VALUATED MATROID INTERSECTION,I: OPTIMALITY CRITERIA∗

KAZUO MUROTA†

Abstract. The independent assignment problem (or the weighted matroid intersection problem)

is extended using Dress-Wenzel’s matroid valuations, which are attached to the vertex set of the

underlying bipartite graph as an additional weighting. Specifically, the problem considered is: Given

a bipartite graph G = (V +, V −; A) with arc weight w : A → R and matroid valuations ω+ and ω− on

V + and V − respectively, find a matching M(⊆ A) that maximizes∑

w(a) | a ∈ M + ω+(∂+M) +

ω−(∂−M), where ∂+M and ∂−M denote the sets of vertices in V + and V − incident to M . As natural

extensions of the previous results for the independent assignment problem, two optimality criteria are

established; one in terms of potentials and the other in terms of negative cycles in an auxiliary graph.

Key words. weighted matroid intersection problem, independent assignment problem, valuated

matroid, combinatorial optimization

AMS subject classifications. 90C35; 90C27, 90B80

December 1994 – January 1995Revision: August 1995 (Version: November 8, 1996)

Address for correspondence:Kazuo MurotaResearch Institute for Mathematical SciencesKyoto University, Kyoto 606-01, Japanphone: +81-75-753-7221, facsimile: +81-75-753-7272e-mail: murota@kurims.kyoto-u.ac.jp

∗ SIAM J. Discrete Mathematics, Vol.9 (1996), 545–561 (Report No. 95837-OR, Forschungsinstitut

fur Diskrete Mathematik, Universitat Bonn, 1995)† Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-01, Japan, e-mail:

murota@kurims.kyoto-u.ac.jp. This work was done while the author was at Forschungsinstitut fur

Diskrete Mathematik, Universitat Bonn.

1

2 K. MUROTA

1. Introduction. The weighted matroid intersection problem and its extensionshas played a major role in the theory of combinatorial optimization (see for instance,Edmonds [7] [8], Faigle [9], Fujishige [14], Lawler [20]). One of its equivalent variantsintroduced by Iri-Tomizawa [17] is the independent assignment problem defined asfollows. We are given a bipartite graph G = (V +, V −;A), matroids M+ = (V +,B+)and M− = (V −,B−), and a weight function w : A → R, where (V +, V −) is thebipartition of the vertex set of G, A is the arc set of G, M+ (resp., M−) is definedon V + (resp., V −) in terms of the family of bases B+ (resp., B−), and R is a totallyordered additive group (typically R = R (reals), Q (rationals), or Z (integers)). Theindependent assignment problem is to find a matching M(⊆ A) that maximizes

w(M) ≡∑

w(a) | a ∈ M(1.1)

subject to the constraint

∂+M ∈ B+, ∂−M ∈ B−,(1.2)

where ∂+M (resp., ∂−M) denotes the set of vertices in V + (resp., V −) incident toM . The independent assignment problem has been shown to be a useful framework inwhich to formulate engineering problems in systems analysis (see, e.g., Iri [16], Murota[22], Recski [30]).

Recently, on the other hand, Dress-Wenzel [5], [6] introduced the notion of valua-tion on a matroid. A valuation on a matroid M = (V,B) is a function ω : B → R whichenjoys the exchange property: For B,B′ ∈ B and u ∈ B − B′ there exists v ∈ B′ − B

such that B − u + v ∈ B, B′ + u − v ∈ B and

ω(B) + ω(B′) ≤ ω(B − u + v) + ω(B′ + u − v).(1.3)

A matroid equipped with a valuation is called valuated matroid.A valuation ω can be induced from a weight function η : V → R and α ∈ R by

ω(B) = α +∑

η(u) | u ∈ B for B ∈ B.(1.4)

Such a valuation will be called separable (called “essentially trivial” in [6]).A (nonseparable) valuated matroid naturally arises from a polynomial matrix with

coefficients from a field. Let A(x) be an m×n matrix of rank m with each entry beinga polynomial in a variable x, and let M = (V,B) denote the (linear) matroid definedon the column set V of A(x) by the linear independence of the column vectors. Thena valuated matroid is obtained if ω(B) for B ∈ B is defined to be the degree in x ofthe determinant of the m × m submatrix with columns in B, i.e.,

ω(B) = degx detA[B].(1.5)

In fact, the Grassmann-Plucker identity implies the exchange property of ω, as pointedout in Dress-Wenzel [5], [6] in a more algebraic term of “field valuation.” Some

VALUATED MATROID INTERSECTION, I 3

examples of valuated matroids of a more combinatorial-geometrical flavor are reportedin Terhalle [33], whereas Example 3.3 in Section 3 below shows another example arisingfrom graphs. Details on valuated matroids are given in Section 3.

