Submodular Functions and Optimization Second Edition
ANNALS OF DISCRETE MATHEMATICS
Series Editor: Peter L. HAMMER Rutgers University, Piscataway, NJ, U.S.A
Please refer to this volume as follows: S. Fujishige: Submodular Functions and Optimization (Second Edition) (Annalsof Discrete Mathematics, Vol. 58) (2005)
58
Submodular Functions and Optimization Second Edition Satoru Fujishige Research Institute for Mathematical Sciences Kyoto University Kyoto, Japan
2005
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V
Preface Submodular functions frequently appear in the analysis of combinatorial systems such as graphs, networks, and algebraic systems, and reveal combinatorially nice and deep structures of the systems. The importance of submodular functions has widely been recognized in recent years in combinatorial optimization and other fields of combinatorial analysis. The present book provides the readers with an exposition of the theory of submodular functions from an elementary technical level to an advanced one. The theory of submodular functions was developed in the earliest stage till 1950's by H. Whitney and W. T. Tutte for matroids, by G. Choquet for the capacity theory and by 0. Ore for graphs. A nourishing stage of the theory came with J. Edmonds' work on matroids and polymatroids in 1960's. Related studies were also made in the theory of characteristic function games by L. S. Shapley and others. Since 1970, applications of (poly-)matroids to practical engineering problems have been extensively made by M. Iri, A. Recski and others, and also theoretical developments in submodular functions by W. H. Cunningham, J. Edmonds, A. Frank, M. Iri, E. L. Lawler, L. Lovasz, A. Schrijver, E. Tardos, N. Tomizawa, D. J. A. Welsh, U. Zimmermann and others. The theory of submodular functions is now becoming mature, but a lot of fundamental and useful results on submodular functions are scattered in the literature. The main purpose of the present book is to put these materials together and to show the author's unifying view of the theory of submodular functions by means of base polyhedra and duality for submodular and supermodular systems. Special emphasis is placed on the constructive aspects of the theory, which will lead us to practical efficient algorithms. No comprehensive survey of submodular functions is aimed at here. I had to omit important results on submodular functions such as a strongly polynomial time algorithm for minimizing submodular functions due to M. Grotschel, L. Lovasz and A. Schrijver. This is mainly because the precise description and validation of the results would require further technical developments outside the mainstream of this book. A sketch of the author's view of submodular functions was given in a survey paper [Puji84c], which was written while I was visiting Professor Bernhard Korte's Institute in Bonn as an Alexander von Humboldt fellow in 1982-83, and laid a basis of the project of writing this book, which I gratefully acknowledge. I also acknowledge that part of my work, upon
VI
which the present book is primarily based, has been supported by grantsin-aid of the Ministry of Education, Science and Culture of Japan. I would like to express my deep sincere thanks to Professor Masao Iri of the University of Tokyo who first drew my attention to the theory of matroids, a promising and enjoyable research field of combinatorics, in 1975 and has since then been keeping giving me invaluable advice and stimulating discussions on submodular functions and other related discrete systems. Without his advice most of my work would not have been accomplished. Thanks are also due to Professor Nobuaki Tomizawa, now at Niigata University, with whom I enjoyed inspiring discussions and joint work. I am also very much grateful to Professor Peter L. Hammer for his encouragement to write this book. I used a preliminary version of this book for a lecture of the Doctoral Program in Socio-Economic Planning at the University of Tsukuba and I thank the students who attended the lecture for their useful comments. I have also benefited from comments and communications received from Bill Cunningham, Tetsuo Ichimori, Naoki Katoh, Kazuo Murota, Masataka Nakamura, Hans Rock and Uwe Zimmermann, to name a few, in the course of my research on submodular functions and writing this book. July 1990
S.F.
