11.18.1)

and hence x is

optimal, Partitions of vectors and matrices are denoted in the usual way. In particular A

j

is the j-th column of A . E denotes the m x m unit

matrix and ( 0 1 ) is a row vector with n zero-entries followed by m one-entries.

(11.28) P r i m a l d u a l s o l u t i o n m e t h o d Step

1

Let u . = vi = e for all i E I, x = -j

Step

2

Let K:= {j E J I

a I.

0

for all j E J .

= el; determine an optimal basis B

225

Algebraic Linear Programs

of the usual linear program

I

min{Zy.

E:=

A K x K + y = b; x

let n E R m be defined by T ~ : =

Step

3

If

Step

4

Let K : =

N

if

Step

= 0

E

z

then

{j

ElB -1

(0 1)[A

stop

EJ\KI

= @ then

T

j

j

0);

. >

01;

a).

(P =

stop T

2

(x is optimal).

n A

Let d:= min { (l/n A

5

> 0; y

K -

) 0

let ui:= ui

*

(n+)iOd

vi:= vi

*

(n-Ii 0 d

aj I

j

€;I;

for all i E I and go to step 2. Provided that (11.27) is always fulfilled we see that all steps of the method are well-defined in the underlying algebraic structure. In particular in the definition of n in step 2 and the definition of d in step 5 we observe the necessity of the assumption that ( R , + , - ) sitions

"*"

and

"0"

is a field. If we replace the compo-

by the usual addition and multiplication

of real numbers then the above method is the usual primal-dual one for linear programming. For the minimization of b o t t l e n e c k

objectives z(x):= maxja I x j

j

> 03

with real cost coefficients a E I R , j

(x E P) j E J , the method reduces

to a method proposed by FRIEZE [1975] and GARFINKEL and

RAO

19761. (11.28) is closely related to the primal-dual method in FRIEZE [1978]. Both methods reduce to the same methods for the

226

Linear AIgebmic Optimization

usual linear program and the bottleneck p r o g r a m u

z:=

min max{a.I x . > 0). 1 1 XEP

The main difference lies in the introduction of a further vector w E Hm in the constraints replacing ( 1 1 . 2 7 ) .

This leads to

a more complicated optimality criterion and a more complicated method in FRIEZE 1 1 9 7 8 1 .

In the following we give a proof of the validity and finiteness of ( 1 1 . 2 8 ) .

Due to the assumption a . > e for all j E J the 1 -

objective function of the ALP is bounded from below by e. Therefore there are only two possible terminations of the method. In the same way as for usual linear programming we see N

that K =

0 in step

4

means that the current value

f

> 0 is the

optimal value of the linear program minizy. I Ax + y = b; x

(11.29)

0, y

2

0) ;

this implies P = 0. Finiteness will follow from the fact that in step

2

a sequence of bases B for ( 1 1 . 2 9 )

It suffices to show that the value

f

is considered.

- * 0.

strictly decreases in an

iteration. For an iteration we always assume K tioned after ( 1 1 . 2 8 )

As men-

we have at first to show that ( 1 1 . 2 7 )

holds after an iteration. We denote the redefined vectors in step

2 by

i,;.

(11.30) P r o p o s i t i o n m If u , v E H + satisfy ( 1 1 . 2 7 ) ATn;

+

then ; , ; E H T

A ~ O ;

-c ~-

~

and

o 3

*

(cf. 11.22). Further W E

(11.39)

0

ow 5

T

A . 0

7

-

w = a(i)

j

G X ) m+ is called

t/

j E J

i - d u a t f e a s i b l e if

a(i)

I f H is a module over R then x-complementarity and i-dual feasibility coincide with dual feasibility (cf. 11.5) and complementarity (cf. 11.10).

235

Algebraic Linear Program

Let D ( X ) denote the set of all i-dual feasible vectors. Then yields bT

composition of (11.39) with N

w E D(h) and therefore with A:=

0

w

5

-T x 0 a for all

X ( y ) we find

This is another dual program f o r A L P . F o r a !-complementary -T vector w we find bT O w = x O a . This concept leads to the

following optimality criterion.

(11.41) Theorem

(Prima

optimality criterion)

Let H be a weakly cance lative d-monoid which is an extended

x E P and x and 1-dual

semimodule over R+. Let

!:=

mentary with respect to

optimal solution o f (11.26) and N

h = b

(1)

T

-T

O W = X

7

implies optimality of

oa). If w is i-comple-

feasible, then

x

is an

-

O a = z .

Proof. Due to the assumptions we find for all y E P. If 1 =

-T

X(x

GT 0 a

= b

T

0

w

5 yT

0

a (1)

then y T O a ( X ) = y T o a for all Y E P . This

x.

Otherwise let Y E P denote an optimal

-T -T solution. Then y O a ( 1 ) = e < x O a yields a contradiction. Now

=

follows f r o m (11.40).

Theorem (11.41) shows that a special type of complementarity and dual feasibility implies optimality without an additional

136

Linear Algebraic Optimization

index condition. Thus ( 1 1 . 3 8 )

can replace ( 1 1 . 1 8 )

if complemen-

tarity is used as optimality criterion. Furthermore it can be seen that if (11.41.1)

is an optimal solution, w 1-dual feasible and

holds then w is 1-complementary. Otherwise if

optimal, w is i-complementary and ( 1 1 . 4 1 . 1 )

x

is

holds then in gene-

ral it is not guaranteed that w is h-dual feasible. In this sense ( 1 1 . 4 1 )

describes only a sufficient optimality criterion

even if a result on strong duality holds. A comparison of ( 1 1 . 4 0 )

with the other duals ( 1 1 . 2 0 )

shows that if the assumptions o f ( 1 1 . 4 1 )

and ( 1 1 . 2 4 )

hold then all optimal

objective values are equal. In this case all three approaches are equivalent. If X-dual feasibility is used as optimality criterion and w

is 1-complementary then it suffices to check (11.42)

-

for x

j

= 0 and A(a.1 1

To use theorem

1. 1. Then

(11.41)

(11.42)

is equivalent to ( 1 1 . 3 9 ) .

in a primal method we have to consider

conditions for the existence of 1-complementary solutions. In primal methods basic solutions play an important r o l e . If B is a feasible basis for P then x (11.43)

B

=

AilbER:.

If we assume

det A - 1 E R B

for all feasible bases of P then the existence of a X-complementary solution with respect to a feasible basic solution

x

is

guaranteed. The existence of an optimal basic solution for bounded, nonempty P follows from theorem ( 1 1 . 3 4 )

in the case

that R is a subfield of I R , i.e. provided that H is a weakly

231

Algebmic Linear Program

cancellative d-monoid which is an extended semivectorspace over R+. Clearly, (11.43) is valid in this case and we can N

assume that P is nondegenerate. Let P denote the non-

w.1.0.g.

empty set of all feasible basic solutions. In the following we develop a primal simplex method in such semivectorspaces.

Let G E v with basis bor of

;if

neighbors of -T x oa

IB

n iI

i . Then

i.

Then

= m -1.

xEF

x E F with

basis B is called a neigh-

Let N ( i ) denote the set of all

is called l o c a t l y o p t i m a l if

T

5 x m a for all XEN(;).

(11.44) Proposition Let H be a weakly cancellative d-monoid which is an extended N

semivectorspace over R+. Then G E P is an optimal solution of the algebraic linear program ( 1 1 . 2 6 )

if and only if

x

is locally

optimal. Proof. We only show the nontrivial if-part. Let

G

be locally

n d

optimal and let x E P denote an optimal solution. Then we know from linear programming that

with uxEIR+. A s (R,+,.)

is a subfield of (lR,+,-) we can show

p x E R + for all xEN(x).

Therefore

NT x oa

with

u:= 1

pX.

*

u

(iTna)

2

( i + u )0 ( i T o a )

Cancellation due to ( 6 . 1 7 . 7 )

yields NT x oa

1. x-T m a . a

From the proof we see that proposition (11.44) may be invalid if the semimodule is not extended. In particular this may happen

Linear A lgebmic Optimization

238

for the linear bottleneck program. Proposition

(11.44)

again

is attained at a

shows that the optimal value of ( 1 1 . 2 6 )

feasible basic solution. We can state a necessary and sufficient optimality conditi.on and another strong duality theorem.

(11.45) Theorem

( L i n e a r programming d u a l i t y theorem)

Let H be a weakly cancellative d-monoid which is an extended -

N

semivectorspace over R+. Let x E P with basis

and

x:=

h(XOa).

Then the following statements hold:

x

(1)

is an optimal solution of ( 1 1 . 2 6 )

w:=

0

a(X),

if and only if

is x-dual feasible:

(2)

there exists an optimal basic solution:

(3)

the optimal values of the considered dual programs ( f l . 2 0 1 , (11.24)

and ( 1 1 . 4 0 )

Proof. Due to ( 1 1 . 4 1 )

coincide with

T.

it suffices to show for ( 1 ) that

x

opti-

mal implies that w is 1-dual feasible. If w is not X-dual feasible then there exists a E Hi

,

ci

B = g\{r}

s

6

> e. Then bT

U

{ s }

with A(a 0

w = xT

0

)

5

x

T

0

w = a

(x) * a

with

a(1) t (xs 0 a ) for x E 'i; with basis

(Existence o f such a basic solution x follows

from linear programming and ( 1 1 . 4 3 ) ) . x

and A

Since b

T

-T

o w = x

m a and

-T T > 0 we find x m a > x O a ( X ) which implies the contradiction

xTOa

> xTOa.

(11.25)

(2)

follows from ( 1 1 . 4 4 )

and ( 3 ) follows from

if the equality constraints of ( 1 1 . 2 6 )

are replaced by

inequality constraints in the usual manner.

Similar to the classical case theorem ( 1 1 . 4 5 ) (11.44)

and proposition

imply the validity of the following primal simplex method

Algcbmic Linear Programs

239

for algebraic programming problems. We give an outline of the method, showing the differences to the classical case.

(11.46) P r i m a l s i m p l e x m e t h o d

€y

Step 1

Find

Step 2

Calculate w:= if A

Step 3

T 1

w

5

Choose s d

6

1 T ( -~ i oa(h)E

5

x

a).

;

a . (1) for all j d 1 with h (as)

-T h(x

h:=

E;

with basis

i

and

with h(a.) 1

Af 0

5

w > as(:);

determine xEN(;)

with basis B containing s ;

redefine

1

x,

and

If the method stops in step

then stop.

and return to step 2 .

2

then w is h-dual feasible and

theorem (11.41) yields optimality. Finiteness of (11.46) can be seen from the proof o f (11.4s). A basis exchange step due to step tion. A s

3

decreases strictly the value of the objective funcis a finite set this implies finiteness. Primal

degeneracy can be handled with some of the usual finiteness rules (cf. BLUM and O E T T L I 119751 or B L A N D [1977]).

If H is a weakly cancellative d-monoid which is an extended semimodule over R + then the primal simplex method can be applied provided that (11.43) is satisfied. In particular, (11.43) holds if the constraint matrix A is t o t a l l y u n i m o d u -

l a r , i.e. or

+1.

A

contains only submatrices o f determinant 0

240

Linear Algebraic Optimization

The primal simplex method is applied to algebraic transportation problems in chapter 12. The optimality criterion (11.42) can in this special case be improved such that it suffices to check a less number of feasibility conditions.

We have developed two solution methods f o r algebraic linear programs which simultaneously solve the corresponding dual programs (11.20), (11.24) and (11.40). These duals have objective function and constraints with values in the semimodule H. Therefore, in general, such problems seem to have a structure differing from algebraic linear programs. We will now consider the solution of such problems. The dual (11.20) of ( 1 1 . 2 6 ) has an objective function f: H Y b

if b

T

ov

5

b

T

T

*

Ov

ou

:H

f(u,v):= b A(b

01

X

T

T

+

H defined by

Ou

o v ) = X(b

T

Ou) =

Xo

and

f (u,v):= e

otherwise. The set of all strongly dual feasible solutions consists in all (u,v) EX:

x

5

HY with

(11.46) (11.47)

f o r all i E I. If H is a vectorspace over R then

is equivalent to

(11.48)

sue WED

with 6:= { W E Hml

A

T

b

T

O w

ow

5

a}. This form l o o k s much more familiar

Algebmic Linear Fmgram

24 1

and does not contain an explicit index condition with respect to the optimal value of the original primal problem. Such a problem can be solved by the application of the generalized simplex methods to ( 1 1 . 2 6 ) . the dual of ( 1 1 . 4 8 ) .

Here we may interpret ( 1 1 . 2 6 )

as

In the general case of a semivectorspace

difficulties arise from the fact that the external composition is only defined on R +

(otherwise

a nontrivial extension to R

implies that H is a modul over R

and that variables in H are

more o r less positive ( A ( a ) > A.

-D

a

-1

does not exist). There-

fore, splitting of variables and matrices leads to constraints of the form ( 1 1 . 4 6 ) .

If the strong index condition ( 1 1 . 4 7 )

be assumed without changing the problem then ( 1 1 . 2 6 ) be interpreted as a suitable dual. If ( 1 1 . 4 7 )

I s

can

again may

not added to

the constraints ( 1 1 . 4 6 ) and is not implicitly implied by these constraints then even weak duality may fall f o r the pair of programs ( 1 1 . 2 6 )

and

SUP f (u.v) (u,v) E D '

(11.48)

with D':={(u,v) E H ~ x+ H A ~ TIO A solution of

(11.48)

u

*

T A-OV

5 A T- O U

*

ATOV

*

a).

is not known in general.

Based on the max-matroid flow min-cocircuit theorem for algebraic flows in HAMACHER [1980a] we consider an example for a problem of the form ( 1 1 . 4 8 ) .

For fundamental definitions and

properties o f matroids we refer to chapter 1 . Let M denote a regular matroid defined on the finite set

I = {0,1,2,

...,m}.

SC resp. SB denotes the set of all circuits

resp. cocircuits. Let A denote the o r i e n t e d i n c i d e n c e matrix of elements and cocircuits, 1.e.

242

Linear Algebraic Optimization

i E B+

1

i E B-

i 6 B m for all i E I and all B E SB. For a given capacity-vector c E H+ the elements of D:= {(t,u)I

T

A+

t o( ) = U

T

t

A-• (u), u z c ,

m u E H + , tEH+j

are called a L g e b r a i c m a t r o i d fZolJs in HAMACHER [1980a].

The

m a s - m a t r o i d flo1.7 p r o b l e m is (11.49)

N

t:=

max (t,u) E D

t .

Now we consider its formal dualization (11.50)

N

z : = min XEP

x

T

O c

with the set of feasible solutions

HAMACHER [1980a] develops an algorithm which determines an optimal solution

for a l l i = 1 , 2 ,

(7,;)

...,m.

of (11.49) with the property

Furthermore the algorithm yields a

constructive proof for the existence of a cocircuit

with O E Z -

such that (11.52)

N

t = (alg,a2;,...,am;)+

T

oc

and N

t

5 (a1B.a2B,...,amB)+

for all B E S B w i t h . O E €3-.