In this paper we consider an extension of the independent assignment problem toits valuated version. Namely we assume that M+ = (V +,B+) and M− = (V −,B−)are equipped with valuations ω+ : B+ → R and ω− : B− → R, and consider theproblem of finding a matching M(⊆ A) that maximizes

Ω(M) ≡ w(M) + ω+(∂+M) + ω−(∂−M)

subject to the constraint (1.2). We shall call this problem the valuated independentassignment problem. This is a proper extension of the independent assignment prob-lem, whereas it obviously reduces to the ordinary independent assignment problem inthe case that ω+ and ω− are trivial valuations that vanish identically on B+ and B−

respectively, and also in a more general case of separable valuations.In the present paper we establish two forms of optimality criteria for the valuated

independent assignment problem by extending in a natural way the two well-knownoptimality criteria for the ordinary independent assignment problem. The first (Theo-rem 4.1) is in terms of potentials, as in Frank [10], and the second (Theorem 4.3) is interms of negative cycles in an auxiliary graph, as in Fujishige [12] (see also Fujishige[14] and Zimmermann [35], which give a similar condition for submodular flows). Thenegative-cycle criterion yields a primal-type cycle-canceling algorithm for solving thevaluated independent assignment problem, to be reported in Part II [25], which isan extension of Fujishige’s [12] and Zimmermann’s [36] for the ordinary independentassignment problem.

The driving wheels for these extensions are the proper generalizations of the fun-damental lemmas on the exchangeability in a matroid to those in a valuated matroid.Among others it should be mentioned that the so-called “no-shortcut lemma” (seeLemma 3.3 in Section 3 for a precise statement) is generalized to what we shall call“unique-max lemma” (Lemma 3.8 in Section 3). When specialized to a valuated ma-troid associated with a polynomial matrix as in (1.5), the “no-shortcut lemma” statesto the effect that a triangular matrix having nonzero diagonal elements is nonsingular,whereas the “unique-max lemma” reveals a stronger property that a square matrixhaving a unique maximum-degree transversal is nonsingular (see Remark 3.2 for moreabout this).

The objective of this paper is twofold. The first is purely theoretical within thefield of combinatorial optimization. As compared with the richness of matroid opti-mizations (greedy algorithm, intersection/union, lexico-optimality, etc., · · ·), not muchis known about valuated-matroid optimizations. All the known results center aroundgreedy procedures for a single valuated matroid (cf. Dress-Wenzel [5], Dress-Terhalle[2], [3], [4], Murota [24]). The present results, along with the algorithms in [25], will

4 K. MUROTA

contribute to the development of the theory of valuated-matroid optimization. Thisline of research is pursued further in the subsequent papers [26], [27], [28]; the optimal-ity criteria are extended to the submodular flow problem in [27], duality theorems areestablished in [26], [28] in relation to convex analysis, and the matroid union operationis extended to valuations in [28].

The second objective is more application-oriented. As explained above, the val-uated matroid is a combinatorial abstraction of polynomial matrices. In view of theprincipal role of polynomial matrices in system engineering (see, e.g., Rosenbrock[31], Vidyasagar [34]) as well as the previous success in application of matroids toit, it is natural to hope for successful applications of valuated matroid to engineer-ing problems. In this connection it should be emphasized that most of the significantapplications of matroid theory have been related, more or less, to the matroid intersec-tion problem. This paper will lay the foundation for future engineering applications.Some applications of the valuated matroid intersection to mixed matrices [22], whichin fact have been the motivation of this paper, are discussed in [29].

VALUATED MATROID INTERSECTION, I 5

2. Problem Formulations. In this section we describe the problem and itsvariants. Suppose we are given a bipartite graph G = (V +, V −;A), valuated matroidsM+ = (V +,B+, ω+) and M− = (V −,B−, ω−), and a weight function w : A → R. Thevaluated independent assignment problem is the following.

[Valuated independent assignment problem]Find a matching M(⊆ A) that maximizes

Ω(M) ≡ w(M) + ω+(∂+M) + ω−(∂−M)(2.1)

subject to the constraint

∂+M ∈ B+, ∂−M ∈ B−.(2.2)

Clearly the two matroids must have the same rank for the feasibility of this problem.It is sometimes convenient to extend the domain of definition of ω+ to 2V +

by simplysetting ω+(B) = −∞ for B ⊆ V + with B 6∈ B+ and similarly for ω−. Then theconstraint (2.2) will be implicit in the objective function Ω(M).

The above problem reduces to the independent assignment problem if the valua-tions are trivial with ω±(B) = 0 for B ∈ B±, and reduces further to the conventionalassignment problem if the matroids are trivial or free with B± = 2V ±

.

Just as the weighted matroid intersection and partition problems may be regardedas special cases of the independent assignment problem, the following three problemsfall into the category of our problem. Suppose now we are given a pair of valuatedmatroids M1 = (V,B1, ω1) and M2 = (V,B2, ω2) defined on a common ground set V ,and a weight function w : V → R.

Intersection problem : Find a common base B ∈ B1 ∩ B2 that maximizesw(B) + ω1(B) + ω2(B). (In case the valuations are separable as (1.4) thisproblem reduces to the usual optimal common base problem.)

Disjoint bases problem : Find disjoint bases B1 and B2 (i.e., B1 ∩ B2 = ∅,B1 ∈ B1 and B2 ∈ B2) that maximize ω1(B1) + ω2(B2).

Partition problem : Find a partition (B, V −B) of V that maximizes ω1(B)+ω2(V − B).