Vll
Preface to the Second Edition When I finished my monograph (the first edition) in 1990, there was a polynomial-time algorithm for minimizing submodular functions by means of the ellipsoid method, due to Grotschel, Lovasz, and Schrijver, and devising a combinatorial polynomial-time algorithm for minimizing submodular functions was still an open problem. Submodular function minimization is so fundamental in the theory of submodular functions and optimization that the monograph could not be completed without treating algorithms for submodular function minimization, but I hesitated to include the ellipsoid method for submodular function minimization due to Grotschel, Lovasz, and Schrijver because of its non-combinatorial feature. Hence in the first edition submodular function minimization was treated algorithmically in a very unsatisfactory way. However, in 1999 the long-standing open problem of submodular function minimization was resolved independently by Satoru Iwata, Lisa Fleischer and myself, and by Lex Schrijver, in different ways though both algorithms were based on the framework of Bill Cunningham. I am very happy to add Chapter VI thereby including the combinatorial strongly polynomial algorithms for submodular function minimization. Moreover, among other related developments after 1990 one of the most important results is the discrete convex analysis due to Kazuo Murota, which has been described in another new chapter, Chapter VII. Chapters VI and VII form Part II of the second edition. Part I includes Chapters I~V, which formed the original edition of the monograph. In Chapters I~V of Part I typos and minor errors have been corrected. I appreciate comments received from readers and my friends and colleagues on the first edition of this monograph. I have also added remarks and references related to the developments after 1990, which are put between brackets [ and ]. I would like to mention here another important recent development related to submodular functions that I could not include in Part II. It is concerned with the connectivity augmentation problem for graphs and its generalization, which was pioneered by T. Watanabe and A. Nakamura [Watanabe+Nakamura87] and further developed by A. Frank, T. Jordan, H. Nagamochi, T. Ibaraki, and others. Readers should be referred to [Frank92, 94b, 05], [Frank+Jordan95], [Frank+KiralyO2], [Jordan95], [Jackson+Jordan05], [Nagamochi+Ibaraki02], [NagamochiOO, 04] and the references therein for later and recent developments. I also would like to
Vlll
mention the source location problem, which is closely related to the connectivity augmentation problem. The source location problem was first considered in [Tamura+Sengoku+Shinoda+Abe92] and [Tamura+Sugawara+ Sengoku+Shinoda98], and was further investigated in [Ito+Uehara+YokoyamaOO], [Ito+IINUY02] and [Arata+Iwata+Makino+Fuji02] for undirected networks, and in [Nagamochi+Ishii+ItoOl], [Ito+MAHIF03] and [Barasz+ Becker+FrankO5] for directed networks (also see the references therein). I am very grateful to Hiroshi Hirai, Satoru Iwata, Tom McCormick, Kazuo Murota, Takeshi Naitoh, Akiyoshi Shioura, Akihisa Tamura, and Zaifu Yang for their useful comments on an earlier version of Chapters VI and VII in Part II, which helped rectifying errors and improved the presentation of Part II. Special thanks are also due to Kazuo Murota, who gave me valuable detailed comments on Chapter VII. I also thank Andras Frank and Tibor Jordan for useful information about recent developments in connectivity augmentation. Kyoto, January 2005
S.F.