T

o c

This implies directly the max-matroid

f l o w - m i n cocircuit theorem in HAMACHER [1980a].

Furthermore it

Algebraic Linear Programs

243

leads to the following strong duality result for (11.49) and (11.50).

(11.53) Theorem

(Matroid f l o w d u a l i t y theorem)

Let H be a weakly cancellative d-monoid which is an extended semimodule over R+. Then there exists an optimal pair of a N

-

N

feasible primal solution (t,u) E D and a feasible dual solution

-

NT

x E P such that t = x

oc.

Proof. At first we show weak duality, i.e. t (t,u) E D , x E P . Clearly t

5 xT O c for all

5 t for all optimal solutions (t,u) N

-

[-:I.

N

of (11.49). W.1.o.g.

we assume that (11.51) holds. x E P implies

the existence of y E R

SB

with Ay

1.

Using distributivity

and monotonicity in the semimodule we find

(11.51) shows that we can cancel the first terms on each side (cf. 6.17.7) which are equal due to the equality constraints N

in the definition of D. Now u semimodule imply

75

Secondly cocircuits

xT

0;

G E SB

5 c and monotonicity in the

5 xT

0

c.

with 0 E Z- correspond to certain

elements of P defined by

x := i

0

N

otherwise.

-

N

Indeed, yB:= - 1 if B = B and y := 0 otherwise, shows feasibility of

F.

B

-T Clearly x o c = (al;,a2;,...,am;)+

implies

T

=

T o c . Therefore (11.52)

YToc. rn

244

Linear Algebraic Optimization

In fact the proof shows that (11.50) has always a zero-one solution which is the incidence vector of the positively directed part of a cocircuit. As in particular we may consider the usual vectorspace

( l R , + , z )over IR

a direct conse-

quence is that all vertices of the polyhedron

are in one-to-one correspondence with the set of all cocircuits B E S B with O E B - for a given regular matroid M.

A detailed study of the matroid flow problem is given in XAMACHER [1980b].

Related problems are discussed in BURKARD

and HAMACHER 119791 and HAMACXER [198Oc].

At the beginning of this section we gave some remarks on problem (11.3) which i s also not equivalent to an algebraic linear program of the form (11.11, in general. (11.3) may be solved via the solution of an index problem of the form

a : = sup X(xTna)

N

(1 1 . 5 4 )

XEP similar to (11.21). Its objective is A(x

If

T

n a ) = max({a(a,)l

-

xJ

.

> 01 u ( ~ ~ 1 )

i s known then ( 1 . 3 ) can be reduced to a problem over the

module

GY. A

Let M:= {j

A(aj)

1. A } .

Then we consider the reduced

problem (11.55)

- 1

q

with PM:= { y E R

I

CMy

2 c}. This is an optimization problem

245

Algebmic Linear h a r a m

over the module

GT

which is equivalent to the algebraic

linear program inf YEPM

y

T

ob

for all j E M. The relationship between ( 1 1 . 3 )

with b := j

and ( 1 1 . 5 5 ) is described by the following theorem.

(11.56) T h e o r e m Let H be a weakly cancellative d-monoid which is an extended semimodule over R+. Let P

* @.Hence x N

N

and ( 1 1 . 5 5 ) are equivalent (q = q').

exists. Then ( 1 1 . 3 )

If x E P is an optimal solu-

tion for ( 1 1 . 3 ) then xM is an optimal solution for ( 1 1 . 5 5 1 .

If

y E P M is an optimal solution of ( 1 1 . 5 5 ) then (y,O) is an opti-

-

mal solution of ( 1 1 . 3 ) .

Thus g: P P M with NT is a bijective mapping. Let x E P with A(x m a ) = A .

Proof. If x E P then x g(x):- x

j

= 0 for all j b M .

N

#u

M

NT T NT Then x T O a < x O a and x ~ O ~ ( =A e) 1 then R' i s

is a linearly ordered vectorspace. If 1R'I

an extended vectorspace over Q; this will be assumed w.1.0.g. in the following. The finite eigenvalue problem consists in the determination of h E R', y E (R')n such that max{a

(11.63)

ij

@ y

for all i E N : = {l,2,

j

I

...,n]

j = 1,2,

...,nI

h

=

yi

with respect to a given n x n matrix

(a ) over R' u ( 0 ) (0 denotes an adjoined zero). The soluij tions of such a problem are characterized in theorem ( 9 . 2 2 ) and

A =

are discussed subsequent to this theorem.

-

GA

denotes the di-

graph with set of vertices N and arc set E defined by (i,j)E E

:

a

i j

* O

-

for i,j E N . If a finite eigenvalue A E R ' exists then

G

A

contains

an elementary circuit p with length l ( p ) and weight w(p) such that

1

=

(l/l(p)) nw(p). Then further

We assume that

G

A

contains at least one circuit. Otherwise the

finite eigenvalue problem has no solution.

Let a denote the vector with components a i j

,

(i,j)E E .

We

consider the algebraic linear program (11.64)

d .,

w:= max XEP

xTna

with set P of feasible solutions described by the linear constraints

(11.65)

25 1

Algebraic Linenr Proflams E

a n d x E Q + . I n m a t r i x f o r m w e d e n o t e t h e s e e q u a t i o n s by C x = c. T h e n t h e d u a l o f (11.64) i s N

(11.66)

min sED

A:=

cTus

w i t h D = { s E (R')n+ll C C

T

0

s

2

T

0 s

2

a).

The inequality constraints

a have the form

(11.67)

si

-1 sj

@

8 s

n+l

> -

a

ij

for a l l ( i , j ) € E . D i s a l w a y s n o n e m p t y a s it c o n t a i n s s d e f i n e d

by si:= e f o r I E N a n d s,+~:=

maxis

ij

I (i,j) E E )

(e d e n o t e s

t h e n e u t r a l e l e m e n t o f R'). If

1 is

a f i n i t e e i g e n v a l u e and y E ( R t I n a f i n i t e e i g e n v e c t o r

is a dual feasible solution with then s = ( Y , X ) ~ cTos =

1=

A(A)

.

Let p denote an elementary circuit in G

A

with

x

=

(1/1 ( p )) O w ( p )

k h e n x E Qy d e f i n e d by

{

=

x.

lj

1/1 (PI

if (irj) E P ,

0

otherwise

,

is a primal feasible solution with x

T

o a = (l/l(p)) nw(p).

-

D u e t o t h e w e a k d u a l i t y t h e o r e m x T O a < cT u s f o r a l l f e a s i b l e pairs

N

N

thus A = w = A(AI =

(x;s);

1.

If no finite eigenvalue exists then the existence of a circuit in GA i m p l i e s

y 2

(11.34) s h o w s

=

M

N

h(A).

2

Further the strong duality theorem

A(A).

In any case A = w can be determined using the methods proposed in this chapter for the solution of linear algebraic programs

.

Linear Akebraic Optimization

252

(cf.

11.28 and 11.46).

As

(11.64) i s a maximization problem

t h e s e methods which a r e f o r m u l a t e d f o r m i n i m i z a t i o n p r o b l e m s h a v e t o be m o d i f i e d i n t h e u s u a l manner ( e . g . r e p l a c e a by ij -1 w a i j f o r 1 1 . 4 6 , e t c . ) . The o p t i m a l v a l u e A i s u s e d i n c h a p t e r 9 e i t h e r t o compute t h e f i n i t e e i g e n v e c t o r s ( i f

1

exists) or to

v e r i f y t h a t t h e f i n i t e e i g e n v a l u e p r o b l e m h a s no s o l u t i o n .

For t h e s p e c i a l c a s e o f t h e a d d i t i v e g r o u p of r e a l numbers t h i s a p p r o a c h is p r o p o s e d by D A N T Z I G , B L A T T N E R a n d R A O 1 1 9 6 7 1 and i s d i s c u s s e d i n CUNINGHAME-GREEN [ 1 9 7 9 ] .

12.

Algebraic Flow Problems

In this chapter we discuss the minimization of linear algebraic functions over the set of feasible flows in a network (algebraic flow problem). Although we assume that the reader is familiar with basic concepts from flow theory we begin with an introduction of the combinatorial structure which is completely the same as in the classical case. Then we develop the primal dual solution method which is a generalization of the min-cost max-flow algorithm in F O R D and F U L K E R S O N 1 1 9 6 2 1 . We discuss three methods for the minimization of linear algebraic functions over the set of feasible flows in a transportation network (algebraic transportation problem). These methods are generalizations of the Hungarian method, the shortest augmenting path method and the stepping stone method.

The underlying d-monoid H is weakly cancellative and an extended semimodule over the positive cone R + of a subring of the linearly ordered field of real numbers (cf. chapter 6). Only in the discussion of the Hungarian method for algebraic transportation this assumption can be weakened. Then H is a d-monoid which is a linearly ordered semimodule over R+.

We begin with an introduction of the combinatorial structure. G = (N,E)

denotes a simple, connected digraph with two special

vertices u and terminal vertex

called 80urce and sink. G contains no arc with

T U

or initial vertex T. W.1.0.g.

G is a n t i 8 y m m e t r i c I i.e. 253

we assume that

Linear Algebraic Optimization

254

B denotes the incidence matrix of vertices and arcs of G , i.e.

{ -: 1

Bih:=

if h

=

if h =

, (j,i) ,

(i,j)

otherwise

E f o r all i E N and f o r all h E E. Then x E R + is called a G if 0

5 B x

f l o w in

= -B x and

Bix = O Then g:= B U x is called the fzow v a l u e of x.

for all iEN\{o,r}.

Let c E ( R + \ {O})E be a given c a p a c i t y v e c t o r . Then a flow x is called f e a s i b t e if x

Cut if o E (i,j) E ( I

S

5 c.

A

partition (S,T) o f N is called a

and T E T . Then the sum of all capacities c . with 11

x J)

n E is called the c a p a c i t y Of t h e c u t

capacity is denoted by c(S,T).

(S,T). The

It is easy to establish the

relation (12.2)

9 =

1

(SXT)n~

xij -

I: (TXS) n E

< c(S,T)

x i’

-

f o r all feasible flows and f o r all cuts in G. A well-known

result of flow theory (cf. FORD and FULKERSON [I96211 is the

max-flow m i n - c u t f

theorem which states that the maximum v a l u e

of a feasible flow is equal to the m i n i m u m c a p a c i t y of a

cut in G.

A

feasible flow of maximum value is called a m a x i -

mum f l o w ; similarly a minimum c u t is defined. In ( 1 1 . 4 9 ) we introduced algebraic flows in regular matroids. A

generalization of the max-flow min-cut theorem for algebraic

flows in regular matroids is developed in HAMACHER

([1980a],

A

[1980c]).

A detailed study of such flows i s given in HAMACHER

1980bl.

There are many procedures for the determination of a flow with m a x i m u m f l o w value. B e s i d e s t h e f l o w a u g m e n t i n g p a t h m e t h o d i n F O R D and FULKERSON [I9621 such methods have been investigated by D I N I C [1970], E D M O N D S a n d K A R P [ 1 9 7 2 1 , Z A D E H [ 1 9 7 2 1 , K A R Z A N O V ([1974],

[19751), EERKASSKIJ 119771, MALHOTRA, PRADMODB KUMAR

a n d M A H E S H W A R I [ 1 9 7 8 ] , G A L I L 1 1 9 7 8 1 , S H I L O A C H 119781 and G A L I L a n d N A A M A D [1979]. E V E N [ 1 9 8 0 ] h a s w r i t t e n a n e x p o s i t i o n o f t h e m e t h o d o f K A R Z A N O V w h i c h h a s a c o m p l e x i t y b o u n d o f O ( NI

3

1;

n u m e r i c a l i n v e s t i g a t i o n s a r e d i s c u s s e d by H A M A C H E R 119791

With respect to a feasible flow x we introduce the increm n t a l d i g r a p h AG = ( N , E U E ) w i t h city vector ACER;'~

i : ={(j,i)I

(i,j) E E ]

and w i t h c a p a -

defined by

A C i s w e l l - d e f i n e d a s R is a s u b r i n g o f IR.

A

f e a s i b l e f l o w Ax

in AG i s c a l l e d a n i n c r e m e n t a z f l o w . T h e n x e A x r d e f i n e d by

for a l l ( i r j ) E E , i s a f e a s i b l e f l o w i n G. F o r c o n v e n i e n c e w e assume that an incremental flow has the property Ax

ij

for all ( i r j ) E E

> O

u i.

*

AX

ji

= O

C o n v e r s e l y t o (12.41, i f

is a feasible

Linear AIgebmic Optimization

256 N

flow i n G with g i n AG s u c h t h a t

2

g t h e n t h e r e e x i s t s a n i n c r e m e n t a l f l o w Ax Ax = x .

X *

t a l f l o w i n Ax t h e n g

+

I f Ag i s t h e v a l u e o f t h e i n c r e m e n -

Ag =

y.

T h u s x i s a maximum f l o w i f a n d

o n l y i f t h e v a l u e o f a maximum i n c r e m e n t a l f l o w i s 0 . L e t p d e n o t e a p a t h i n G from

t o T o r a c i r c u i t i n G.

0

Then

p ( 6 ) with

(12.5)

for a l l of

P . . ( 6 ) := 11

( i , j ) E E and 0

1. 6

(i,j) Ep

,

otherwise

,

E R is a flow i n G.

The c a p a c i t y 6 ( p )

p i s d e f i n e d by

5

Then p ( 6 ) i s a f e a s i b l e f l o w i f 0

6

5

6 ( p ) . Such a f e a s i b l e

f l o w i s c a l l e d a p a t h f l o w o r a c i r c u i t flow.

The f l o w v a l u e

o f a p a t h f l o w i s 6 a n d t h e f l o w v a l u e o f a c i r c u i t f l o w is 0. E a c h f l o w x h a s a r e p r e s e n t a t i o n a s t h e sum o f c e r t a i n p a t h flows and c i r c u i t f l o w s

(cf.

r x =

(12.6)

BERGE [ 1 9 7 3 ] ) , i . e . S

P=l

Pp(6 1 +

z

p=r+1

pp(Gp)

w h e r e t h e f i r s t s u m c o n t a i n s a l l p a t h f l o w s a n d t h e s e c o n d sum c o n t a i n s all c i r c u i t f l o w s .

S i m i l a r l y e a c h i n c r e m e n t a l f l o w Ax

i n AG i s t h e sum o f i n c r e m e n t a l p a t h flot)s a n d i n c r e m e n t a l

c i r c u i t flows r

(12.7)

The s e t P

cl

S

of a l l f e a s i b l e flows i n G with c e r t a i n fixed flow

257

Algebraic Flow Roblems

value g E R (0 5 g

5

f) can be characterized by linear con-

straints. Let b € R Y be defined by b all i L { c r I ? } .

0

=

-b

?

= g and bi = 0 for

Then

(12.8) For cost coefficients a

ij

( i , j ) E E we consider the

EH,

alge-

b r a i c f l o w problem (12.9)

N

z:=

inf XEP

xTma. 9

H is a weakly cancellative d-monoid which is an extended semi-

module over R+.