As a matter of course, the disjoint bases problem for more than two valuated matroidscan also be formulated as a valuated independent assignment problem. The partitionproblem is an intersection problem in disguise, since it is the intersection problem forM1 and (M2)∗, the dual of M2, whose valuation is defined by (ω2)∗(B) = ω2(V −B).

Remark 2.1. The valuated independent assignment problem can easily be gen-eralized to an independent linkage-type problem (cf. Fujishige [13], Iri [15]). The un-derlying bipartite graph is replaced with an arbitrary (directed or undirected) graphhaving specified entrance and exit vertex sets, on which valuated matroids are defined,and matchings are replaced by linkings from the entrance to the exit. The optimality

6 K. MUROTA

criteria of the present paper, mutatis mutandis, are easily shown to remain valid forthis linkage-type problem. 2

In the ordinary independent assignment problem the constraint imposed on amatching M is more often that ∂±M be independent in M± than that ∂±M be abase in M± . This motivates us to consider the following problem parametrized by aninteger k:

[VIAP(k)]Maximize

Ω(M,B+, B−) ≡ w(M) + ω+(B+) + ω−(B−)

subject to the constraint that M is a matching of size k, and

∂+M ⊆ B+ ∈ B+, ∂−M ⊆ B− ∈ B−.

In fact, the primal-dual type incremental algorithm of [25] consists of solving thisproblem successively for k = 0, 1, 2, · · ·. The optimality criteria for VIAP(k) arederived in Section 5.

3. Properties of a Valuated Matroid.

3.1. Examples. The first two examples are already mentioned in Introduction.Example 3.1. Let M = (V,B) be a matroid. For η : V → R and α ∈ R,

ω(B) = α +∑

η(u) | u ∈ B (B ∈ B)

is a matroid valuation. Such ω is called a separable valuation. 2

Example 3.2. Let A(x) be an m × n matrix of rank m with each entry being apolynomial (or rational function) in a variable x, and let M = (V,B) denote the (linear)matroid defined on the column set V of A(x) by the linear independence of the columnvectors. Then ω : B → Z defined by ω(B) = degx detA[B] (B ∈ B) is a matroidvaluation (see Dress-Wenzel [6] for the proof), where degx(f/g) = degx f − degx g fortwo polynomials f and g in x. An example of nonseparable valuation of this kind isprovided by

A(x) =

(x + 1 x 1 1

1 1 1 2

).

2

Example 3.3. In addition to the above two constructions that can be foundin the literature [6] we point out here another instance of (nonseparable) valuatedmatroid that arises from the minimum cost of a linking in a graph. Let G = (V ,A) bea directed graph having no self-loops, and S and T be disjoint subsets of the vertexset V . By L we denote (the arc set of) a Menger-type vertex-disjoint linking from S

VALUATED MATROID INTERSECTION, I 7

to T , and by ∂+L the set of its initial vertices (in S); put U = V − (S ∪T ). As is wellknown, B = ∂+L | L ∈ L, where L denotes the family of maximum linkings, definesa matroid M = (S,B). Given a cost function c : A → Z such that every cycle has anonnegative cost,

ω(B) = −min∑a∈L

c(a) | ∂+L = B,L ∈ L (B ∈ B)

is a matroid valuation.To see this first note that, by the max-flow min-cut theorem, we may assume

∂−L = T , where ∂−L designates the set of the terminal vertices of L (in T ). Considera rational function matrix, say A(x) = (Aij(x)), in variable x with the row set indexedby T ∪ U and the column set by S ∪ U defined by

Aij(x) =

1 (i = j ∈ U)αijx

−c(j,i) ((j, i) ∈ A)

where αij | (j, i) ∈ A is an algebraically independent set of real numbers. Thenwe have ω(B) = degx detA[B ∪ U ], which is a version of Example 3.2. An exampleof nonseparable valuation of this kind is provided by G = (V , A) with V = S ∪ T ,S = s1, s2, s3, s4, T = t1, t2, A = (s1, t1), (s2, t2), (s3, t1), (s3, t2), (s4, t1), (s4, t2),c(a) = 0 except for c(s3, t2) = −1 and c(s4, t2) = −2. See also [27, Example 2.3], inwhich a flow-type generalization is given. 2

3.2. Basic properties. This subsection describes some relevant results of Dress-Wenzel [5], [6] on the maximization of a matroid valuation.

Let M = (V,B, ω) be a valuated matroid of rank r. For B ∈ B and v ∈ V −B wedenote by C(B, v) the unique circuit contained in B + v (= the fundamental circuitof v relative to B). For B ∈ B, v ∈ V − B and u ∈ C(B, v) we define

ω(B, u, v) = ω(B − u + v) − ω(B).(3.1)

For convenience we set

ω(B, u, v) = −∞ for u 6∈ C(B, v).

This is also a consequence of our former convention to put ω(B′) = −∞ for B′ 6∈ B.The following lemma is most fundamental, showing the local optimality implies

the global optimality.Lemma 3.1 ([5], [6]). Let B ∈ B. Then ω(B) ≥ ω(B′) for any B′ ∈ B if and

only if

ω(B, u, v) ≤ 0 for any (u, v) with u ∈ C(B, v).(3.2)

8 K. MUROTA

Proof. The original proof is by induction on |B − B′|. An alternative proof isgiven later in Remark 3.1.