IX
Contents
Preface Preface to the Second Edition
v vii
PART I
1
Chapter I. Introduction
3
1. Introduction
3
1.1. Introduction
3
1.2. Mathematical Preliminaries
4
(a) Sets
4
(b) Algebraic structures
5
(c) Graphs
9
(d) Network flows
13
(e) Elements of convex analysis and linear inequalities
15
Chapter II. Submodular Systems and Base Polyhedra 2. From Matroids to Submodular Systems
21 21
2.1. Matroids
21
2.2. Polymatroids
25
2.3. Submodular Systems
33
3. Submodular Systems 3.1. Fundamental Operations on Submodular Systems
45 45
(a) Reductions and contractions by sets
45
(b) Reductions and contractions by vectors
46
(c) Translations and sums
51
(d) Other operations
53
X
3.2. Greedy Algorithm
55
(a) Distributive lattices and posets
55
(b) Greedy algorithm
58
3.3. Structures of Base Polyhedra
66
(a) Extreme points and rays
66
(b) Elementary transformations of bases
70
(c) Tangent cones
72
(d) Faces, dimensions and connected components
75
3.4. Intersecting- and Crossing-Submodular Functions
86
(a) Tree representations of cross-free families
87
(b) Crossing-submodular functions
91
(c) Intersecting-submodular functions &
3.5. Related Polyhedra
101 102
(a) Generalized polymatroids
102
(b) Polypseudomatroids
106
(c) Ternary semimodular polyhedra
112
3.6. Submodular Systems of Network Type
122
Chapter III. Neoflows 4. The Intersection Problem 4.1. The Intersection Theorem
127 127 127
(a) Preliminaries
128
(b) An algorithm and the intersection theorem
131
(c) A refinement of the algorithm
136
4.2. The Discrete Separation Theorem
140
4.3. The Common Base Problem
142
XI
5. Neoflows
145
5.1. Neoflows
145
(a) Submodular
flows
145
(b) Independent
flows
146
(c) Polymatroidal
flows
147
5.2. The Equivalence of the Neoflow Problems
148
(a) From submodular flows to independent
flows
148
(b) From independent flows to polymatroidal
flows
149
(c) From polymatroidal flows to submodular
flows
150
5.3. Feasibility for Submodular Flows
153
5.4. Optimality for Submodular Flows
155
5.5. Algorithms for Neoflows
167
(a) Maximum independent
flows
167
(b) Maximum submodular
flows
172
(c) Minimum-cost submodular 5.6. Matroid Optimization
flows
175 188
(a) Maximum independent matchings
188
(b) Optimal independent assignments
194
Chapter IV. Submodular Analysis 6. Submodular Functions and Convexity
199 199
6.1. Conjugate Functions and a Fenchel-Type Min-Max Theorem for Submodular and Supermodular Functions 199 (a) Conjugate functions
199
(b) A Fenchel-type min-max theorem
201
6.2. Subgradients of Submodular Functions
203
(a) Subgradients and subdifferentials
203
(b) Structures of subdifferentials
209
XI1
6.3. The Lovasz Extensions of Submodular Functions 7. Submodular Programs
211 216
7.1. Submodular Programs — Unconstrained Optimization . . . . 216 (a) Minimizing submodular functions
217
(b) Minimizing modular functions
223
7.2. Submodular Programs — Constrained Optimization
228
(a) Lagrangian functions and optimality conditions
229
(b) Related problems
234
(b.l) The principal partition
234
(b.2) The principal structures of submodular systems .. 245 (b.3) The minimum-ratio problem
Chapter V. Nonlinear Optimization with Submodular Constraints 8. Separable Convex Optimization
248
253 253
8.1. Optimality Conditions
253
8.2. A Decomposition Algorithm
257
8.3. Discrete Optimization
260
9. The Lexicographically Optimal Base Problem
261
9.1. Nonlinear Weight Functions
262
9.2. Linear Weight Functions
264
10. The Weighted Max-Min and Min-Max Problems
269
10.1. Continuous Variables
269
10.2. Discrete Variables
272
11. The Fair Resource Allocation Problem
273
11.1. Continuous Variables
273
11.2. Discrete Variables
274
Xlll
12. The Neoflow Problem with a Separable Convex Cost Function
280
PART II
285
Chapter VI. Submodular Function Minimization
287
13. Symmetric Submodular Function Minimization: Queyranne's Algorithm
287
14. Submodular Function Minimization
290
14.1. The Iwata-Fleischer-Fujishige Algorithm
293
(a) A weakly polynomial algorithm
293
(b) A strongly polynomial algorithm
300
(c) Modification with multiple exchanges
303
(d) Submodular functions on distributive lattices
305
14.2. Schrijver's Algorithm
308
14.3. Further Progress in Submodular Function Minimization
313
Chapter VII. Discrete Convex Analysis
315
15. Locally Polyhedral Convex Functions and Conjugacy
315
16. L- and L''-convex Functions
319
16.1. L- and L''-convex Sets
319
16.2. L- and L''-convex Functions
322
16.3. Domain-integral L- and L^-convex Functions
326
17. M- and M^-convex Functions
331
18. Conjugacy between L'/L^-convex Functions and M'/M^-convex Functions 338 19. The Discrete Fenchel-Duality Theorem