A

solution x o f

(12.9)

is called a minimum C o s t

f l o w . W . 1 . o . g . we assume that g = f, i.e. we solve a max-flow min-cost problem. The classical case is obtained from ( 1 2 . 9 ) f o r the special vectorspace

( J R , + 1 5over ) the real field. A

combinatorial generalization of the classical case is discussed in B U R K A R D and H A M A C H E R [ 1 9 7 9 ] .

They develop finite procedures

for the determination of a maximum matroid flow (cf. theorem 11.53) with minimum cost in IR. It is well-known that B i s a totally unimodular matrix (e.g. LAWLER [ 1 9 7 6 1 ) . Thus ( 1 1 . 4 3 ) plex method ( 1 1 . 4 6 )

is satisfied and the primal sim-

can be applied for a solution of the

algebraic flow problem ( 1 2 . 9 ) .

If the semimodule considered

is not extended then the primal simplex method may terminate

in a locally optimal solution. Later on in this chapter we will discuss the application of the prfmal simplex method to the special case o f algebraic transportation problems (cf. 12.33);

for classical transportation problems this method is

known to be the most efficient solution procedure.

258

Linear Akebmic Optimization

In the following we develop a primal dual method for the solution of the algebraic flow problem which generalizes the min-cost max-flow algorithm in FORD and FULKERSON 119621. T h i s method has previously been considered in BURKARD 119761,

BURKARD, HAMACHER and ZIMMERMANN, U. [1977b]).

([1976I, [1977a1 and

Based on Involved combinatorial arguments BURKARD,

HAMACHER and ZIMMERMANN, U .

[1976] proved the validity of

this method. Now we can provide a much simpler proof based on the duality theory in chapter 11. h'e

assume that a

ij

E H + for

all ( i , j ) E E . This assumption allows an obvious initial solution and guarantees that the objective function is bounded from below by e. Self-negative cost coefficients ( a , .< e) 11

will play a role in the solution of (12.9) only if (12.10)

a(z) N

=

xo .

(12.10) holds if and only if there exists a maximum flow (with value f) in the partial graph G' = (N,E') with

This can easily be checked using one o f the usual maximum flow algorithms. w

If x ( z ) = h o then it suffices to solve the algebraic flow problem in the graph G'. This is a problem in the module

over R

GA 0

and can be solved similarly to the classical min-cost max-flow problem. For example, the necessary modification in the min-cost max-flow method in FORD and FULKERSON 119621 mainly consists in replacing usual additions by *-compositions and usual compari-

.

sons by 5 - c o m p a r i s o n s in G AO

Akebraic Flow Pmblem

Otherwise A(;)

> A.

259

and without changing the objective

function value of any maximum flow we can replace a by ai,:=

i j

< e

e.

In order to apply the duality theory of chapter 1 1 to the algebraic flow problem we assign dual variables u,v and t to the constraints Bx to ( 1 1 . 6 )

2

and ( 1 1 . 1 7 )

b, -Bx

2

2

-b and -x

-c. According

(u,v,t) is strongly dual feasib1.e if

for all (i,j) E E and i f

for all i E N . The complementarity condition ( 1 1 . 1 4 . 1 )

has the

form (12.13)

( c - x ) T ~ t= e = x T o a

with reduced cost coefficients (12.14)

As

ui

-

* v, * aij

=

ai,

a

defined by

* u, * vi * tij .

in the discussion of the modification ( 1 1 . 2 8 ' ) and ( 1 1 . 2 7 ' )

of the general primal dual method we consider only the case that all dual variables are elements of A

u

U (el for a certain

index p E A (cf. chapter 6 for the decomposition of H). Then in ( 1 2 . 1 3 ) and ( 1 2 . 1 4 ) is replaced by a (11). ij i, For flow problems it suffices to consider only dual variables

a

v v

j'

j E N with va = e. A pair (x;v) of a feasible flow x and

E (a,, u {ellN with va

if

- ( z ) is

= e and p c h

called p-compatibte

260

f o r all

Linear Algebmic Optimization

(i,j) E E .

If (x;v) i s U-compatible then w e can define

=

a strongly dual feasible solution (u,v,t) by u

1

(12.16)

f o r all

if vj >aij ( p ) * v i and v

E

j

e and

=:E

*a

ij

(i,j)E E. All dual variables are elements of H

P

(11) * v i

u

,

{el

and (x;u,v,t) satisfies the complementary condition ( 1 2 . 1 3 ) with reduced cost coefficients defined by (12.17)

ui*v

j

* aij

= a

i j

(11) * u j * v i

* tij

for all ( i , j ) E E . (12.13) i s a sufficient condition for (11.18.1). Therefore theorem ( 1 1 . 1 8 ) applied to

shows that x is an optimal solution of ( 1 2 . 9 ) if x i s a feasible solution with flow value g .

(12.18) P r o p o s i t i o n Let (x;v) be v-compatible and let g denote the flow value of x. Then x is a minimum cost flow of value g and ( 11

Proof:

qnvT = x

T

0

a(ll)

*

cTtYt.

Due to our remarks it i s obvious that x is a minimum

cost flow. From theorem (11.18) we get (' - b )T+ n v = x T o a The definition of b in ( 1 2 . 8 )

*

( - b )T- n v

*

cT n t .

and va = e lead to ( 1 ) .

Algebraic Flow h b l e m s

261

The method proceeds In the following manner. The initial compatible pair is (x;v) with x

=

0 and v

=

.e

(?J =

Ao).

Then we

try to increase the value of x without violation of compatibility. If this is impossible then we try to increase the dual variables v without violation of compatibility. In this way the current flow i s of minimum cost throughout the performance of the method. We will show that the procedure leads to a flow x of maximum value in a finite number of steps. Then due to proposition

(12.18)

x is an optimal solution of ( 1 2 . 9 )

with

g = f. At first we try to change x using an incremental flow Ax such that ( x , x o Ax) is p-compatible, too. An arc (i,j E E is called

a d m i s s i b l e if (12.19)

an arc (i,j ) E

E

is called admissible if ( j ,I) E E is admissible.

Let 6 G = ( N , 6 E ) denote the partial graph of admissible arcs. Then feasible flow 6x in

AG

containing all

is called the a d m i s s i b i l i t y g r a p h . A

6G

6G I s

called an a d m i s s i b l e i n c r e m e n t a t flow.

Clearly the admissibility graph depends on the current flow x and the current variables v.

(12.20) P r o p o s i t i o n Let ( x ; ~ )be p-compatible, let g denote’the flow value of x and let 6 x be an admissible incremental flow of value 69. Then (1)

(x. 6x;v)

(21

(X

is p-compatible,

6xlT oa(p) = xT oa(p)

*

( 6 9 0vT).

262

Linear Algebraic Optimization

Proof:

( 1 ) Since 6x is an incremental flow x o 6x is a feasible

flow of value g + 69. Admissibility of 6x shows that x is changed only in admissible arcs. Thus (x o 6x;v) is v-compatible. ( 2 ) Proposition (12.18)

leads to

These equations in the module G

over R imply (2).

v

m

Proposition ( 1 2 . 2 0 ) shows that augmentation of x with an admissible incremental flow 6x does not violate compatibility. 6x can be determined by a maximum flow procedure. In particular, this is a purely combinatorial step with complexity bound O(lNI

3

)

if we use the method of KARZANOV [ 1 9 7 4 1 .

Secondly we discuss the case that the value of a maximum admissible incremental flow is 0. Let (S,T) be a minimum cut of 6G. Then (12.21)

Ac

ij

for all (i,j) E (S

x

= 0

T) fl 6E. (12.21) follows from the max-flow

min-cut theorem. If Acij = 0 for all (i,j) E (S

x

T)

n

(E U

i)

then

( S I T )is a minimum cut in AG of value 0 o r , equivalently, (S,T)

is a minimum cut in G of value f. Then x is a maximum flow in G and the procedure terminates. Otherwise we know that each

flow x in G of flow value y > g has either (12.22)

N

x

rs

for Some (r,s) E F : =

> x

rs

{ ( i , j )I

x

ij

C

c

ij

n

(S x T)

or

263

Algebraic Flow h b l e m s M

x

(12.23)

f o r some

< x

rs

{ ( i , j )I

( r , s )E B:= v

rs

o
0;

T:= N \ S ; B:=

6G

( U E S

rs rs

for

I

i j

< c

il

1 n

(SXT);

(TXS);

(exit 2).

(r,s) E F U B from ( 1 2 . 2 4 ) ;

( r , s )E F U B } ;

)i(a).

R e d e f i n e a l l d u a l v a r i a b l e s by a n d go t o s t e p 1 .

v

j

:=

if j ES,

vJ * a

i f j E T,

2 64

Linear Algebm'c Optimization

Due to our previous remarks it is clear that all steps are well-defined. In particular P = (12.25).

*

If F U B

0 in step

0

in the first call of

2 then

IS1 increases after a

revision of the dual variables in step 4 at least by one. Thus the dual revision procedure terminates after at most IN1

-

1 iterations either at exit 1

( 6 G contains an incremen-

tal flow of nonzero value) or at exit 2

(x is a maximum flow

in G ) . Each iteration contains at most O ( l N I tions, O(lNI 2 )

*-compositions and O ( l N I 2 )

2

usual opera-

5 - comparisons.

(12.26) P r o p o s i t i o n Let (x;v) be u-compatible and let g denote the value of x

< f). If the revision procedure leads to revised dual vari-

(g

ables denoted by (x;;)

(1)

with revised index

i then

is ;-compatible.

Proof: It suffices to prove ( 1 ) for one iteration in the dual revision procedure. The definition ( 1 2 . 2 4 ) U-compatibility implies h ( Q )

-

u 5 )i(z). Otherwise N

a =

.

a

rs

2

P in step 3 .

of a

rs

If p =

together with

i then

( u ) > e for some (r,s) E F. Thus

Since a , . = t for (i,j) E B we find B = 0. Therefore rs 11 ij * N each flow x in G with flow value q > q has x > x for some ij ij (i,j) E F. Thus It remains to prove the inequalities 5 A(;).

a = a

(12.15)

u

for (x;;).

(T x S ) ]

n

It suffices to consider arcs (i,j) E [ ( S x T )

E. If (i,j) E F U B then a

5

a

ij

implies the

corresponding inequality. If (i,j) E (S x TI X F then x and

G 3.

> a.

j -

5 a

ij

11

*vi.

*ii .

ij

=

c

ij

If (i,j) E ( T x S ) X B then x . = 0 and ij w

265

Algebraic Flow h b l e m s

Proposition ( 1 2 . 2 6 ) shows that a dual revision step does not violate compatibility provided that g

C

f. In the last step

it may happen that before actually finding a minimum cut LJ is raised beyond A(:).

Therefore if a minimum cut is detected

then an optimal solution of the possible dual programs (cf. chapter 1 1 ) has to be derived from the values of the dual variables before entering the dual revision procedure for the last time. Optimality of x is guaranteed in any case. By alternately solving a max-flow problem in the admissibility graph and revising dual variables we construct a minimum cost flow x E P f . We summarize these steps in the following algorithm.

( 1 2 . 2 7 ) Primal dual m e t h o d f o r a l g e b r a i c f l o w s x

Step 2

Determine a maximum flow 6x of value 6g in 6 G ;

R

0; v

e; z:= e; p : = h o .

Step 1

x:= x o 6x; z : = z Step 3

*

-

(69 o v T ) ; v:= v.

Revise dual variables by means of ( 1 2 . 2 5 ) ;

-

if ( 1 2 . 2 5 ) terminates at exit 2 then v:= v stop; otherwise go to step 2 .

At termination x is an optimal solution of ( 1 2 . 9 ) for g = f and v leads to an optimal solution (u,v,t) of the possible dual programs (cf. chapter 1 1 ) .

z is the optimal value; the

recursion for z in step 2 follows from proposition ( 1 2 . 2 0 ) . We remark that explicit knowledge of the ordinal decomposition (cf. chapter 6) of the semimodule is not necessary. The recursion for the index p is deleted and ars(p) in step 4

Lineor Algebraic Optimization

266

of the dual revision procedure is determined by

a

(v):= rs

1 ,"

if v T * a r s > a if v

T

*a

In step 5 the dual variables v.(u) for i E

rs S

if a * v i >

This follows from proposition

rs

= v

'

.

can be found by a

,

(6.17). Admissibility can be

found similarly. It remains to show that the primal dual method ( 1 2 . 2 7 ) is finite.

( 1 2 . 2 8 ) Theorem Let H be a weakly cancellative d-monoid which is an extended semimodule over R+. Then the primal dual method

(12.27) is

finite. Proof: The performance of step 1, step 2 and step 3 is finite. Therefore it suffices to prove that step 3 in ( 1 2 . 2 7 ) is carried out only'a finite number of times. At termination of step 2 the admissibility graph with respect to (x;v) does not contain any path p from a to

T

of capacity 8(p) > 0. If the current

flow x is not a maximum flow then at termination of step 3 the admissibility graph with respect to (x;v) does contain such a path p .

Then v

-

< v

.

With respect to x the arcs of p are

partitioned in F U B defined by

267

Algebraic Flow Problems

The number of partitions is finite as well as the number of different paths from U to T in

G.

Thus it suffices to prove

that p did not appear in a previous iteration of the primal dual method with the same partition of its arcs. Admissibility and

U

=

(12.29)

i

where

(v)

(i,j) E B

e lead to

;*

s=v

denotes the composition of all weights a

ij

(i) with

((i,j) E F).

Now assume that p previously appeared in an admissibility graph N

N

with respect to (x;v), with capacity z(p) > 0 and with the same partition of its arcs. Similarly to ( 1 2 . 2 9 ) N

v

(12.30)

*T=Y

with respect to weights a

> e implies

Now

we get

7

N

ij

( P I . Further

> e and from

T 5

1. v T


0

*

(i,j) admissible

for all i E I, j E J . Let (S,T) be a cut determined in step 1 and step 2 of the dual revision procedure ( 1 2 . 2 5 ) .

then

S x

Let

T = L U K . Further Q = (T x S ) f l E satisfies Q:=

( T n I)

x

(Sfl J)

(cf. Figure 7 ) .

Figure 7

The cut (SIT) in a transportation network

Linear Algebraic Optimization

274

Due to the construction of (S,T) and ( 1 2 . 4 1 ) x (12.42)

This

oi

= c

i

x I. T

= cj

x

= o

i j

we find

for all (0.i) E K

,

for all ( j , ~ E) K

,

for all ( i ,j ) E L U Q.

leads to a simple finiteness proof.

(12.43)

Proposition

Let H be a d-monoid which is a linearly ordered semimodule over R+. Then the Hungarian method terminates after finitely many

steps with a maximum flow x. Proof: Due to ( 1 2 . 4 2 )

the value g of the current flow x satis-

f ies (12.44)

g =

1 c. TflI

+

z

snJ

c

j

in step 3 of the Hungarian method. There are only finitely many flow values of such a form.

As

the flow value is strictly in-

creased in an iteration of the Hungarian method after finitely many iterations a maximum flow x is constructed and termination occurs.