For the maximization of ω the greedy algorithm of [5] starts with an arbitrarybase B0 = u1, u2, · · · , ur ∈ B and repeats the following for k = 1, 2, · · · , r: Findvk ∈ V − Bk−1 such that

ω(Bk−1 − uk + vk) ≥ ω(Bk−1 − uk + v) (∀v ∈ V − Bk−1)

and put Bk = Bk−1 − uk + vk. Then Br can be shown to be optimal. In this way anoptimal base (maximizing ω) can be found with r(|V | − r) + 1 function evaluations ofω.

For η : V → R we define ω[η] : B → R (or 2V → R ∪ −∞) by

ω[η](B) = ω(B) +∑

η(u) | u ∈ B.(3.3)

This is again a valuation on M. This operation is called a similarity transformation.A valuation ω is separable (or “essentially trivial” in the terminology of [6]) if andonly if ω[η](B) = constant (∀B ∈ B) for some η : V → R.

M∗ = (V,B∗, ω∗) defined by

B∗ = B | V − B ∈ B, ω∗(B) = ω(V − B)

is again a valuated matroid, called the dual of M = (V,B, ω).

3.3. Further exchange properties. We shall establish a number of lemmasconcerning basis exchanges in a single valuated matroid. They will play the key rolesthroughout this paper.

For B ∈ B and B′ ⊆ V we consider the exchangeability graph, denoted G(B,B′),in the usual sense in matroid theory. Namely, G(B,B′) is a bipartite graph having(B − B′, B′ − B) as the vertex bipartition and (u, v) | u ∈ B − B′, v ∈ B′ − B, u ∈C(B, v) as the arc set. The following fact (Brualdi [1]) is well known in matroidtheory.

Lemma 3.2. Let B ∈ B. If B′ is also a base, then G(B,B′) has a perfectmatching.

The converse of the above statement is not always true. A partial converse is thekey property underlying the (weighted or unweighted) matroid intersection algorithmand is known as “no-shortcut lemma” (for this name we refer to Kung [19]; see Iri-Tomizawa [17, Lemma 2], Krogdahl [18], Lawler [20, Lemma 3.1 of Chap. 8] andSchrijver [32, Theorem 4.3]). It can be stated as follows, in a form suitable for itsextension to a valuated matroid.

Lemma 3.3 (“no-shortcut lemma”). Let B ∈ B and B′ ⊆ V with |B′| = |B|.If there exists exactly one perfect matching in G(B,B′), then B′ ∈ B.

VALUATED MATROID INTERSECTION, I 9

To capture the exchangeability with valuations, we need quantitative extensionsof the above statements. To this end we attach “arc weight” ω(B, u, v) of (3.1) toeach arc (u, v) of G(B,B′) and denote by ω(B,B′) the maximum weight of a perfectmatching in G(B,B′) with respect to the arc weight ω(B, u, v). Lemma 3.2 is extendedas follows.

Lemma 3.4 (“upper-bound lemma”). For B,B′ ∈ B,

ω(B′) ≤ ω(B) + ω(B,B′).(3.4)

Proof. For any u1 ∈ B − B′ there exists v1 ∈ B′ − B with

ω(B) + ω(B′) ≤ ω(B − u1 + v1) + ω(B′ + u1 − v1),

which can be rewritten as

ω(B′) ≤ ω(B, u1, v1) + ω(B′2)

with B′2 = B′ + u1 − v1. By the same argument applied to (B,B′

2) we obtain

ω(B′2) ≤ ω(B, u2, v2) + ω(B′

3)

for some u2 ∈ (B − B′) − u1 and v2 ∈ (B′ − B) − v1, where B′3 = B′

2 + u2 − v2 =B′ − u1, u2 + v1, v2. Hence

ω(B′) ≤ ω(B′3) +

2∑i=1

ω(B, ui, vi).

Repeating this process we arrive at

ω(B′) ≤ ω(B) +m∑

i=1

ω(B, ui, vi) ≤ ω(B) + ω(B,B′),

where m = |B − B′| = |B′ − B|, B − B′ = u1, · · · , um, B′ − B = v1, · · · , vm.Remark 3.1. Lemma 3.4 (“upper-bound lemma”) gives an alternative proof for

the optimality condition given in Lemma 3.1. The necessity of (3.2) is obvious. Forsufficiency take any B′ ∈ B and consider G(B,B′). The condition (3.2) is equivalentto all the arcs having nonpositive weights. Hence ω(B,B′) ≤ 0, which implies ω(B′) ≤ω(B) by Lemma 3.4. 2

In Lemma 3.4 it is natural to ask for a (sufficient) condition under which thebound (3.4) is tight. Comparison of Lemma 3.3 and Lemma 3.4 will suggest

[Unique-Max Condition]There exists exactly one maximum-weight perfect matching in G(B,B′).

10 K. MUROTA

In what follows we shall show that this is indeed a sufficient condition for the tightness(see Lemma 3.8).