341
XIV
20. Algorithmic and Structural Properties of Discrete Convex Function
344
20.1. L- and iAconvex Functions
344
20.2. M- and M^-convex Functions
345
20.3. Proximity Theorems
351
21. Other Related Topics
356
21.1. The M-convex Submodular Flow Problem
356
21.2. A Two-sided Discrete-Concave Market Model
357
22. Historical Notes
360
References
365
Index
389
1
PART I
Part I consists of Chapters I~V that formed the main body of the first edition of this monograph. Comments and remarks added in the second edition are given within brackets [ and
This Page is Intentionally Left Blank
3
Chapter I. Introduction In this chapter we give a short introduction of the structure of this book and briefly review elements of algebra, graphs, networks and linear inequalities for readers' convenience.
1.
Introduction
1.1.
Introduction
In 1935 H. Whitney [Whit35] introduced the concept of matroid as an abstraction of the linear dependence structure of a set of vectors. Several systems of axioms for defining a matroid are now known, each of which is simple but substantial enough to yield a deep theory in Combinatorial Optimization and to have a lot of applications in practical engineering problems (see [Iri83], [Iri+Fuji81], [Murota87, 00a], [Recski89]). Matroidal structures are closely related to a class of efficiently solvable combinatorial optimization problems; a careful examination of an efficiently solvable problem often reveals a matroidal structure which underlies the problem. In 1970 J. Edmonds [Edm70] combined the matroid theory with polyhedral combinatorics and lead us to the concept of polymatroid. A polymatroid polyhedron, called an independence polyhedron, is expressed by a system of linear inequalities with {O,l}-coefficients and the right-hand sides given by a submodular function which is the rank function of the polymatroid. The relation between matroids and polymatroids is similar to that between matchings in bipartite graphs and flows in networks. The rank function of any polymatroid is a monotone nondecreasing submodular function on a Boolean lattice 2E for a finite set E. The monotonicity of the rank function does not play any essential role in characterizing the combinatorial structure of the polymatroid polyhedron since the monotonicity is not invariant with respect to translations of the polyhedron. The concepts of submodular and supermodular systems [Fuji78b,84c] naturally come up with this observation. The rank function of a submodular (or supermodular) system is a submodular (or supermodular) function on a distributive lattice (or a ring family), a sublattice of a Boolean lattice. The duality is denned between a submodular system and a supermodular system, which dissolves the clumsy definition of polymatroid duality [McDiarmid75]. Submodular systems are not only theoretical generalizations
4
I. INTRODUCTION
of matroids and polymatroids but also significantly extend the applicability in practical problems. In Chapter II we first introduce the concepts of submodular and supermodular systems and their associated base polyhedra by following the historical generalization sequence of matroids, polymatroids and submodular systems. We then examine algorithmic aspects of submodular systems and basic structures of base polyhedra. In Chapter III we consider a class of network flow problems with submodular boundary constraints, which we call the neoflow problem. It includes the (poly-)matroid intersection problem of J. Edmonds [Edm70], the submodular flow problem of J. Edmonds and R. Giles [Edm+Giles77], the independent flow problem of the author [Fuji78a] and the polymatroidal flow problem of R. Hassin [Hassin78,82] and E. L. Lawler and C. U. Martel [Lawler+Martel82b]. Submodular functions are discrete analogues of convex functions. In Chapter IV we develop a theory of submodular functions from the point of view of convex analysis [Rockafellar70], which we call the submodular analysis. We will make clear the close relationship between the submodular analysis and the results obtained in Chapter III. Finally we consider nonlinear optimization problems with submodular constraints in Chapter V. A decomposition algorithm is shown for a separable convex optimization problem over a base polyhedron and it lays a basis for the algorithms of the other problems such as the lexicographically optimal base problem, the weighted max-min (min-max) problem and the fair resource allocation problem. We also consider a neoflow problem (the submodular flow problem) with a separable convex cost function. 1.2.