For a discussion of optimality of the final flow x a combina-

torial property of maximum flows in transportation networks is o f particular interest. A maximum flow

X

saturates all arcs

(U,i), i E I and all arcs ( j , ~ )j, E J. Therefore ( 1 2 . 4 4 ) g = 1 K (12.44)

ij

.

Then ( 1 2 . 2 )

r L x ij

=

applied to f-g

+

leads to

I:

Q

-

x

ij

'

implies

A4pebraic Flow Problem

275

Another important combinatorial property is

C (i,j)E

EI

o

n (T x

c xij3

#.

SI =

Therefore in the dual revision procedure ( 1 2 . 2 5 )

the determina-

tion of a can be simplified. We get (12.45)

a = min{aijI

Due to ( 1 2 . 4 0 )

v,

*aij:=

aij(v) * v i , ( i , j ) E L I .

reduced cost coefficients

a ij

may be defined as

solutions of (12.46)

vj

* aij:=

a

ij ( u ) * v i

for all (i,j) E I X J . Nevertheless these reduced cost coefficients are not uniquely determined by ( 1 2 . 4 6 ) . then we define

-

a

ij

a ij

=

e a s usual. Otherwise v

j

If v . = a 7

< aij

* vi

and

In general d-monoids,reduced cost coeffi-

cients are chosen as proposed in BURKARD [ 1 9 7 8 a ] ; a

( v ) *vi

is uniquely determined if H is weakly cancellative (cf.

proposition 4 . 1 9 . 3 ) .

-

(y)

i. j

initially

a. Then after determination of a from a = min{a.,

(12.47)

13

I

(i,j) E

LJ

in the dual revision procedure the reduced cost coefficients are redefined by if (i,j) E Q (12.48)

I

if (i,j) E L and a

* E : = a- ij

otherwise. In particular

E:=

-

e if a = aij. It is easy to see that ( 1 2 . 4 6 )

is always satisfied. Even ( 1 2 . 4 8 ) all reduced cost coefficients. determined.

'

If we add

dual method ( 1 2 . 2 7 )

a

=

does not uniquely determine

In any case p = A ( a ) is uniquely

a in the initial step 1 of the primal

then the following modification of the

276

Linear A(gebroic Optimization

dual revision procedure is well-defined.

(12.49)

Modified dual r e v i s i o n procedure

Step 1

Determine the connected component S ( u E S C_ N ) of the partial graph of AG which contains all arcs with A c , . > 0; if 11

Step 2

T:=

N \ S ., L:=

if L Step 3

Step 4

E S then return (exit 1). ( S n I ) x ( T n J ) ; Q:=

(TflI) x (SflJ);

then return (exit 2 ) .

=

a:= min{a. . / 11

p:=

T

(i,j) E L I ;

A(al.

Redefine all dual variables by

j

v .:= 1

Vj(U)

1 v 3.

i a

if j E s , if j E T ,

all reduced cost coefficients by

(12.40)

and go to

step 3 .

(12.49)

specifies ( 1 2 . 2 5 )

for algebraic transportation problems

in such a way that we can give a proof of the optimality of the final U-compatible pair constructed by the following Hungarian method.

(12.50)

H u n a a r i a n method w i t h d u a l v a r i a b l e s

Step 1

x

Step 2

Determine a maximum flow 6x of value 6 9 in 6G; x:=

0; v

x

0

3

6x;

e; z : = e ;

z:= z

*

u

(6g

= A

0;

-

a:= a -

v T ) ; v:= v .

277

Akebraic Flow Roblems

Step 3

Revise dual variables and reduced cost coefficients by means of ( 1 2 . 4 9 ) ; if ( 1 2 . 4 9 )

terminates at exit 2 then v:=

stop;

otherwise go to step 2 . The final p-compatible pair (x;v) is shown to be optimal in the proof of the following strong duality theorem.

( 1 2 . 5 1 ) Theorem

( T r a n s p o r t a t i o n d u a l i t y t h e o r e m 11)

Let H be a d-monoid which is a linearly ordered semimodule mn over R+. For a cost coefficient vector a E H + there exists a feasible optimal pair (y;w,s) for the algebraic transportation (y;w,s) i s complementary and satisfies

problem.

cTow J

J

=

*

yT0a

cTos I I '

Further z defined recursively in step 2 of the Hungarian method (12.50)

is the optimal value of the algebraic tr.ansportation

problem. Proof: We introduce an auxiliary variable B in the Hungarian method. Initially B : = (12.52)

B

B:=

*

e. Then add

[ (f - g)

0

a1

in step 4 of the modified dual revision procedure ( 1 2 . 4 9 ) . g denotes the flow value of the current flow x. We remark that A(B)

= p.

(12.53)

We will prove that ~

*

T

oy a = y T n a

is satisfied for all Y E P Hungarian method ( 1 2 . 5 0 ) .

T

throughout the performance of the In the initial step ( 1 2 . 5 3 )

i s ob-

278

Linear AIgebmic Optimization

-

viously satisfied since a = a. It suffices to consider an iteration in the modified dual revision procedure. Let ( 1 2 . 5 3 )

-

be satisfied before step 3 and denote the revised variables N

-

N

by v , a , p and

7.

yT o

(12.54)

i

Let y E P T . = yT

Q

o

iQ *

Then

5

A

T y L o aL

*

yH

(2)

T

0

5

(yT

0

a) leads to

-

aH

Due to ( 1 2 . 4 4 ) we get

(12.54) and ( 1 2 . 4 8 ) this leads to

Together with (12.55)

Then

7

=

yT0a = y B *[(f-g)

“f-g)

Ll:

o a ]

proves

0

a1

.

(12.53) with respect to the rede-

fined variables. Let (y;w,s) denote the final pair derived from the final compatible pair (x;v) of the modified primal dual -T

method. Then y E P T and y

-

-T y o a = ~ ( 1 2.56)

for all Y E P

< T’

Hence

O a = e. Therefore

-T * oy a =

ii

i * y T o a = yTna

y

is optimal.

The equation ( 1 ) is derived from the equations w

j

*a

i j - aij

*

‘i

by collecting all these equations after composition with

i.

The recursion for the optimal value is derived f r o m ( 1 2 . 5 6 )

-

shows B = y- T O a (another recursion!).

which

NOW let (a , g u ) be the

u

constructed sequence of variables a in step 3 of (12.49) and corresponding current flow values 9 ; then 5 satisfies (cf. 12.52)

219

Akebraic Flow Problem

*

B =

p=1,2, ...fr

Let 69p : = g p - g p - l for p = 2.3

[(f-gp) o aU 1

,...,r

.

and let v :

,

p = 1,2

,...,r

denote the corresponding sequence of values of the dual variable v

. Then

v1 = e and T

vp = a for p = 2,3

,...,r. B =

*a2

* ... * a P-1

Thus

*...,

~=2,3,

69

r

1!

mvy = z .

If we are not interested in the optimal dual variables then we may delete dual variables from (12.50). In this case it is necessary to identify admissible arcs without explicit use of dual variables. Due to the remarks on admissibility before (12.39) we may consider all arcs ( O , i ) , i E I and (j

f

~

)

jEJ

as admissible throughout the performance of the Hungarian method without changing the generated sequence of compatible pairs. Now ( i f j )E I

x

J is admissible (cf. 12.19 and 12.46) if and only

if (12.57)

vj

* aij

vj

=

-

Then ( j , i ) is admissible, too. If H is weakly cancellative then this is equivalent to (12.58)

8

*iij

=

a ij

=

e as well as to

B

where B is recursively defined by (12.52). BURKARD [1978a] uses (12.58) for a definition of admissibility. In general d-monoids H it is possible that these definitions lead to differing admissibility graphs. Let (1,j ) € I

x

J be B-admissible

if (12.58)

280

Lineor Algebmic Optimization

is satisfied. Then (j,i) is called 5 - a d m i s s i b Z e , too. Let 6G(B)

denote the partial subgraph of AG containing all arcs

from ( E U E ) \ ( ( I

J ) u ( J x 1)) and all 5-admissible arcs. This

x

leads to the following variant of the Hungarian method (12.50).

(12.59)

Hungarian method w i t h o u t dual variables

Step 1

x

Step 2

Determine a maximum flow 6x of value 69 in 6 G ( B ) ;

=

0; g : = 0;

x:= x Step 3

m

B:=

6x; g:= g

-

e ; a:= a ; f:= Tici.

+ 69.

Revise reduced cost coefficients and 5 by means of (12.60);

if ( 1 2 . 6 0 ) terminates at exit 2 then stop; otherwise g o to step 2 .

(12.60) R e v i s i o n o f r e d u c e d c o s t c o e f f i c i e n t s Step 1

Determine the connected component

S

(0

E S C_ N ) of the

partial graph of 6 G ( B ) which contains all arcs with Ac if Step 2

ij

> 0;

T E S

T:= N \ S ;

then return (exit 1 ) . L:=

(S n I) x (T fI J ) ; Q:=

if L = Ql then return (exit 2 ) . Step 3

a:= min{aij

I

(i,j) E L);

(T n I ) x ( S n J ) ;

28 1

Algebraic Flow Problems

Step 4

Redefine all reduced cost coefficients by ij a

if (i,j) E Q

*a

I

if (i,j) E L and u

*

E:=

aij

I

otherwise

ij

and go to step 1 .

Obviously all steps in the Hungarian method are well-defined. The reduced cost coefficients (i,j) E I

x J

a ij

satisfy

a ij -> # e for

all

throughout the performance of the Hungarian method.

Further (12.61)

x

ij

*

> 0

for all (i,j) E I

x J

(i,j) is B-admissible

i s always satisfied. This follows from the

transformation of the reduced cost coefficients in step 4 of (12.60). X 4

ij

= 0

-

a

ij

is increased only on arcs (i,j)E Q ; therefore

for ( i , j ) E Q and dom B

(cf. 4 . 1 8 . 2 )

5

dom(6

*

(f-g) O a )

lead to ( 1 2 . 6 1 ) .

The cardinality of

S

is increased

at least by one after such a transformation. Thus ( 1 2 . 6 0 ) minates after at most (n + m

-

1)

transformations. The same argu-

ment as used in the proof of proposition ( 1 2 . 4 3 )

x.

Then due to ( 1 2 . 6 1 )

-T D a - = e for y

y:=

IXJ'

can be proved in a similar manner as before. Thus -T y ma = 0

5 O*yToa

for all y E PT since

a E HYn.

shows that

is finite and leads to a maximum

the Hungarian method ( 1 2 . 5 9 ) flow

ter-

=

y

Hence

Toa is optimal.

(12.53)

282

Linear Algebraic Optimization

(12.59)

is developed in B U R K A R D 11978a1. T h e particular case

of algebraic assignment p r o b l e m s

(ck = 1 , k € I and R = Z ) is

solved in the same manner in B U R K A R D , HAHN and Z I M M E R M A N N , U. [19771.

It should be noted that ( 1 2 . 5 9 )

and ( 1 2 . 5 0 )

lead to

the same sequence of flows x if H is weakly cancellative. I n the Hungarian method can be interpreted as a

the form ( 1 2 . 5 9 )

sequence of (admissible) transformations o f the reduced cost

a

coefficients

a

-

+

such that y

for all Y E P

T'

T

*

o a = y

-

yToa

The composition o f a finite number of (admissible)

transformations is again an (admissible) transformation. The transformations are constructed in a systematic manner such that a final admissible transformation a

+

mn a E H + is generated

satisfying y for all Y E P

T

T

*

oa = 0

yToa

and f3 =

8

*

-T for some y € a . A s we have shown before such

are optimal.

In discrete semimodules ( R = Z Z ) we get similar complexity bounds as for the algebraic network flow problem. In particular for algebraic assignment problems an optimal solution is found in O(n

4

1 usual operations, O(n

4

)

*-compositions and O(n

4

)

5

-

comparisons. In the classical case the Hungarian method [ ( 1 2 . 5 0 )

a s well as

( 1 2 . 5 9 1 1 reduce to the Hungarian method for transportation

Algebraic Flow Problem

283

problems (cf. MURTY 119761) which is developed for assignment problems in KUHN [19551. For bottleneck objectives it reduces to the second (threshold) algorithm of GARFINKEL and RAO [1971]. In this case the algorithm remains valid even in the case of finite capacities (c C -1 ij

on arcs (i,j) E I

XJ.

This follows

from its threshold character (cf. BURKARD [1978a], EDMONDS and FULKERSON [1970]). We will discuss two further methods for a solution of algebraic transportation problems. Both methods are well-defined in d-monoids H which are linearly ordered semimodules over R+; on the other hand a Proof of validity and finiteness is only known in the case that H is a weakly cancellative d-monoid which i s an extended semimodule over R+. NOW we develop an algorithm proposed in BURKARD and ZIMMERMANN,U. [1980]. The method proceeds similar to the Hungarian method by

alternately flow augmentations and dual revision steps. Different from the Hungarian method flow augmentation is performed along certain shortest paths. The method consists of m stages (k = l,Z,

...,m).

In the k-th stage the flow x satisfies

={c i

x

01

for all i E I and 0

5

0

xk

C

i f i C k i f i > k

ck. In order to augment such a flow x

we consider paths in the r e d u c e d i n c r e m e n t a l g r a p h AG

k

with

set of arcs {(u,k)) U I x J

u

( ( j , i ) l (i,j)EIxJ,

xij

>ol u

{ ( j , r ) I XjT c C j L

If xk < ck then AGk contains at least one elementary path p

from u to T. Then 6(p) > 0 (capacity of p), p(6(p))

is an

Linear Algebraic Optimization

284

incremental path flow, and we may augment x to

xo

p ( 6 (p))

.

At the k-th stage the current flow x and the current dual variables v

j E N will be a l m o s t p - c o m p a t i b l e ,

j

i.e. all

conditions (cf. 12.15) are satisfied with the possible exception of v At

> v . which is necessary while x

a -

1

ui

< ci for i

5 k.

the end of the k-stage x o k = ck. If we augment x by p ( 6 ( p ) )

then almost u-compatibility is violated, in general. Due to the choice of p it will be possible to revise the dual variables i n such a manner that the new current x and v are almost p-compatible, again.

For a determination of p we assign weights a to the arcs of AGk.

Since x and v are almost v-compatible the reduced cost

coefficients

i,

can be defined by

1j

for all ( i , j ) E I

x

J. Let zij:=

a ij

for (i,j) E I

x

N

J and a

.= e

ij'

otherwise. We remind that by definition x.

11

for all (i,j) E I X J

> O

-

a

-b

(cf. 12.46).

ij

= e

Let w(p) denote the weight of N

a path p in AGk with respect to weights a .

Let N':=

...,k}

{If2,

lj

U J U

IT).

(cf. chapter 8 ) .

Shortest paths from u to i E N '

can be determined using the algebraic version of ( 8 . 4 ) with (8.10) since

yijE H +

together

for all arcs (i,j) of AGk.

The following shortest path method is a suitable version of (8.4)

applied to AGk. During its performance a label [ni; p(i)]

is attached to vertices i E N ' . r i

is the weight of the current

Algebraic Flow Roblems

285

d e t e c t e d s h o r t e s t p a t h from a t o i and p ( i ) i s t h e p r e d e c e s s o r o f i on t h i s p a t h .