First we note the following fact, rephrasing the unique-max condition in terms of“potential” or “dual variable”.

Lemma 3.5. Let B ∈ B and B′ ⊆ V with |B′ − B| = |B − B′| = m.(1) G(B,B′) has a perfect matching if and only if there exist p : (B −B′)∪ (B′ −

B) → R and indexings of the elements of B−B′ and B′−B, say B−B′ = u1, · · · , umand B′ − B = v1, · · · , vm, such that

ω(B, ui, vj) − p(ui) + p(vj)

= 0 (1 ≤ i = j ≤ m)≤ 0 (1 ≤ i, j ≤ m)

(3.5)

(2) The pair (B,B′) satisfies the unique-max condition if and only if there existp : (B −B′)∪ (B′ −B) → R and indexings of the elements of B −B′ and B′ −B, sayB − B′ = u1, · · · , um and B′ − B = v1, · · · , vm, such that

ω(B, ui, vj) − p(ui) + p(vj)

= 0 (1 ≤ i = j ≤ m)≤ 0 (1 ≤ j < i ≤ m)< 0 (1 ≤ i < j ≤ m)

(3.6)

Proof. This is an immediate corollary of the complementary slackness well knownin matching theory (see, e.g., Lawler [20], Lovasz-Plummer [21]). Let M = (ui, vi) |i = 1, · · · ,m be a maximum-weight perfect matching and p be an optimal potential(or dual variable). Then ω(B, u, v) − p(u) + p(v) ≤ 0 for all arcs (u, v). Call an arctight if this inequality holds true with equality and define G∗ to be the subgraph ofG consisting of tight arcs. The complementary slackness says that maximum-weightperfect matchings in G(B,B′) are in one-to-one correspondence with perfect matchingsin G∗.

Lemma 3.6. Let B ∈ B and u, u, v, v be four distinct elements with u, u ⊆ B,v, v ⊆ V −B, and let B′ = B−u, u+v, v. Assume that M = (u, v), (u, v)is the unique maximum-weight perfect matching in G(B,B′).

(1) B′ ∈ B and ω(B′) = ω(B) + ω(B,B′).(2) For B = B − u + v we have

ω(B, u, v) = ω(B, u, v),

ω(B, u, u) = ω(B, u, v) − ω(B, u, v),

ω(B, v, v) = ω(B, u, v) − ω(B, u, v).

Proof. (1) Putting B∗ = B − u + v we see

ω(B∗) + ω(B) = ω(B, u, v) + ω(B, u, v) + 2ω(B) = ω(B,B′) + 2ω(B).(3.7)

VALUATED MATROID INTERSECTION, I 11

By applying the exchange axiom (1.3) to (B, B∗) with u ∈ B − B∗ we have

ω(B∗) + ω(B) ≤ ω(B∗ − v′ + u) + ω(B + v′ − u)

for some v′ ∈ B∗ − B = u, v. Combining this with (3.7) we obtain

ω(B,B′) + 2ω(B) ≤ ω(B∗ − v′ + u) + ω(B + v′ − u).(3.8)

Suppose that v′ = u. Then

RHS of (3.8) = ω(B∗ − u + u) + ω(B + u − u)

= ω(B − u + v) + ω(B − u + v)

= ω(B, u, v) + ω(B, u, v) + 2ω(B).

This means that M ′ = (u, v), (u, v) is also a maximum-weight perfect matchingin G(B,B′), a contradiction to the uniqueness of M .

Therefore we have v′ = v in (3.8), and then

RHS of (3.8) = ω(B∗ − v + u) + ω(B + v − u) = ω(B) + ω(B′).

Hence follows ω(B) + ω(B,B′) ≤ ω(B′). The reverse inequality has already beenshown in the “upper-bound lemma” (Lemma 3.4). Note that B′ ∈ B follows fromω(B′) 6= −∞.

(2) By straightforward calculations as follows:

ω(B, u, v) = ω(B − u + v − u + v) − ω(B − u + v)

= ω(B′) − ω(B) − ω(B, u, v)

= ω(B,B′) − ω(B, u, v)

= ω(B, u, v),

ω(B, u, u) = ω(B − u + v) − ω(B − u + v)

= ω(B, u, v) − ω(B, u, v),

ω(B, v, v) = ω(B − u + v) − ω(B − u + v)

= ω(B, u, v) − ω(B, u, v).

Lemma 3.7. Let B ∈ B and B′ ⊆ V with |B′| = |B|. If there exists exactlyone maximum-weight perfect matching M in G(B,B′), then for any (u, v) ∈ M thefollowing hold true.

(1) B ≡ B − u + v ∈ B.(2) There exists exactly one maximum-weight perfect matching in G(B, B′).(3) ω(B, B′) = ω(B,B′) − ω(B, u, v).Proof. (1) This is obvious.