Mathematical Preliminaries
(a) Sets We denote the set of reals by R, the set of rationals by Q and the set of integers by Z. We also denote the set of nonnegative elements of R (Q, Z) byR+ (Q+, Z+). For any finite set X we denote its cardinality by |X|. When X is a subset of a set Y, we write X C Y, and when X is a proper subset of Y (i.e., X CY and X ^ Y), we write X C Y. It should be noted that this is different from the conventional notation.
1.2. Mathematical Preliminaries
5
For subsets X and Y of a set E, when X n Y = 0, we say X and Y are disjoint, and when I U F = £ (or ( £ - X) fl ( £ - F) = 0), we say X and Y are codisjoint. A set of disjoint nonempty subsets Bj (i € /) of a set E is called a partition of I? if (Jig/ -^i = -£" If {-^i | z € / } is a partition of E, we call {.E — Bi \ i £ / } a copartition of £\ For sets ^4 and _B we call {(1, a) | a € ^4} U {(2, 6) | b € £>} the direct sum of A and B and denote it by A 0 i?. Also, we call {(a, b) \ a € A, b £ B} the direct product of ^4 and B and denote it by A x £?. For a mapping / : A —> £> we often express / as (/(a) | a € A). For example, a family T of sets JQ (i € /) is written as T = (JQ | i € /) and a matrix M with a row index set /, a column index set J and an (i, j)-element Mij (i £ I, j £ J) is expressed as M = [M^ \ i G /, j € J ) . The set of all the mappings from A to B is denoted by BA. The characteristic vector of a subset A of an underlying set E is the mapping %^ : E —> {0,1} such that XA(e) = 1 for e e A and x^(e) = 0 for e e E - A. (b) Algebraic structures For a set A we call a binary relation, denoted by ^ , on A a partial order or simply an order on A if it satisfies (i) (reflexivity) Va € ^4: a ^ a, (ii) (antisymmetry) a ^< b, b ^< a =^
a = b,
(iii) (transitivity) a ^< b, b ^ c ^=> a ^< c. We call the pair (A, R such that Va € A: c(a) < 2E, is called the dependence function (see [Fuji78a]). For x € P( + )(p) and e € sat(x), dep(x, e) = {ef \ e' E E, 3a > 0: x + a(x e - Xe') € P (+) (/))},
(2.15)
where note that for e' 6 dep(x, e) — {e} we have x(e') > 0. We call an ordered pair (e, e') such that e! € dep(x-, e) — {e} an exchangeable pair associated with x. The dependence function is a generalization of a fundamental circuit of a matroid. For x 6 P(_|_)(p) and e 6 E — sat(x) define 6(x, e) = max{a | a € R, x + a\e € P(+)(/?)} (> 0),
(2-16)
which is called the saturation capacity associated with x and e. The saturation capacity is also expressed as c(x, e) = minjppO - x{X)\ e € X C £ } .