(12.61)

Shortest paths i n A G ~

Step 1

L a b e l v e r t e x k w i t h [ e ; o ] and l a b e l a l l v e r t i c e s j E J

[ zk j ; k ] ;

with Step 2

I':= {1,2,

Determine h E J ' II

h

= minIn

If

x

hr

;

J':= J.

with j

I

j EJ')

and r e d e f i n e J':= Step 3

...,k - l }

J'

{h}.

< c h t h e n go t o s t e p 6 ;

l e t S ( h ) : = { i E 1'1 x . > 01; ih i f S(h) =

$ ,t

t h e n go t o s t e p 2 ;

o t h e r w i s e l a b e l a l l v e r t i c e s i E S ( h ) w i t h [ n h ; h ] and redefine I':= I ' \ S ( h ) . Step 4

For a l l

( i , j ) E S ( h ) x J' do

if

then l a b e l vertex j with

Step 5

Go t o step 2.

Step 6

Label

'I

N

[ni*a i j;I].

w i t h [ n h ; h ] and d e f i n e

ni:=

=h

f o r a l l v e r t i c e s i E I U J which s a t i s f y ai >

kh

o r which

are not labeled.

A t t e r m i n a t i o n t h e s i n k T i s l a b e l e d and n N

s h o r t e s t p a t h p from a t o

'I

is t h e w e i g h t o f a

which c a n e a s i l y be f o u n d by back-

tracing using the predecessor labels.

I f t h e weight of a

286

Linear Algebraic Optimization

shortest path from is equal to n after O(lNI O(lNI

2

)

2-

2

)

i'

to i E N '

0

Otherwise

is not greater than =

TIi

TI

usual operations, O(lN1

2

then it

algorithm terminates )

*-compositions and

comparisons.

(12.62)

Augmenting p a t h method

Step 1

k:= 1 ; x

Step 2

. The

TI

0; v

3

e; U:=

Determine weights n .

I'

N

p with capacity ~(61, u:=

-

N

Ao.

iEN\{o)

using the shortest path method ( 1 2 . 6 1 ) .

* n T 1.

Step 3

x:=

Step 4

Redefine the dual variables by

step 5

I f xok < c

x o

k

and the shortest path

A(v

T

go to step 2 ;

if k = m stop; k:= k+l and go to step 2 .

All steps in ( 1 2 . 6 2 )

are well-defined in a d-monoid H provided

that (x;v) is almost U-compatible throughout the performance of (12.62).

The necessary determination of the reduced cost coeffi-

cients is not explicitly mentioned.

As

in the Hungarian method

it is possible to derive a recursion for the reduced cost coefficients; then the dual variables can be deleted from the algorithm with the exception of v T . The determination of

TI

i

in

step 2 is also possible using cuts and dual revisions as in the

287

Algebraic Flow Problems

Hungarian method; but the shortest path method leads to a better complexity bound: O ( l N 1 2 ) instead of O(lNI

3

)

with

respect to all types of operations considered (cf. 1 2 . 2 5 ) . This is due to the fact that we assign nonnegative weights M

to the arcs of AGk which admits the application of a aij fast shortest path procedure. It should be noted that in the

case

T

= e the subsequent determination of shortest paths

of zero-weight can be replaced by the determination of a maximum incremental flow in AGk using only arcs with weight 'c; ij=e For a proof of finiteness and validity of the augmenting path method we assume that H is a weakly cancellative d-monoid which is an extended semimodule over R+.

A

proof for general

d-monoids which are linearly ordered semimodules is not known.

(12.63) P r o p o s i t i o n The augmenting path method ( 1 2 . 6 2 )

generates a sequence of

almost v-compatible (x;v). Further x is of minimum cost among all flows

N

with xoi = x

Proof: In step 1

oi

for all i E I.

x a 0 and v

e are Ao-compatible. Thus it

suffices to prove that an iteration does not violate almost u-compatibility. If at the end of the k-th stage (x;v) is almost u-compatible then the same holds at the beginning of the k+lth stage since xdk = ck. Thus it suffices to discuss. an iteration at the k-th stage. Let (x;v) denote the current pair in step 2 and let (x'Iv') denote the redefined variables. We assume that (x;v) is almost p-compatible and we will prove that (x';v') is almost p'-compatible

(11'

=

A(vr * a r ) ) .

288

Linear Algebraic Optimization

At first we consider the validity of the inequalities in (12.15).

Obviously, these inequalities are satisfied on arcs

(U,i), 1

5

i < k and ( j , ~ ) , j E J in the same way as in the

Hungarian method

-

v. * a , . = a

(12.64)

11

3

ij

due to our assumption. If Otherwise

71,

1

> n

Let (i,j) E I

(cf. 1 2 . 3 9 ) .

i

x J.

Then

(PI *vi

TI,

3

implies n

5

71.

>

TI

then v ! < a . . ( P ' ) *v; 3

i'

-

-

13

Therefore rri

.

is the value

of a shortest path in AGk and TI * a > n Hence v . * T I i ij - j' 7 < v + a . . * a = a ( 1 1 ) * v . * a i . u 5 P ' implies v . < a . . ( p ' ) * v i . - j 11 i ij 3 13 Now let x!

ij

> 0. Then either x

ij

> 0 or (i,j) €,;

path from a to T. If x . . > 0 then v . = a 7

13

-

a , . = e. Therefore ?ri =

TI

11

ij

the shortest

( p ) * v i implies

which leads to v ! = a . . ( u ' ) * v i . We 3

j

13

r e m a r k that N

x T o a = e.

(12.65)

Otherwise ( i , j ) € c . Then = a , . ( P ) *vi* 1 3

TI^.

u

5

TI,

3

=

a 11 , . * n .1

P' leads to v! = a 3

Secondly we remark v' = e and v'E(H u' a

u ' 5 x(z). N

(12.65)

which implies ij

(11')

*v;

v ,

1

.

* T ,

3

u {ellN. Now we will show

shows that x is a minimum cost flow with N

respect to weights a. Let

be a flow in G with

7ai

=

x' for ai

N

all i E I. Then x = x D A X where Ax is an incremental flow in hGk.

Due to ( 1 2 . 7 )

we know that Ax has a representation as

sum of incremental path flows and circuit flows r

Ax =

x

p=l

S

pP(bp) +

and weights with respect to

2

x

p=r+l

satisfy

pp(dp)

289

Algebraic Flow Problems

If

i s o f minimum c o s t among a l l s u c h f l o w s t h e n

too.

Since w(p)

N

5

N

-

w(p 1 f o r a l l p = 1 , 2 , P

..., r

we f i n d t h a t x '

i s o f minimum c o s t w i t h r e s p e c t t o a , t o o .

G

Now l e t N

t o a. i

E

b e a maximum f l o w i n G o f minimum c o s t w i t h r e s p e c t

N

x contains a p a r t i a l flow

x

5

with

= x'

ai

oi

for a l l

I . Then

By t h e o r e m ( 1 2 . 3 8 ) t h e a l g e b r a i c t r a n s p o r t a t i o n p r o b l e m h a s an optimal solution yEPT. cT o v J

Therefore

*

J

GT 0 ; 5

5 yToa

yTo; yT 0;

*

c;ov,

(12.64) leads t o

*

T

cI o v I =

N

2

*

c

T I D V I

-

implies

(x')Toz
e and

Choose ( r ls ) E K with ars :=

max{ai,l

(if,) E K , vj:= ai, *aij(u) e v i l

and determine the new basis by introducing x

rs

into

the basis; redefine

i,

and

u and qo to step

2.

We remark that with these modifications of the general primal simplex method ( 1 1 . 4 6 ) it is possible to avoid an explicit use of the associated negative part of the groups variables in ( 1 2 . 7 5 )

G

lJ

.

All

are elements of the given semimodu1.e H.

The choice of the nonbasic variable in step 3 which is introduced into the basis leads in the classical case to the

298

Linear Algebmic Optimization

classical stepping stone method; for bottleneck objectives the method is equivalent to those of BARSOV 119641, SWARCZ (119663, 119711) and HAMMER

(119691, 119711) in the sense that

it generates the same sequence of basic solutions provided that the starting solutions are the same. In all these methods the entering nonbasic variable is determined investigating certain cycle sets defined with respect to the bottleneck values in the current basic solution. The above proposed rule using dual variables looks more promising as well from a theoretical as from a computational point of view.

Finally we give some remarks on particular algebraic transportation problems which have been considered in more detail. We forgo an explicit discussion of the classical problem and refer for this purpose to standard textbooks. Algebraic assignment problems were solved first in BURKARD, HAHN and ZIMMERMANN, U.

119771 using a generalized Hungarian

method. The augmenting path method was directly developed for algebraic transportation problems by BURKARD and ZIMMERMANN, U. 119801; that paper contains a short discussion of the specific

case of algebraic assignment problems, too. The primal simplex method for algebraic transportation problems is given in ZIMMERMANN, U.

[1979dl without explicit discussion of algebraic

assignments. FRIEZE [1979] develops a method for the algebraic assignment problem similar to the classical method of DINIC and KRONROD 119691. This method is of order O(n 3 ) similar to the augmenting path method. The solution of algebraic transportation problems is used for

Algebraic Flow Problems

299

deriving bounds in a branch and bound method for the solution of certain algebraic scheduling problems in BURKARD [ 1979c1.

Particular objective functions differing from the classical case which have found considerable interest in the literature are bottleneck objectives and lexicographical objectives (cf. 11.58 and 11.59). For lexicographical objectives a FORTRAN program is developed from the algebraic approach in ZIMMERMANN, U. 119761. Although the program is structured in order to show the impact of the algebraic approach it is easy to derive an efficient version for lexicographic objectives, in particular. Computational experience is discussed, too. Bottleneck assignment and bottleneck (or time) transportation problems have been considered by many authors. The following list will hardly be complete but hopefully provides an overview which seems to be necessary since even today similar approaches are separately published from time .to time. Such problems were posed and solved by BARSOV 119641 (1959 in russian). He proposed a primal solution method. Independently GROSS [1959] stimulated by FULKERSON, GLICKSBERG and GROSS [19531 investigated the same objective function for assignment problems. In the following years GRABOWSKI ([1964]., [1976]), SWARCZ "19661,

[19711) 'and

HAMMER ([1969], [1971]) developed solution methods. In particular HAMMER ([1969],

[1971]), SWARCZ [19711, SRINIVASAN and THOMPSON

[1976], ZIMMERMANN, U.

([1978a],

[1979d]) consider primal methods

equivalent to BARSOV'II method. They only differ in the method for the determination of the pivot element. GARFINKEL 119711 and

3 00

Linear Algebraic Optimization

G A R F I N K E L and R A O [1971] discuss primal dual methods based

on the threshold algorithm in E D M O N D S and F U L K E R S O N [1970]. The augmenting path method is finally developed within the scope of an algebraic approach by B U R K A R D and Z I M M E R M A N N , U. [19801. Further discussions can be found in S L O M I N S K I

(119761,

i19771, [19781). Computational experience is discussed by P A P E and S C H o N [1970], B U R K A R D [1975a], S R I N I V A S A N and T H O M P S O N Z I M M E R M A N N , U.

([1978b],

[19761, D E R I G S and

[19791), F I N K E and S M I T H 119791, and

D E R I G S [1979d]. F O R T R A N programs of the augmenting path method

are contained in D E R I G S and Z I M M E R M A M N , U.

([1978b],

[19791).

Computational efficiency has shown to be highly dependent on the choice of initial heuristics; by now, augmenting path methods and primal methods have been developed to nearly the same computational efficiency. Both seem to be faster than versions of the Hungarian method (also called primal dual or threshold method).

13.

Algebraic

Independent Set Problems

In this chapter we discuss the solution of linear algebraic optimization problems where the set of feasible solutions corresponds to special independence systems (cf. chapter 1 ) . In particular, we consider matroid problems and 2-matroid intersection problems; similar results for matching problems can be obtained. We assume that the reader is familiar with solution methods for the corresponding classical problems (cf. LAWLER [ 1 9 7 6 ] ) . Due to the combinatorial structure of these problems it suffices to consider discrete semimodules ( R = 72 1 ;

in fact, the

appearing primal variables are elements of {O,l}.

We begin with a discussion of linear algebraic optimization problems in rather a general combinatorial structure. In chapter 1 we introduce independence systems F which are subsets of the set P(N) of all subsets of N:= assume that F is norrnaz, i.e.

{1,2,

...,n}.

W.1.o.g.

we

{i) E F for all i E N . By defini-

tion (1.23) F contains all subsets J of each of its elements IEF.

In the following we identify an element I of F with its

i n c i d e n c e v e c t o r x E {0,1)n defined by (13.1)

x

j

= 1

CI*

jEI

for all j E N . Vice versa I = I ( x ) is the s u p p o r t of x defined by I = {j E N 1 xj = 1). A

linear description of an independence system F can be derived

in the following way. Let H denote the set of all flats (Or closed sets) with respect to F and let r: P ( N ) 30 1

+

Z + denote the

301

Linear Algebraic Optimization

rank function of F. A denotes the matrix the rows of which are the incidence vectors of the closed sets C E H ; b denotes a vector with components b

:=

r(C), C E n . Then the set P of

all incidence vectors of independent sets is (13.2)

P = {x E Z:

I

Ax

5 b}.

We remark that such a simple description contains many redundant constraints; for irredundant linear descriptions of particular independence systems we refer to G R ~ T S C H E L 119771.

Let H be a linearly ordered, commutative monoid and let a E H

n

.

We state the following linear algebraic optimization problems: (13.3)

minfxToal ~ E P I ,

(13.4)

max{xToal

(13.5)

min{xT

0

a1 x E P ~ I ,

(13.6)

max{xT

0

a1 x E P ~ I ,

with Pk:=

~ E P I I

I x E P I E x . = k) for some k E N . 7

(13.3) and (13.5) can

trivially be transformed into a problem of the form (13.4) and (13.6) in the dual ordered monoid. Since P and Pk are finite sets all these problems have optimal solutions for which the optimal value is attained. Theoretically a solution can be determined by enumeration; in fact, we are only interested in more promising solution methods.

An element in P of particular interest i s the lexicographically

maximum Vector ;(PI. defined by

A related partial ordering

on 22

n

+

is

303

Algebraic Independent Set Problems

(13.7)

X'y

f o r x,y E

k 1 j=l

VkEN:

k

z:.

( 1 3 . 8 ) Proposition I f x E P i s a maximum o f P 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 i n g (13.7)

t h e n x = x(P).

Proof:

I t s u f f i c e s t o show t h a t x

x,yEP.

>

Let x

f o r some k E N .

y.

= yi

Then x i

Yk

Therefore

k

k j=1

which shows t h a t x

k

y.