12 K. MUROTA

(2) Using the notation in Lemma 3.5 we have M = (ui, vi) | i = 1, · · · ,m and(u, v) = (uk, vk) for some k. Put

Bij = B − ui + vj = B − ui, u + vj , v

for i 6= k, j 6= k. It then follows from (3.4) and (3.6) that

ω(B, ui, vj)

= ω(Bij) − ω(B)

≤ ω(B,Bij) − ω(B, uk, vk)

= max (ω(B, uk, vk) + ω(B, ui, vj), ω(B, ui, vk) + ω(B, uk, vj)) − ω(B, uk, vk)

≤ [p(ui) + p(uk) − p(vj) − p(vk)] − [p(uk) − p(vk)]

= p(ui) − p(vj),

where the second inequality is strict for i < j. For i = j, on the other hand, bothinequalities are satisfied with equalities, since G(B,Bii) has a unique maximum-weight perfect matching (ui, vi), (u, v) and Lemma 3.6 implies ω(B, ui, vi) =ω(B, ui, vi) = p(ui) − p(vi). Thus, the potential p for (B,B′) serves as a certificate ofthe unique-max condition also for (B, B′).

(3) ω(B, B′) =∑

i 6=k (p(ui) − p(vi)) = ω(B,B′) − ω(B, u, v).We are now in a position to state the main result of this section, “unique-max

lemma,” which is a quantitative extension of “no-shortcut lemma.”Lemma 3.8 (“unique-max lemma”). Let B ∈ B and B′ ⊆ V with |B′| = |B|. If

there exists exactly one maximum-weight perfect matching in G(B,B′), then B′ ∈ Band

ω(B′) = ω(B) + ω(B,B′).(3.9)

Proof. By induction on m = |B − B′|. The case of m = 1 is obvious. So assumem ≥ 2. Take any (u, v) contained in the unique maximum-weight perfect matching,and put B = B − u + v. (B, B′) satisfies the unique-max condition by Lemma3.7(2), and we have

ω(B′) = ω(B) + ω(B, B′)

by the induction hypothesis. By Lemma 3.7(3) we see

ω(B, B′) = ω(B,B′) − ω(B, u, v)

while ω(B) = ω(B) + ω(B, u, v) by definition. Hence follows (3.9).Remark 3.2. Some remark is in order on the relation between “no-shortcut con-

dition” (= uniqueness of a perfect matching in G(B,B′)) and “unique-max condition”

VALUATED MATROID INTERSECTION, I 13

(= uniqueness of the maximum-weight perfect matching in G(B,B′)). Obviously theformer implies the latter, and not conversely in general. For a separable valuation (cf.(1.4) and Section 3.2), however, these two conditions are equivalent, and consequently“unique-max lemma” reduces to “no-shortcut lemma.” See also Frank [10, Lemma 2]in this connection. 2

Remark 3.3. An alternative proof of “unique-max lemma” was suggested byAndras Sebo after the submission of the first draft. This proof makes use of “no-shortcut lemma” in contrast to the above proof. Let p : (B − B′) ∪ (B′ − B) → R beas in Lemma 3.5 and extend it to p : V → R by defining p(u) = +M for u ∈ B ∩ B′

and p(v) = −M for v ∈ V − (B ∪ B′) with a sufficiently large M > 0. It follows fromthe exchange property (1.3) that the family of the maximizers of ω[p]:

B = B ∈ B | ω[p](B) ≥ ω[p](B′′) (B′′ ∈ B)

forms the basis family of a matroid, say M = (V,B). We claim that B ∈ B.To see this, first note that ω[p](B′′) − ω[p](B) ≤ ω(B′′) − ω(B) − M ≤ 0 unlessB ∩ B′ ⊆ B′′ ⊆ B ∪ B′. If B ∩ B′ ⊆ B′′ ⊆ B ∪ B′, on the other hand, we haveω[p](B′′) − ω[p](B) ≤ 0 by “upper-bound lemma” and the inequality

ω[p](B, u, v) = ω(B, u, v) − p(u) + p(v) ≤ 0 (u ∈ B − B′′, v ∈ B′′ − B).

We also claim that the exchangeability graph G(B,B′) in M has a unique perfectmatching, since B − ui + vi ∈ B (1 ≤ i ≤ m) and B − ui + vj 6∈ B (1 ≤ i < j ≤ m)by (3.6). By applying “no-shortcut lemma” to the given pair (B,B′) in the matroidM = (V,B), we obtain B′ ∈ B, which means ω[p](B′) = ω[p](B), i.e., ω(B′) =ω(B) +

∑mi=1 p(ui) −

∑mi=1 p(vi) = ω(B) + ω(B,B′). 2

The following lemma is used in Part II in justifying a variant of the cycle-cancelingalgorithm.

Lemma 3.9. Under the same assumption as in Lemma 3.8, let p, ui and vj be asin Lemma 3.5. Then

ω(B′, vj , ui) ≤ p(vj) − p(ui) (1 ≤ i, j ≤ m).

Proof. Putting B′ij = B′ − vj + ui and using Lemma 3.4, Lemma 3.8 and (3.6) we

see

ω(B′, vj , ui) = ω(B′ij) − ω(B′) ≤ ω(B,B′

ij) − ω(B,B′)

≤

∑k 6=i

p(uk) −∑k 6=j

p(vk)

−[

m∑k=1

p(uk) −m∑

k=1

p(vk)

]= p(vj) − p(ui).