(2.17)
For any a such that 0 < a < c(x, e) we have x + ot\e G P(+)(p). For x G P(+)(p), e G sat(x) and e! G dep(x, e) — {e}, define c(x, e, e') = max{a | a G R, s + a(x e - Xe') G P(+)(p)} (> 0),
(2.18)
which is called the exchange capacity associated with x, e and e'. (See Fig. 2.2.) The exchange capacity is also expressed as c(x, e, e') = mm{p(X) - x{X)
e e I C £ , e' £ X},
(2.19)
where note that x(e') > c(x, e, e'). For any a such that 0 < a < c(x,e,e') we have x + a(xe — Xe') £ P(+)(p)A vector v G R B such that x < v for any x- G ?(+)(/?) is called a dominating vector of (E,p). Note that u € R is a dominating vector of (E, p) if and only if v(e) > p({e}) for all e G -E. For a dominating vector v of the polymatroid P = (E, p) define pi,: 2E —> R by p^ ) (X) = U(X) + / ; ( E - X ) - p ( J E )
(KB).
(2.20)
2.2. Polymatroids
29
Figure 2.2: Exchange capacity c(x, 1,2) and saturation capacity c(y,2).
Figure 2.3: The duality in polymatroids.
30
II. SUBMODULAR
SYSTEMS AND BASE
POLYHEDRA
Then P? s = (E,p*,^) is a polymatroid and is called the dual polymatroid of P = (E,p) with respect to v ([McDiarmid75]). (See Fig. 2.3.) For most polymatroids there is no reasonable and physically meaningful way of choosing a dominating vector v and the arbitrariness of dual polymatroids remains, though the choice of v = (p({e}) \ e € E) may be reasonable. For matroidal polymatroids, the matroid duality corresponds to the polymatroid duality with respect to the vector 1 of all the components being equal to 1. Examples of a Polymatroid
(1) Multiterminal flows: Consider a capacitated flow network N = (G = (V,A), c,S+,S~) with the underlying graph G = (V, ^4), a nonnegative capacity function c: A — R + , the set S+ of entrances and the set S~ of exits, where S+, S~ C V and S+ n S~ = 0. We assume that there is no arc entering S+ or leaving S~. A function ip: A —> R is a feasible flow in M if VaeA:
0 < ip(a) < c(a),
Vv € V - (S+ U 5 " ) : &/>(«) = 0,
(2.21) (2.22)
where dip: V —> R is the boundary of (/? denned by 3p( V ) = ^
^(a) -
^
R_|_ is called an entropy function. Entropy function h is a monotone nondecreasing submodular function with /J(0) = 0, i.e., (E, h) is a polymatroid. The submodularity of h is equivalent to the nonnegativity of conditional mutual information. See [Fuji78c] for polymatroidal problems in the Shannon information theory. (4) Convex games: Consider a characteristic-function game (see, e.g., [Shubik82]). Let E = {1, 2, ,n} be a set of n persons, called players. A characteristic function v is a nonnegative function defined on the set of coalitions which are subsets of E, where we assume u(0) = 0. A characteristic-function game (E,v) is called a convex game ([ShapIey71]) if the characteristic function v is supermodular, i.e., VX, Y C E: v(X) + v(Y) < v(X 1>Y) + v(X D Y).
(2.27)
The core of the game (E, v) is the set of payoff vectors defined by {x | x € R B , VX C E: x(X) > v(X),x(E) = v(E)}. Define the function v&: 2E
(2.28)
R by
v*(E-X) = v(E)-v{X) (XCE).
(2.29)
Then we can show that i># is a polymatroid rank function and that the core given by (2.28) coincides with the base polyhedron B(u#) of the polymatroid (see Lemma 2.4). [Also see [Ichiishi81].]
32
II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA
A permutahedron [(5) P e r m u t a h e d r a : Let E = {1,2, Consider a nondecreasing concave function g : R —> R with g(0) = 0 and define a function p : 2E —> R by
p(X)=g(\X\)
(XQE).