We i n t r o d u c e two v e c t o r s x

+

a n d xk d e r i v e d f r o m x E ZZ

i f a

>

j -

e

,

(13.9) otherwise

(13. 1 0 )

:=

{z.l

if j

5

I

j (k),

otherwise

f o r a l l j E N a n d f o r a l l k E 22+

w i t h j ( k ) : = max{jENI

j Z

'

x. zk}. i=1 The r e l a t i o n s h i p o f t h e s e v e c t o r s a n d t h e p a r t i a l o r d e r i n g (13.7) i e described i n t h e following proposition.

Linear Algebraic Optimization

3 04

(13.11) P r o p o s i t i o n Let ii be a linearly ordered, commutative monoid. Hence H is a

linearly ordered semimodule over

LI =

+ xx.

and

3

e

(1)

Proof:

(2)

V

xy

=

j'

Z + .

Let x,y E

with

P,:

Then

.

5 (x+lT0a

( 1 ) is an immediate implication of (13.9).

y 5 x implies

k(x'):= rnin{jl

yj

*

x'.

-

Exv = v . j

Assume y

*

X I : =

xv and let

Then 6:= x ' - y k > 0. Define x" by k

XI).

j

6

i f j = k ,

x " :=

if j = k+l, otherwise

Then y 5 x " and due to 6 O a

k+l

-
k(x') after a finite number of steps we find y = x " . (31

If

r

5

P

5

y then xr j

- xj

= 0 for all j with a

j

< e. There-

f o r e xr < xs for all j E N implies the claimed inequality. If 1 - j

;

S for all j with a . 2 e. Further x 2 xj xj I for all j E N implies x r O a < xs O a f o r all j with a < e. j j - j j j This proves the claimed inequality.

LI

5 s 5

(4)

r then xr =

I

From (2) we get y T o a 5 ( x " ) ~O a . From ( 3 ) we know

Algebraic Independent Set Pmblems

305

( ~ ~ ) 5~ (o ~ a+ ) ~ ofor a all k E Z + . F o r k = u we find the claimed inequality.

The importance of proposition

( 1 3 . 1 1 ) for the solution of

the linear algebraic optimization problems (13.4) and ( 1 3 . 6 ) can be seen from the following theorem which is directly implied by ( 1 3 . 1 1 ) and (13.8).

(13.12) T h e o r e m Let H be a linearly ordered, commutative monoid. Hence H is a linearly ordered semimodule over Z + .Let

x

denote the lexico-

graphically maximum solution in P and let k E N with 1

< -

max(Ex.1 1

xEPI.

5 k 5

~f

...

(1)

a 1 -> a 2 ->

(21

P has a maximum with respec

> -

an ' to the partial o r ering

(13.7)

;+lToa

= max{xTnal

~ E P I ,

x -k T o a = maxfxT m a 1 XEP,}.

Theorem ( 1 3 . 1 2 ) gives sufficient conditions which guarantee that a solution of (13.4) and (13.6) can easily be derived from the lexicographically maximum solution

x

in P. It is well-known

(cf. LAWLER [ 1 9 7 6 ] ) that the following algorithm determines the lexicographic maximum x(P); ej vector o f 23

n

.

, jE

N denotes the j-th unit

306

Linear Algebraic Optimization

(13.13)

Greedy algorithm

Step 1

x

Step 2

If x + e . E P 1

Step 3

If j = n then stop; j:=

0; j:= 1 .

j + 1

then

x:= x + e . 7

.

and go to step 2.

If we assume that an efficient procedure is known for checking x i e . E P in step 2 then the greedy algorithm is an efficient 3

procedure for the determination of x(P).

I t is easy to modify

the greedy algorithm such that the final x is equal to

Condition ( 1 3 . 1 2 . 1 )

x+

-k or x

.

can obviously be achieved by rearranging

the components of vectors in Z n .Let Il denote the set of all permutations n: N

-B

N. For n E Il we define a mapping

n:

Zn+Zn

by

which permutes the components of x E 22 inverse permutation of n). to achieve (13.12.1)

If

TI

accordingly (n-'

is the

is the necessary rearrangement

then fi[Pl:= {i(x)I

x E P 1 is the corres-

ponding new independence system. We remark that +[PI does not necessarily have a maximum with respect to the partial ordering

(13.7) if P satisfies (13.12.2).

If ;[PI

satisfies ( 1 3 . 1 2 . 2 )

for all n E TI then the lexicographically maximum solution ;(;[PI) leads to an optimal solution if

307

Algebraic Independent Set Problems

Such independent systems are matroids as the following theorem shows which can be found in ZIMMERMANN, U.

119771. Mainly

the same result is given in GALE [1968]; the difference lies in the fact that GALE explicitly uses assigned real weights.

(13.14) T h e o r e m Let T(P) denote the independence system corresponding to P. Then the following statements are equivalent: (1)

T(P) is a matroid,

(2)

for all n E I I there exists a maximum of ;[PI

with

respect to the partial order (13.7). Proof: (1)

*

(2).

If T(P) is a matroid then T(n[P])

is a matroid

since the definition of a matroid does not depend on a permutation of the coordinates in P. Thus it suffices to show that P contains a maximum with respect to the partial ordering (13.7). Let

=

;(PI

Suppose y

&

be the lexicographical maximum of P and let y E P.

x.

Then there exists a smallest k E N such that k

k

I: Y j >

I:

j=l Let J : = {jI y j =. 1, j

5

k);

j=l

xj

.

-

I:= {jl x

=

j

1, j < k}. Then

IJl > 111 and due to (1.24) there exists an independent set IU

{ u ] for some

x'

%

( 2 ) =+

x

p E J L I . The incidence vector x' of I satisfies

contrary to the choice of

-

X.

(1). Assume that T(P) is not a matroid. Then there exist

two independent sets 1,J with 111

C

IJI such that for all

j E J X I the s e t I U {j} is dependent. We choose n E l l such that

n[I] < n[J\I]

< n[N\ (IUJ)]

.

3 08

Linear Algebraic Optimization

Let u denote the incidence vector of J. If ;(PI

contains a

maximum n(y) with respect to the partial ordering (13.7) then proposition (13.8) shows that the lexicographical maximum

-

- -

x = X ( n ( P ) ) satisfies

= n(y).

yi =

Therefore i(u)

0

for all i E J \ I .

that ;[PI

Then y

i

1 for all i E I and

=

i(y) which implies

does not contain a maximum with respect to the parti-

cular ordering (13.7).

Due to the importance of the lexicographical maximum x ( P ) we describe a second procedure for its determination. Let

-

5 be a partial ordering on Zn. Then we define

5'

x

(13.15)

y

O(y)

5

on 22

n

by

G(x)

for a € ll with o ( i ) = n - i + 1 for all i E N .

In particular we

may consider the partial orderings i'and 4 ' . We remark that x 5' y implies x -s'

y.

(13.16)

M o d i f i e d greedy a l a o r i t h m

Step 1

x

Step 2

If x + e

Step 3

If j = 1 then stop; j:=

0;

j

-

j:= n.

j

EP

then

xi= x + e

j'

1 and go to step 2 .

The final vector x in this algorithm is denoted by x ' ( P ) . Since the application of this algorithm to P is equivalent to the application of the greedy algorithm to ;[PI

we get x ' ( P ) =;(:[PI)

(13.15) shows that x' Is the minimum of P with respect to 4'

.

.

Algebraic Independent Set Problem

1

L e t r:= max{Ix. 1

p*:=

309

x E PI and d e f i n e Iy€{0,11"1

3 xEP,:

y

5

1

- X I .

T h e n P* c o r r e s p o n d s t o a c e r t a i n i n d e p e n d e n c e s y s t e m T ( P * ) .

In p a r t i c u l a r , i f T ( P ) i s a m a t r o i d t h e n T ( P * ) i s t h e d u a l matroid

( c f . c h a p t e r 1 ) . T h e r e f o r e w e c a l l P* t h e d u a l of P.

L e t s:= m a x { I y i l

Y E P * ) . Then r + s = n.

(13.17) Proposition I f P h a s a maximum w i t h r e s p e c t

-

then x(P) = 1 Proof:

S i n c e ;(PI

t i a l ordering x

x

i

- x'

t o t h e p a r t i a l ordering (13.7)

(P*).

is t h e maximum o f P w i t h r e s p e c t t o t h e pa.r-

(13.7),;EPr.

Thus 1 - , € P i .

Let xEPr.

Then

means by d e f i n i t i o n k

r

-

j=l xJ

for a l l kEN.

k




I:

-

j -k+ 1 x J f o r a l l k = O,l,.

. ., n - 1

i s t h e maximum o f P

r

i

j

-

j

= Ex

j

= r . Thus

n E x j j=k+l

i ( x ) . Therefore

which shows

w i t h respect t o

i'. Hence

x

i s the maxi-

mum o f P r w i t h r e s p e c t t o 4 ' . T h u s

1 - x 6' 1 - x for a l l x E P Let y € P*xPs. implies

and 1

-x

i s t h e minimum of P* w i t h r e s p e c t t o 4 ' . 9

Then t h e r e e x i s t s

P:.such

that y

5

which

4' y . T h e r e f o r e 1 - x is t h e minimum o f P* w i t h

-

r e s p e c t t o 6'. H e n c e 1 - x = x ' ( P * ) .

Linear Algebraic Optimization

310

Proposition ( 1 3 . 1 7 )

shows that the lexicographical maximum

x(P) can be derived a s the complement of x'(P*I determined by ( 1 3 . 1 6 )

which is

applied to P*. The application of the

modified greedy algorithm to P * is called the d u a l g r e e d y

a z g o r i t h m . This is quite a different method for the solution of the optimization problems ( 1 3 . 4 )

and ( 1 3 . 6 ) .

In particular,

checking x + e . E P in step 2 of the greedy algorithm is reI placed by checking x + e . E P * in the dual greedy algorithm. 3

Therefore the choice of the applied method will depend on the computational complexity of the respective checking procedures.

For a more detailed discussion on properties o f related partial orders we refer to ZIMMERMANN, U.

([19761,

[19771).

The greedy

algorithm has been treated before by many authors. KRUSKAL 1 1 9 5 6 1 applied it to the shortest spanning tree problem, RADO 1 1 9 5 7 1 realized that an extension to matroids is possible.

WELSH 1 1 9 6 8 1 and EDMONDS 1 1 9 7 1 1 discussed it in the context of matroids. The name 'greedy algorithm' is due to EDMONDS. Related problems are discussed in EULER 1 1 9 7 7 1 ;

a combinatorial

generalization is considered in FAIGLE 1 1 9 7 9 1 . For further literature, in particular for the analysis of computational complexity and for related heuristics we refer to the bibliographies of KASTNING 1 1 9 7 6 1 and BAUSMANN 1 1 9 7 8 1 . For examples of matroids we refer to LAWLER [ 1 9 7 6 1 and WELSH 1 1 9 7 6 1 .

The following allocation problem is Solved in ALLINEY, BARNABEI and PEZZOLI [ 1 9 8 0 1 for nonnegative weights in the group ( I R , + , z ) . Its solution can be determined using a special (simpler) form of the greedy algorithm ( 1 3 . 1 3 ) .

Algebraic Independent Set Problems

(Allocation Problem)

Example A

31 1

system with m units (denoted a s set T) is requested to serve

n users (denoted as set S). It i s assumed that each user can engage at most one unit. Each user i requires a given amount a

i

of resources and each unit j can satisfy at most a given required amount b unit j iff ai

5 b

with vertex sets

of resources. Thus user i can be assigned to j' S

Let G = (S U T , L ) denote the bipartite graph and T and edge set L where (i,j) E L iff

< b.. A matching I is a subset of L such that no two diffei I rent edges in I have a common endpoint. A set A 5 S is called a

assignable if there exists a matching I such that

A =

{il

(i,j)EI).

The set of all assignable sets is the independence system of a matroid

(cf. EDMONDS and FULKERSON 1 1 9 6 5 1 or WELSH [ 1 9 7 6 ] ) .

In

particular, all maximal assignable sets have the same cardinality. We attach a weight c i E H + to each user i where (A,*,() a linearly ordered commutative monoid.

(13.12)

is

and ( 1 3 . 1 4 )

show that we can find an optimal solution of the problem max{xTUcI

x E PI by means of the greedy algorithm ( 1 3 . 1 3 )

notes the set of all incidence vectors of assignable sets).

(P de-

Due

to the special structure o f the matroid it is possible to simplify the independence test ( x + e E P ) in step 2 o f the greedy j algorithm. In fact, we will give a separate proof o f the validity o f the greedy algorithm for the independence system T(P). Together with theorem ( 1 3 . 1 4 )

we find that T(P) is a matroid.

= max{ckl k E S 1 . If Ti:= {j € T I ai 5 b 1 = 0 then there i j exists no unit which can be assigned to user i. Then G ' denotes

Let c

the subgraph generated by (S\{i))

U T. If we denote the optimal

Linear Algebraic Optimization

312

value of an assignable set of (11

z ( G ' )

If Ti

$

= z G)

0 then let b . 3

by

z(G)

then

-

min{b

=

G

k

I k E T i } . Let I denote a match( I J , E~ ) I}.

ing corresponding to an optimal assignable set {vl

Every such matching is called an "optimal matching". We claim (i,j) E I. Let V:= { u , p l

that we can assume w.1.o.g. If i , j 6 V then I U {(i,j)}

( I J , ~ )E

I}.

is an optimal matching, too. If

i E V , j e V then we replace (i,k) E I by

(i,j). If i @ V , j E V

then we replace (k,j) € 1 by (i,j); since c . is of maximum value the new matching is optimal, too. If i,j E V but , E I by (i,j) and ( p , v ) . then ,we replace (i,v), ( I Jj)

that

( I J , ~ )E

Let G ' Then I

L since a

We remark

5 bv. Thus we can assume (i,j) E I.

< b.

u -

1

U (T'{j}).

denote the subgraph generated by (S'{i}) { (i,j) 1 is a matching in G '

.

Therefore z

(G)

On the other hand, if I' is an optimal matching in I' U { (i,j)) is a matching in G and thus Therefore in the case Ti (2) ( 1 ) and

z ( G ' )

(2)

*

ci =

$

(i,j) a I

z ( G ' )

*

c

< i -

5 G ' ,

z (G')

* ci.

then

z ( G ) .

@ we find

z ( G ) .

show the validity of the following variant of the

greedy algorithm. W e assume c 1 > c2 >

0;

Step 1

I =

Step 2

If T.:= {b

T:= { 1 , 2 ,

i:= 1 ;

k

1 -

I:= ~ U { ( i , j ) l

;

j

=

Cn.

...,m}.

I a . < bk, k E T }

find j E T . such that b

T:= ~'{j}.

. - -->

=

0 then go to step

min Ti

;

3;

313

Algebraic Independent Set Problems

Step 3

If i = n then stop; i:= i

+

1 and go to step 2 .

At termination I is an oFtimal matching which describes the assignment of the users i of an optimal assignable set A* = {il

(i,j) E 11 to units j. It should be clear (cf. theo-

rem (13.12)) that {ii

( i , j ) € 1 1 is optimal among all assign-

able sets of the same cardinality at any stage of the algorithm and for arbitrary c E H i

..