14 K. MUROTA

4. Optimality Criteria.

4.1. Theorems. Two optimality criteria are given for the valuated indepen-dent assignment problem (2.1)–(2.2) on G = (V +, V −;A) with valuated matroidsM+ = (V +,B+, ω+), M− = (V −,B−, ω−), and weight function w : A → R, where R

is a totally ordered additive group (e.g., R = R, Q, or Z). Both of these criteria arenatural extensions of the corresponding results for the ordinary independent assign-ment problem, which have been extended also for the submodular flow problem (seeFrank [10] [11], Fujishige [12] [14], Zimmermann [35]). The proofs are postponed toSection 4.2.

The first theorem refers to a “potential” function. It may be emphasized that inthe case of R = Z the integrality of p is a part of the assertion.

Theorem 4.1. (1) An independent assignment M in G is optimal for the valuatedindependent assignment problem (2.1)–(2.2) if and only if there exists a “potential”function p : V + ∪ V − → R such that

(i) w(a) − p(∂+a) + p(∂−a)

≤ 0 (a ∈ A)= 0 (a ∈ M)

(ii) ∂+M is a maximum-weight base of M+ with respect to ω+[p+],(iii) ∂−M is a maximum-weight base of M− with respect to ω−[−p−],

where p± is the restriction of p to V ± and ω+[p+] (resp., ω−[−p−]) is the similaritytransformation defined in (3.3); namely,

ω+[p+](B+) = ω+(B+) +∑

p(u) | u ∈ B+ (B+ ⊆ V +),

ω−[−p−](B−) = ω−(B−) −∑

p(u) | u ∈ B− (B− ⊆ V −).

(2) Let p be a potential that satisfies (i)–(iii) above for some (optimal) independentassignment M = M0. An independent assignment M ′ is optimal if and only if itsatisfies (i)–(iii) (with M replaced by M ′).

The optimality condition for the intersection problem deserves a separate state-ment, in a form of Frank’s weight splitting [10], though it is an immediate corol-lary of the above theorem. Recall that the intersection problem is to maximizew(B) + ω1(B) + ω2(B) for a pair of valuated matroids M1 = (V,B1, ω1) and M2 =(V,B2, ω2) and a weight function w : V → R.

Theorem 4.2. A common base B of M1 = (V,B1, ω1) and M2 = (V,B2, ω2)maximizes w(B) + ω1(B) + ω2(B) if and only if there exist w1, w2 : V → R such that

(i) [“weight splitting”] w(v) = w1(v) + w2(v) (v ∈ V ),(ii) B is a maximum-weight base of M1 with respect to ω1[w1],(iii) B is a maximum-weight base of M2 with respect to ω2[w2],

where ω1[w1] (resp., ω2[w2]) is the similarity transformation defined in (3.3).

To describe the second criterion we need to introduce an auxiliary graph GM =(V , A) associated with an independent assignment M . We put B+ = ∂+M , B− =

VALUATED MATROID INTERSECTION, I 15

∂−M and denote by C±(·, ·) the fundamental circuit in M±. The vertex set V of GM

is given by V = V + ∪ V − and the arc set A consists of four disjoint parts:

A = A ∪ M ∪ A+ ∪ A−,

where

A = a | a ∈ A (copy of A),

M = a | a ∈ M (a: reorientation of a),

A+ = (u, v) | u ∈ B+, v ∈ V + − B+, u ∈ C+(B+, v),

A− = (v, u) | u ∈ B−, v ∈ V − − B−, u ∈ C−(B−, v).

In addition, arc length γM (a) (a ∈ A) is defined by

γM (a) =

−w(a) (a ∈ A)w(a) (a = (u, v) ∈ M, a = (v, u) ∈ M)−ω+(B+, u, v) (a = (u, v) ∈ A+)−ω−(B−, u, v) (a = (v, u) ∈ A−)

(4.1)

where ω+(B+, u, v) and ω−(B−, u, v) are defined as in (3.1). We call a directed cycleof negative length a negative cycle.

Theorem 4.3. An independent assignment M in G is optimal for the valuated in-dependent assignment problem (2.1)–(2.2) if and only if there exists in GM no negativecycle with respect to the arc length γM .

Remark 4.1. The exchangeability graphs G(B+, V +−B+) for M+ and G(B−, V −−B−) for M− introduced in Section 3.3 are embedded in GM . Namely, they can beidentified with the subgraphs (V +, A+) and (V −, A−) respectively; note, however, thatthe arc weight is the negative of the arc length. 2

Remark 4.2. The optimality criterion in Theorem 4.2 can be reformulated as aFenchel-type duality between the matroid valuations and their conjugate functions, asreported in [26]. It is also mentioned that Theorem 4.2 is extended for the submodularflow problem in [27]. 2

4.2. Proofs. We are to prove the equivalence of the following three conditionsfor an independent assignment M :

(OPT) M is optimal.(NNC) There is no negative cycle in GM .(POT) There exists a potential p with (i)–(iii) in Theorem 4.1.