Then (E, p) is a polymatroid whose rank function value p(X) depends only on the size of X. In particular, when p is given by 1*1
p(X) = '£(n-i + l) %=\
(XQE),
where p(0) = 0, the base polyhedron B(/o) is called a 'permutahedron (or permutohedron). Every permutation or linear ordering (CTI,(T2, ,o~n) of integers 1, 2, , n can be identified with the vector (cri, 02, , an) in R B . We can show that the set of such vectors for all permutations of 1, 2, , n is exactly the set of all extreme points of the permutahedron (we can see this fact through the discussions in Section 3.2). Note that for the permutahedron the slope g(k) — g(k — 1) decreases by one for k = 1, 2, , n. From a general nondecreasing concave function g with g(0) = 0 we have
2.3. Submodular Systems
33
a nonincreasing sequence > 0,2 > > an with a^ = g(k) — g(k — 1) for k = 1, 2, , n. Then the base polyhedron denned from g has extreme bases, each being a vector {aa^ \ k = 1, 2, , n) (G R S ) for a permutation a of 1, 2, , n, where an extreme base means an extreme point of the base polyhedron. The concept of permutahedron can be traced back to [Schoutel3] (also see [Berge68], [Yemelichev+Kovalev+Kravtsov81], [Ziegler95] and [Borovik +Gelfand+WhiteO3]). It may be worth pointing out that permutahedra are special cases of base polyhedra arising from multiterminal flows described above and are zonotopes, where a zonotope is an affrne transformation of a unit hypercube.]
2.3.
Submodular Systems
Let E be a nonempty finite set and T> be a collection of subsets of E which forms a distributive lattice with set union and intersection as the lattice operations, join and meet, i.e., for each X, Y G V we have XUY, XC\Y € V. Let / : T> —» R be a submodular function on the distributive lattice P , i.e., VX, Y € V: f(X) + f(Y) > f(X UY) + f(X D Y).
(2.30)
We have the following fundamental lemma concerning submodular functions. Lemma 2.1 ([Ore56]): Let f: T> —> R he an arbitrary submodular function on a distributive lattice P C 2 B . Then the set of all the minimizers of f given by V0 = {X I l e P , f(X) = min{/(y) \Y eV}} forms a sublattice ofV, i.e., for any X, Y G T>0 we haveXUY,
(2.31)
Xt~)Y e T>0.
(Proof) For any X, Y € Vo, f(X) + f(Y)>f(XUY) + f(XnY),
(2.32)
where min{/(X U Y),f(X D Y)} > f(X) = f(Y). Hence we must have f(X UY) = f(X n Y) = f(X)(= f(Y)), i.e., XUY, X n Y G Vo. Q.E.D.
34
II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA
For a submodular function / on a distributive lattice V C 2E with 0, E € P and /(0) = 0, we call the pair (P, /) a submodular system on E, and / the rank function of the submodular system (see [Fuji78b, 84c]). We call f(E) the rank of (P, / ) . Define a polyhedron in R B by P(/) = {x I x € RE, VX € D: z p Q < /(X)}.
(2.33)
We call P(/) the submodular polyhedron associated with submodular system (V,f). Also define B(/) = {x\xe
P(/), x(£7) = /(£?)}.
(2.34)
We call B(/) the base polyhedron associated with submodular system (P, / ) . (See Fig. 2.4.)
Figure 2.4: Submodular polyhedron P(/) and base polyhedron B(/). A vector in the base polyhedron B(/) is called a base of (P, / ) and a vector in the submodular polyhedron P(/) is called a subbase of (P, / ) . The saturation function, the dependence function, the saturation capacity and the exchange capacity introduced for polymatroids can easily be extended for submodular systems.
2.3. Submodular Systems
35
Lemma 2.2: For any subbase x G P(/) define V(x) = {X | X G V, x(X) = f(X)}.
(2.35)
Then T)[x) is a sublattice ofT>. (Proof) This follows from Lemma 2.1 since T>(x) is the set of minimizers of Q.E.D. the nonnegative submodular function / — x: V — R. The unique maximal element ofD(x) is denoted by sat(x). sat: P(/) —> 2E is the saturation function. For any subbase x G P(/) and e 6 sat(z), define V(x, e) = {X | e € X G £>, x(X) = /(X)}.