(iE.5)

satisfying c 1

5 cn . In particular, we may consider (IR,+ , L ) and

min,l). Then for ciEIR, i

E S

2

c2

2

..

(1R U Em},

we find

for all assignable sets A with

IAl

=

lA*l;

i.e. the sum as

well as the minimum of all weights is maximized simultaneously.

In the remaining part of this chapter we discuss linear algebraic optimization problems for combinatorial structures which do not contain a maximum with respect to the partial ordering (13.7), in general. We assume that H is a weakly cancellative d-monoid. Hence w.1.o.g.

H is an extended semimodule over Z +

(cf. 5.16 and chapter 6). F r o m the discussion in chapter 1 1 we know that max- and min-problems are not equivalent in d monoids due to the asymmetry in the divisor rule (4.1). In

Linear Algebraic Optimization

314

chapter 1 1 we developed reduction methods for both types of algebraic optimization problems which reduce the original problem to certain equivalent problems in irreducible sub-d-monoids of H ; since H is weakly cancellative these sub-d-monoids are parts of groups. Let ( H A ; A E h ) denote the ordinal decomposition of H. Then

for an incidence vector x of an independent set ( A o : =

min A ) .

According to ( 1 1 . 2 1 ) and ( 1 1 . 5 4 ) the optimization problems (13.3)-

-

( 1 3 . 6 ) are reduced to optimization problems in H

(13.18) (13.19)

u1 u2

=

min{A(xTma)

I

~ E P ,I

=

max{A(xTna)

I

x~ P I

(13.20)

p 3 = minlA(x

(13.21)

p4

u with

,

a) I x E pkI,

T

= maxrA(xTna)l

~ E P ~ I .

Since o E P the reduction ( 1 3 . 1 8 ) of ( 1 3 . 3 ) is trivial:

u 1 - Ao.

The reduction ( 1 3 . 1 9 ) of ( 1 3 . 4 ) is also simple; since the corresponding independence system is normal the unit vectors e

j

,

j E N are elements of P. Therefore

uz

=

maxIA(aj)

I

j EN).

For the determination of u 3 we assume that it is possible to compute the value of the rank function r : P(N)

+

Z+

for

a given subset of PJ by means of an efficient (polynomial time) procedure. This is a reasonable assumption since we will consider only such independence systems for which the classical

Algebmic Independent Set Problems

315

combinatorial optimization problems can efficiently be solved; then an efficient procedure for the determination of an independent set of maximum cardinality in a given subset of N is known, too. In chapter 1 the closure operator u: P ( N )

-D

P(N)

is

defined by

A

subset I of N I s called c l o s e d if U(1) = I. Clearly, if the

rank function can efficiently be computed then we may efficiently construct a closed set J containing a given set I. J is called a m i n i m a l c l o s e d c o v e r of I. Every set I C_ N has at least one minimal closed cover J , but in general J is not unique. Let s : = k - r ( J )

2

0 (cf. 13.5 and 13.6).

Then we define the

b e e t r e m a i n d e r s e t AJ by i f s = O , (13.22)

AJ:=

{il,i2,...,is}

otherwise,

...

.

5 < a. The It i2 following method for the determination of p 3 is a refinement

where N L J = {il,i2,...,it}

and a

5

a

of the threshold method in EDMONDS and F U L K E R S O N 119701.

(13.23)

R e d u c t i o n m e t h o d f o r (13.5)

Step 1

u:=

Step 2

If r(K) 2 k then stop;

max{A(a.)I

I

j€A@};

K:= {j

€NI

A(a,)

determine a minimal closed cover J of K.

5

p}.

Linear Algebraic Optimization

316

In the performance of (13.23) we assume k method terminates in O(n) steps.

A

5 r(N). Then the

similar method for the deri-

vation of thresholds for Boolean optimization problems in linearly ordered commutative monoids is proposed in ZIMMERMANN, U. [ 1978cl.

(13.24)

Proposition

u in

The final parameter

(13.23) i s equal to

u 3 in

(13.20).

Proof: Denote the value of u before the last revision in step 3 by

i.The

corresponding sets are denoted by

r ( K ) < k. Since Pk

*

@ we conclude

.

u3 >

k , 5 and

Aj.

Then

Further r ( ? ) = r ( K )

shows that an independent set with k elements has at least s = k

- r(k)

elements in N L j . Thus "3 > maxE)i(aj)

I

jE

which implies u 3 2 u. Now r ( K )

A ~ I

5 k shows

p3

5 u.

W e remark that for bottleneck objectives (cf. 11.58)

the re-

duction method (13.21) leads to the optimal value of (13.5) since A(x

T

a) = xT m a in this case.

In the determination of u 4 we consider certain systems F j

, jEN

derived from the underlying independence system F. We define (13.25)

FJ:= {I E F I

I

u

{j} E F , j

d I)

for j E N.

Algebraic Independent Set Problems

317

Obviously F j is an independence system for each j E N . Its rank function is denoted by r j

a

1

> -

a2 >

.

W.1.o.g.

we assume

- - .-> a n

in the following method.

(13.26)

Reduction method f o r ( 1 3 . 6 1

Step 1

w:=

Step 2

If r ( N )

Step 3

w:=

1. W

w

+

2

k

-

1 then stop ( p = X(a,)).

1;

to step 2 .

go

Finiteness of this method is obvious.

(13.27)

Proposition

The final parameter w in ( 1 3 . 2 6 ) satisfies p 4 = h(aw) for p 4 in ( 1 3 . 2 1 ) . Proof: Let

V

denote the final parameter in ( 1 3 . 2 6 ) .

Then there

exists an independent set I E F of cardinality k such that j E I. Therefore h ( a w ) r

w-1

5

pa,

If v = 1 then equality holds. Otherwise

(N) < k-1 shows that there exists no I E F of cardinality k

such that j €.I. Thus p 4

5

A(aw).

Again, we remark that for bottleneck objectives the reduction method ( 1 3 . 2 7 )

leads to an optimal value of ( 1 3 . 6 ) .

After determination of the corresponding index optimization problems ( 1 3 . 3 )

-

(13.6)

u the algebraic

are reduced according to

L i n w Algebraic Optimization

31R

(11.23)

and ( 1 1 . 5 5 ) . The sets of feasible solutions P and P k

are replaced by P : = {xMI x E P M

and x

(Pk)M:= CxMl x E P k and x M(p)

with M =

= {j E N 1

X(a.1 7

5 u).

= 0 for all j E M ) ,

j

= 0

j

for all j E M }

I f F denotes the correspond-

ing independence system then ( 1 3 . 2 8 )

shows that the reduced sets

of

feasible solutions correspond to the restriction F

M,

i.e.

M

of F to

(13.28)

which is again a normal independence system. Theorems ( 1 1 . 2 5 ) and ( 1 1 . 5 6 )

show that ( 1 3 . 3 )

-

(13.6)

are equivalent to the

following reduced l i n e a r a l g e b r a i c o p t i m i z a t i o n probZems T

(13.29)

min{x

(13.30)

max{x

(13.31)

minIx

(13.32)

max{xT

T T

o a ( ~ ) ~XIE P # I

,

oa(u)MI XEP,}

,

0

a(v),l

xE

a(ujMI x E (pklM)

with respect to M = M ( p ) and p = p i , i = 1 , 2 , 3 , 4 defined by (13.18)

-

(13.32)

are problems in the extended module G

p = pi

, i

(13.21).

Since H i s weakly cancellative ( 1 3 . 2 9 ) 1!

over ZZ

-

for

= 1,2,3,4.

Next we consider two particular independence systems. Let F denote the intersection F

n

F 2 o f two matroids F 1 and F

2

where F 1 and F 2 are the corresponding independence systems. An element I E F is called an i n t e r s e c t i o n . Then ( 1 3 . 3 )

-

(13.6)

319

Algebraic Independent Set Problem

are called a l g e b r a i c m a t r o i d i n t e r s e c t i o n p r o b l e m s . In particular, ( 1 3 . 5 )

and ( 1 3 . 6 )

are called a l g e b r a i c m a t r o i d k - i n t e r -

s e c t i o n p r o b l e m s . Let (V,N) denote a graph. Then I

N is

called a m a t c h i n g if no two different edges in I have a common endpoint. We remark that in a graph (i,j) and ( j , i ) denote the same edge. The set of all matchings is an independence system (13.3)

F. Then

-

(13.6)

In particular, ( 1 3 . 5 )

are called a t g e b r a i c matching p r o b t e m s .

and ( 1 3 . 6 )

are called a t g e b r a i c k-match-

ing probtems.

The classical matroid intersection problem is well-solved; efficient solution methods are developed in LAWLER ( [ 1 9 7 3 ] ' , [19761), [1979].

IRI and TOMIZAWA [ 1 9 7 6 1 , FUJISHIGE 1 1 9 7 7 1 and EDMONDS An augmenting path method and the primal dual method

are described in the textbook o f LAWLER [ 1 9 7 6 ] ;

FUJISHIGE [ 1 9 7 7 ]

considers a primal method. The classical matching problem is well-solved, too; an efficient primal dual method f o r its solution is given in EDMONDS [ 1 9 6 5 1 and is described in the textbook of LAWLER 1 1 9 7 6 1 . A primal method is considered in CUNNINGHAM and MARSH 1 1 9 7 6 1 . The necessary modification of the primal dual method for the classical versions of ( 1 3 . 5 ) (13.6)

is discussed in WHITE [ 1 9 6 7 ] .

and

For the classical matroid

intersection problem such a modification is not necessary since in the shortest augmenting path method optimal intersections of cardinality 0 , 1 , 2 , . . .

are generated subsequently (cf.

LAWLER [ 1 9 7 6 ] ) .

In particular, efficient procedures for the determination of a

Linem Algebraic Optimization

320

matching

( m a t r o i d i n t e r s e c t i o n ) o f maximum c a r d i n a l i t y i n a N a r e d e s c r i b e d i n LAWLER

g i v e n s u b s e t N'

[1976].

Thus t h e

e v a l u a t i o n o f t h e rank f u n c t i o n i n step 2 of t h e r e d u c t i o n method

is possible i n polynomial t i m e f o r both pro-

(13.23)

blems. Let F be t h e

(cf.

13.25)

w h e r e N'

set of a l l m a t c h i n g s i n t h e g r a p h

is t h e s e t of a l l m a t c h i n g s i n t h e g r a p h

is t h e set of

common w i t h j sections.

( V , Nj )

a l l edges which have no e n d p o i n t i n

(jEN). L e t F = F 1 n F 2 be t h e set of a l l inter-

Then F j

= Fi

nF;.

a l l i n d e p e n d e n t s e t s of of

(V,N). T h e n F J

F u r t h e r F'

1

( a n d F:)

a certain matroid

t h e o r i g i n a l m a t r o i d t o N'

{ j),

cf.

is t h e set of

(called contraction

WELSH

[1976]).

Thus the

e v a l u a t i o n of t h e r a n k f u n c t i o n r v , V E N i n s t e p 2 o f t h e r e d u c t i o n method

(13.26)

is p o s s i b l e i n polynomial t i m e

for both

problems. The r e s t r i c t i o n FM ( c f .

1 3 . 2 8 ) l e a d s t o t h e set o f a l l match-

i n g s i n t h e g r a p h (V,M) and t o t h e i n t e r s e c t i o n o f t h e t w o restricted matroids blems

(13.29)

-

( F 1 l M f l( F 2 ) M . T h e r e f o r e t h e r e d u c e d p r o -

( 1 3 . 3 2 ) are a l g e b r a i c matching

section) problems provided . t h a t matching

-

( 1 3 . 6 ) are algebraic

(matroid i n t e r s e c t i o n ) problems.

It remains t o

modules.

(13.3)

(matroid inter-

solve t h e reduced problems i n t h e respective

A l l methods f o r t h e s o l u t i o n o f

the classical matroid

i n t e r s e c t i o n problem are v a l i d and f i n i t e i n modules.

A re-

f o r m u l a t i o n of t h e s e m e t h o d s i n g r o u p s c o n s i s t s o n l y i n rep l a c i n g t h e u s u a l a d d i t i o n of r e a l numbers by t h e i n t e r n a l composition i n t h e u n d e r l y i n g group and i n r e p l a c i n g t h e u s u a l l i n e a r o r d e r i n g o f t h e r e a l numbers by t h e l i n e a r o r d e r ing i n t h e underlying group.

Optimality o f t h e augmenting p a t h

32 1

Algebraichiependent Set Problems

method is proved by KROGDAHL (cf. LAWLER 119761) using only combinatorial arguments and the group structure (i.e. mainly cancellation arguments) of the real additive group. Thus his proof remains valid in modules. Optimality of the primal method in FUJISHIGE [I9771 is based on similar arguments which remain valid in modules, too. Optimality of the primal dual method is based on the classical duality principles. From chapter 1 1 we know how to apply similar arguments in modules. In the following we develop these arguments explicitly; for a detailed description of the primal dual method we refer to LAWLER 119761. Let Ax

5 b and

xx

5

be the constraint systems (cf. 13.2) with

respect to the restricted matroids. Then (13.33)

In modules it suffices to consider max-problems since a minproblem can be transformed into a max-problem replacing the cost coefficients by inverse cost coefficients. We assign dual variables ui and vk to the closed sets 1 and k of the restricted matroids. Then (u,v) is dual feasible if A

(13.34) N

where a = a(v),

T

*

ou

NT A ov

and where

2

N

a, ui

2

e l vk

2

e

u denotes the respective index of

the reduction. The set of all dual feasible (u,v) is denoted by DM. Then the algebraic dual of (13.30) (13.35)

min{b

T

ou

*

gTovl

is

(u,v) E DH).

In the usual manner we find weak duality, i.e. x T O z 5 bT O u

NT b Ov.

Linem Algebraic Optimization

322

The corresponding complementarity conditions are

(13.37)

*

ui > e

(Ax)i = b i

(13.38) N

for all j E M and all closed sets i and k. If a j E M then x

0, u

E

e, v

=

< e for all

1 -

e is an optimal pair of primal

and dual feasible solutions. Otherwise the initial solution x s 0 , v s e , ui = e for all i

*

M and

u : = maxIZ. I j E M ) M

3

is primal and dual feasible. Further all complementarity conditions with the possible exception of (13.39)

are satisfied. We remark that bM is the maximum cardinality Of an independent set in one of the restricted matroids. Such a pair (x;u,v) is called compatible, similarly to the primal dual method for network flows. Collecting all equations ( 1 3 . 3 6 ) after external composition with x (

j

we find

13.40)

for compatible pairs. The primal dual method proceeds in stages. At each stage either the primal solution is augmented or the values of the dual variables are revised. At each stage no more than 21MI dual variables are permitted to be non-zero. Throughout the performance of the method the current pair (x;u,vl is compatible. This is achieved by an alternate solution of

323

Algebraic Independent Set Pmblems

(13.41)

max{x

T

O

-

a1 x E P M

(x;u,v) compatible)

I

for fixed dual feasible solution (u,v) and of minjb

(13.42)

T

* FT 0 vI

0u

(u,v) E DM

, (x;u,v) compatible}

f o r fixed primal feasible solution x. From

that ( 1 3 . 4 1 )

i s equivalent to

(13.43)

max{ 1

x

jEM

and that ( 1 3 . 4 2 ) (13.44)

1

I

xEPM

I

(13.40)

we conclude

(x;u,v) compatible)

is equivalent to

(u,v) E D M

min{uMI

,

(x;u,v) compatible).

An alternate solution of these problems is constructed in the same way as described in LAWLER [ 1 9 7 6 1 for the classical case. In particular, the method is of polynomial time in the number of usual operations, *-compositions and (-comparisons.

The

final compatible pair (x;u,v) is complementary. Thus it satisfies xT

0:

= bT

u

* FT 0 v;

then weak duality shows that

(x;u,v) is an optimal pair. Therefore this method provides a constructive p r o o f of a duality theorem for algebraic matroid intersection problems.

5 c and Cx 5 c be the constraint systems (cf. N

Let Cx

M

13.2)

with respect to the two matroids considered. Then (13.45) A

closed set I of one of the matroids with I C_ M is closed,

with respect to the restricted matroid, too. We remark that c, = b

N

j

u

(c, = b

j

)

f o r all j E M .

We assign dual variables u i , v k to the closed sets i and k of the matroids. Then (u,v) is dual feasible with respect to

Linear Algebraic Oprimization

37-4

maxfx

(13.4')

T

oa(p2)1 X E P )

if

*

cTou

(13.46) The set of

-T

c

2

o v

a l l dual feasible

a l g e b r a i c dual o f

(13.4)

min{cTou

2

a ( u 2 ) , ui

2

e, v k

e

( u , v ) i s d e n o t e d by D 2 .

. Then t h e

is

*

-T

c

( u , v ~E D ~ I .

0v1

Algebraic dual programs with r e s p e c t t o (13.3')

minix

T

oa(ul)

I

XEPI

a r e c o n s i d e r e d i n c h a p t e r 11. with respect t o

(13.3') e

An o b j e c t i v e

if

za(ul)

X(ui)

(u,v) i s strongly dual feasible

*

C

5 u1 ,

~

h(vk)

*

-T

c

O

o ~v

,

5 u l , e 5 ui

function is defined according t o

,

e z vk '

(11.19).

( 1 3 . 4 7 ) Theorem ( M a t r o i d i n t e r s e c t i o n d u a l i t y theorem) L e t H be a weakly c a n c e l l a t i v e d-monoid.

Hence H i s an e x t e n -

d e d s e m i m o d u l e o v e r Z + .T h e r e e x i s t o p t i m a l f e a s i b l e p a i r s ( x ; u , v ) f o r t h e a l g e b r a i c matroid i n t e r s e c t i o n problems and

(13.4')

which a r e complementary a n d s a t i s f y

(11

e = x T o a * c

(2)

( - c l T 0 u ( u1 1

(31

cTou

Proof: -

(13.3')

T

*

-T

o u * c - T

(-c)

(for 13.3')

o v

ov(v,) = x

T

ma

TT0v = xT0a

( 1 ) . The m i n - p r o b l e m

( f o r 13.3'1, (for 1 3 . 4 ' ) .

( 1 3 . 2 9 ) i s t r a n s f o r m e d i n t o a max-

problem o f t h e form ( 1 3 . 3 0 ) r e p l a c i n g a

j

(v,)

by i t s i n v e r s e i n

325

Algebmic Independent Set Problems

for a l l j E M = M(pl).

t h e group G

- - -

(x;u,v) denote the

Let

p1

f i n a l p a i r generated i n the application of the primal dual Then f o r j E M :

m e t h o d t o t h i s max-problem. a.(pl) 7 when A a n d

-1

5

(

~

~

*0 . i;i T o v )j ?

a r e s u b m a t r i c e s o f C and

( c o l u m n s j E M and rows

o f c l o s e d s e t s 1 5 M a n d row a ( M ) w i t h t h e c l o s u r e f u n c t i o n of t h e respective matroid). L e t if ] E M , x j:=

then x E P .

otherwise

Further l e t if i $ M u := i

I

if i = U 1 ( M ) ,

e

otherwise

where a l is t h e c l o s u r e f u n c t i o n o f t h e m a t r o i d F 1 and l e t i f k 5 M v

k

I

i f k = a2(M),

:=

otherwise

where a 2 is t h e c l o s u r e f u n c t i o n o f t h e m a t r o i d F 2 .

e for a l l j E M .

*

aj

T (C n u

*

ZTov)

j

N o w A ( a . ) > u1 f o r j 6 M shows

I

e

Therefore

5

Then

5

a(pl)

*

cTou

*

-T

c

~v

(u,v) is strongly dual f e a s i b l e ,

m e n t a r y and ( I ) and

(2)

. ( x ; u , v ) is c o m p l e -

are satisfied.

(31 Let (x;u,v) denote the f i n a l p a i r generated i n the applic a t i o n o f t h e p r i m a l d u a l method t o

(13.30).

Since N = M ( p 2 )

Linear Algebraic Optimization

326 we find

Therefore (u,v) is dual feasible, (x;u,v) is complementary and (3) is satisfied.

A valuable property o f compatible pairs leads to the solution

of algebraic matroid k-intersection problems. Again we consider a solution in the respective group. Then

1

(PkIM = ( x E Z f

Ax

5

N

b, Ax

5

z,

x

jEM

x

j

= kl.

We assign a further dual variable X to E x . = k . Then (u,v,x) is 1

dual feasible if (13.48)

A

T

nu

*

-T A

o v

*

[A]

zZ,

ui

2 e , vk 2 e.

The dual variable X is unrestricted in sign in the respective group. The set o f all dual feasible (u,v) is denoted by (DkIM. Then the algebraic dual o f (13.32) is

Again we find weak duality xTO;5b

T

Ou

*

*

-T b Ov

(koX)

for all primal and dual feasible pairs

(x;u,v,X). We apply the

primal dual method to (13.301,but now for v

v

v

A sequence of compatible pairs ( x ;u ,v

EX'!

I

= v

for v =

o,I,...

)

u:= u 4 and M:= M(u4). is generated with

.

We modify the dual revision procedure slightly in order to admit negative uM (cf. LAWLER [1976], p. 347: let 6 = min{6 6v.6wl).

The stop-condition uM = 0 is replaced by Ex

j

= k.

U

,

327

Algebmic Independent Set Problems

T h e n i t may h a p p e n t h a t uM b e c o m e s n e g a t i v e d u r i n g t h e p e r formance o f t h e p r i m a l d u a l method. t y c o n d i t i o n s remain v a l i d .

and l e t A:=

Hence

u

M'

(x;u',v,A)

Now r e p l a c e u by u '

{:

u' := i

Then

A l l other compatibili-

i

$

MI

i

=

M

s a t i s f i e s (13.pB) and

(x;u',v,h)

i s o p t i m a l f o r (13.32)

primal feasible, i.e.

i f xx

= k.

w e f i n d a s t r o n g d u a l i t y theorem.

W e a s s i g n d u a l v a r i a b l e s ui

and

j

5

k.

(13.49) i f x i s

S i m i l a r l y t o theorem

(13.47)

From (13.45) w e g e t

a n d vk t o t h e c l o s e d s e t s o f t h e

m a t r o i d s a n d two f u r t h e r d u a l v a r i a b l e s A t o Ex

d e f i n e d by

t o xx

I n t h e c a s e o f a max-problem w i t h

> k and A+

j -

= M\N(p4)

$

@

w e have t o a d j o i n t h e i n e q u a l i t y

< o

t - X

j -

M

with assigned dual variable Y. PI:= k {X€Pk[ 1 ; (u,v,A+,A-,Y)'is (13.6')

Then x,

f 01.

dual feasible with respect t o max{x

T

aa(p4)

I

XEP;I

if

ui where

Y

2

e , Vk

2

e l A-

2

e,

denotes t h e vector with j-th

A+

:e ,

Y

2

e

I

c o m p o n e n t y f o r j €;

and

328

Linear Algebraic Optimizarwn

j-th component e otherwise. Without this additional dual variable y which does not appear in the objective function it is necessary that at least one of the other dual variables has the index A(max(a.1 7

j

€:I)

> p4.

This leads to a duali-

ty gap which can be avoided by the introduction of y:= max{a j

€GI.

I

j

The set o f all dual feasible solutions according to

(13.50)

is denoted by D;.

f: Di

H can be defined with respect to (13.6') similarly as

+

in ( 1 1 . 1 9 ) .

Let a:= k O A -

6:= c T O u

Then a dual objective function

and

*

-T c Ov

*

(kOA+)

.

Then f is defined by

if a where

5 B or m

A(a) = A ( E ) = Ao.

Otherwise let f(u,v,A+,A-,y):=

-

denotes a possibly adjolnt maximum o f H. The dual of

(13.6') is

Dual programs with respect to

are defined as in chapter 1 1 .

(u,v) is strongly dual feasible

if

e

5 A- , e 5

A+

,

e

5 u i , e 5 vk

.

An objective function is defined according to ( 1 1 . 1 9 ) .

329

Algebmic Independent Set Pmblerns

(13.51) Theorem

(Matroid k-intersection duality theorem)

Let H be a weakly cancellative d-monoid. Hence H is an extended semimodule over Z+

.

There exist optimal feasible pairs

(x;u,v,A+,A-) and (x;u,v,A+,A-,Y) for the algebraic matroid k-intersection problems ( 1 3 . 5 ' ) and ( 1 3 . 6 ' ) which are complementary and satisfy T

*

-T

c

Ov

T (-c) Ou(v3)

*

(-c)~OV(U)=xToa

(1)

koA- = x

(2)

ko6(U ) 3

(3)

xToa

(41

xToa = cT0u

with 6 ( p 3 ) : = A

*

Oa

*

*

cTou

*

(koA-) =cTOu

*

(A+)

* -1

-T

c

*

Ov

and

*

(for 1 3 . 5 ' )

(kOAy)

N

3

-T

c

*

Ov

(for 1 3 . 6 ' 1 ,

(koA+)

(for 1 3 . 6 ' )

ko€(p4)

€(u4)

:=

A+

(for 1 3 . 5 ' 1 ,

*

(A_)

-1

.

Proof: The proof of ( 1 ) and ( 2 ) is quite similar to the proof of ( 1 ) and ( 2 ) in theorem ( 1 3 . 4 7 ) . The value of the variable in the final pair generated by the primal dual method is assigned to A +

or A -

in an obvious manner.

The proof of ( 3 ) and ( 4 ) follows in the same manner as the proof of ( 3 ) in theorem ( 1 3 . 4 7 ) if we choose y = max{a

j

I

jEN)

;

the variable y does not appear in the objective function.

If we are not interested in the determination of the solutions of the corresponding duals then we propose to use the augmenting path method instead of the primal dual method for a solution of the respective max-problem in a group. This method generates subsequently primal feasible solutions x

I

v=1,2,

...

330

Linear Algebraic Optimization

of cardinality V which are optimal among all intersections of the same cardinality. It should be noted that such solutions are not necessarily optimal among all intersections of the same cardinality with respect to the original (not reduced) problem; optimality with respect to the original problem is implied only for those v which satisfy

where u denotes the index used in the reduction considered. The solution of algebraic matroid intersection (k-intersection) problems is previously discussed in ZIMMERMANN, U. [1978b],

([1976],

[1978c], [1979a]) and DERIGS and ZIMMERMANN, U. [1978a].

A solution of the reduced problems in t h e case of matching problems can be determined in the same manner. Again it suffices

to consider max-problems. For a detailed description of the primal dual method we refer to LAWLER [1976]. We may w.l.0.g. assume that the respective group G is divisible (cf. proposition 3.2). Hence G is a module over Q. The primal dual method remains valid and finite in such modules. Again a reformulation of the classical primal dual method for such modules consists only in replacing usual additions and usual comparisons in the group of real numbers by the internal compositions and comparisons in the group considered. A l l arguments used for a proof of the validity and finiteness in the classical case carry over to the case of such modules. Optimality follows from similar arguments as in the case of matroid intersections.

Algebraic Independent Set Roblems

33 1

Let A denote the incidence matrix of vertices and edges in the underlying graph (V,N). Let S k be any subset of V of odd cardinality 2 s k + l . Then

is satisfied by the incidence vector x of any matching. We

represent all these constraints in matrix form by Sx (13.52)

n P = { x E Z , I Ax

5

8.

Then

5 1, Sx 5 9 1 .

For the reduced problem with M = M ( p ) we find (13.53)

P

M

=

Ex E 22: I AMx

We assign dual variables u

i

5 1, SMx 5

s].

to the vertices i E V and dual

variables vk to the odd sets S k . Then (u,v) is dual feasible if (13.54)

A

T nu M

*

2

s;ov

N

a

N

where a : = a(llIj for j E M = M(p). The set of all dual feasible j

solutions is denoted by DM. In the usual manner we find weak duality, i.e.

-

xToY
e

I .

E x

j i j

=1,

for all vertices V and for all odd sets T k Z N .

Let (13.58)

V(x):= l i E V l

r jx ij

< 11,

Linem Algebraic Optimization

332

i.e. V(x) is the set of all vertices which are not endpoint N

of some edge in the matching corresponding to x. If a for all (i,j) E M then x

*

0, u

e, v

e

9

ij

< -

e

is an optimal pair

of primal and dual feasible solutions. Otherwise let 6:=

(maxIY;.. I

(I/z)

(i,j) E M I )

11

which is well-defined in the module considered. The initial solution x

0, v

E

2

6 is primal and dual feasible,

e and u

satisfies (13.55) and (13.57) and all dual variables vi for i E V ( x ) have an identical value

rl.

Such a pair is called

c o n p a t i b z e , similarly to previously discussed primal dual methods. From (13.55) we conclude xToy

*

(1

-

AMx)

T

O u = lT0u

*

sTOv

for a compatible pair (x;u,v). Since all variables u . with ( 1 -AMxIi

(13.59)

+

0 have identical value rl we get

x

T

oa

*

( I V I- 2 zMxij)

0 0

= 1

T

ou

*

sTov.

Similar to the primal dual method for matroid intersections the primal dual method for matchings alternately proceeds by revisions of the current matching and the current dual solution without violation of compatibility. This is achieved by an alternate solution o f (

13.60)

T maxfx

O

a1 x E P M

, (x;u,v) compatible}

for fixed dual solution (u,v) and of (13.61)

T rninI1 n u

s

T

OvI

(u,v) E D M ,

(x;u,v) compatible)

for fixed primal solution x. From (13.59) w e conclude that

333

Algebmic Independent Set Problems

(13.60) is equivalent to (13.62)

maxI1 x

M ij

I

xEPM

,

( x ; u r v ) compatible)

and that (13.61) is equivalent to (13.63)

(U,V) E D M

min{nI

;

(x;u,v) compatible)

where TI is the common value of all variables u

i with i E V ( x )

(cf. 13.58). Finiteness and validity of the method can be shown in the same manner as described in LAWLER [19761 for the classical case. In particular, the method is of polynomial time in the number of usual operations, *-compositions and