(2.36)
Then X>(x, e) is a sublattice of T>. (Note that D(x, e) is the set of minimizers of the nonnegative submodular function / - i o n the distributive lattice T>(e) = {X | e 6 X € T>}.) The unique minimal element of T>(x,e) is denoted by dep(x,e). For any e 6 E — sat(x) we define dep(x, e) = 0. dep: P(/) x E —> 2E is the dependence function. For any x G P(/) and e € E — sat(x) the saturation capacity c(x,e) is denned by c(s, e) = min{/(X) - x(X) \ e G X G V}. (2.37) For a nonnegative a, we have x- + a\e G P(/) if and only if 0 < a < c(x, e). Moreover, for any x G P(/), e G sat(x) and e' G dep(x, e) — {e} the exchange capacity c(x, e, e') is denned by c(x, e, e) = min{/(X) - x{X) \ e G X G V, e , / ) such that x < y. A function g: V —> R on the distributive lattice P is called a supermodular function if — g is a submodular function, i.e., V X, Y G 2?: 0(X) +5 ( y ) < 5 (X U F ) + 5 (X n y ) .
(2.39)
If 0, E G D and g(0) = 0, the pair (T>, g) is called a supermodular system on £ . Define P(ff) = {x | x E KE, VX e V: x(X) > g(X)},
(2.40)
B(g) = {x | x- e P(g), x(E) = g{E)}.
(2.41)
P(p) is called the supermodular polyhedron and B(g) the frase polyhedron associated with the supermodular system CD.g). A vector in P(g) is called a superbase and a vector in B(g) a 6ase of CD,g). A function which is submodular and at the same time supermodular is called a 'modular function. For a modular function x: V —> R, P(x) should be considered as either a submodular polyhedron or a supermodular polyhedron according as we consider x as a submodular function or a supermodular function. There will be no confusion by the use of this notation. If we consider the dual order 2, /2) on S such that B(/) = B(/ 2 ). (2.51) Moreover, if f is integer-valued, so is fiTheorem 2.6 [Fuji84b]: (i) Let f be an intersecting-submodular function on an intersecting family T C 2 £ with $, E e F and /(0) = 0. The rank function /1 of the submodular system (Pi, /1) in (i) of Theorem 2.5 is given as follows. For each Y C E define /p(y)
= m i n { ^ / p Q ) I {Xi I i <E / } : a partition of Y, iei
Vi € /: Xi € ^ } , (2.52) wnere /p(Y") = +co if the minimum in (2.52) is taken over the empty set. Then, f\ is the restriction of fp on V1 = {Y \ Y C E, fp(Y) < +00}. (ii) Let f be a crossing-submodular function on a crossing family J- C 2E with 0, E € T and /(0) = 0. Define fp(E)
= min{J2 f(Xi) I {X{ \ i € / } : a partition of E, iei
Vi € /: Xi € .F}, (f*)p(E)
= ma^{YJf*(Xl)
(2.53)
| {Xi \ i £ I}: a partition of E,
iei
ViG / : E-Xi
ef},
(2.54)
40
II. SUBMODULAR SYSTEMS AND BASE POLYHEDRA
where f#(X) = f(E)-f(E-X)
{E-XEF).
(2.55)
Then B(/) defined by (2.50) is nonempty if and only if f(E) = fp(E) = (f*)p(E)
(2.56)
or equivalently
£/ # (^) Y, ^ a )' aeA+U
(2-66)
aeA-U
+
where A and A~ are with respect to G'. (2.66) is equivalent to x € B(KC)C). Therefore, there exists a feasible circulation ip in A/7. Restricting ip to A, we obtain a required feasible flow in M whose boundary is equal to x. The converse, d