We prove (OPT) ⇒ (NNC) ⇒ (POT) ⇒ (OPT) and finally the second part of Theorem4.1. We abbreviate γM to γ whenever convenient.

(OPT) ⇒ (NNC): Suppose GM has a negative cycle. Let Q (⊆ A) be the arc setof a negative cycle having the smallest number of arcs, and put

B+ = B+ − ∂+a | a ∈ Q ∩ A+ + ∂−a | a ∈ Q ∩ A+,(4.2)

16 K. MUROTA

B− = B− − ∂−a | a ∈ Q ∩ A− + ∂+a | a ∈ Q ∩ A−,(4.3)

where B+ = ∂+M and B− = ∂−M as before.Lemma 4.4. (B+, B

+) and (B−, B−) satisfy the unique-max condition in M+

and M− respectively.Proof. We prove the claim for (B+, B

+) by adapting Fujishige’s proof techniquedeveloped in [12] (which can be found also in [14, Lemma 5.4]).

Take a maximum-weight perfect matching M ′ = (ui, vi) | i = 1, · · · ,m (wherem = |B+ − B

+|) in the exchangeability graph G(B+, B+) for M+ as well as the

potential function p in Lemma 3.5. Then M ′ is a subset of

A∗ = (u, v) | u ∈ B+ − B+, v ∈ B

+ − B+, ω+(B+, u, v) − p(u) + p(v) = 0.

Put Q′ = (Q − A+) ∪ M ′, where M ′ is now regarded as a subset of A+ as in Remark4.1. Q′ is a disjoint union of cycles in GM with its length

γ(Q′) = γ(Q) + [γ(M ′) − γ(Q ∩ A+)](4.4)

being negative, since −γ(M ′) is equal to the maximum weight of a perfect matchingin G(B+, B

+) and Q ∩ A+ can be identified with a perfect matching in G(B+, B+).

The minimality of Q (with respect to the number of arcs) implies that Q′ itself is anegative cycle having the smallest number of arcs.

Suppose, to the contrary, that (B+, B+) does not satisfy the unique-max condi-

tion. Since (ui, vi) ∈ A∗ for i = 1, · · · ,m, it follows from Lemma 3.5 that there aredistinct indices ik (k = 1, · · · , q; q ≥ 2) such that (uik , vik+1

) ∈ A∗ for k = 1, · · · , q,where iq+1 = i1. That is,

ω+(B+, uik , vik+1) = p(uik) − p(vik+1

) (k = 1, · · · , q).(4.5)

On the other hand we have

ω+(B+, uik , vik) = p(uik) − p(vik) (k = 1, · · · , q).(4.6)

It then follows thatq∑

k=1

ω+(B+, uik , vik+1) =

q∑k=1

ω+(B+, uik , vik)

(=

q∑k=1

[p(uik) − p(vik)]

)

i.e.,

q∑k=1

γ(uik , vik+1) =

q∑k=1

γ(uik , vik).(4.7)

For k = 1, · · · , q, let P ′(vik+1, uik) denote the path on Q′ from vik+1

to uik , and letQ′

k be the directed cycle formed by arc (uik , vik+1) and path P ′(vik+1

, uik). Obviously,

γ(Q′k) = γ(uik , vik+1

) + γ(P ′(vik+1, uik)) (k = 1, · · · , q).(4.8)

VALUATED MATROID INTERSECTION, I 17

A simple but crucial observation here is that( q⋃k=1

P ′(vik+1, uik)

)∪ (uik , vik) | k = 1, · · · , q = q′ · Q′

for some q′ with 1 ≤ q′ < q, where the union denotes the multiset union, and thisexpression means that each element of Q′ appears q′ times on the left hand side. Henceby adding (4.8) over k = 1, · · · , q we obtain

q∑k=1

γ(Q′k) =

q∑k=1

γ(uik , vik+1) +

q∑k=1

γ(P ′(vik+1, uik))

=

[ q∑k=1

γ(uik , vik+1) −

q∑k=1

γ(uik , vik)

]+ q′ · γ(Q′)

= q′ · γ(Q′) < 0,

where the last equality is due to (4.7). This implies that γ(Q′k) < 0 for some k, while

Q′k has a smaller number of arcs than Q′. This contradicts the minimality of Q′.

Therefore (B+, B+) satisfies the unique-max condition. Similarly for (B−, B

−).Lemma 4.5. For a negative cycle Q in GM having the smallest number of arcs,

M = (M − a ∈ M | a ∈ Q ∩ M) ∪ (Q ∩ A)

is an independent assignment with Ω(M) ≥ Ω(M) − γM (Q) (> Ω(M)).Proof. Note that B

+ = ∂+M , B− = ∂−M for B

+, B− defined in (4.2), (4.3),

and recall the notation B+ = ∂+M , B− = ∂−M . By Lemma 4.4 and Lemma 3.8(“unique-max lemma”) we have

ω+(B+) = ω+(B+) + ω+(B+, B+) ≥ ω+(B+) − γ(Q ∩ A+),

ω−(B−) = ω−(B−) + ω−(B−, B−) ≥ ω−(B−) − γ(Q ∩ A−).

Also we have

w(M) = w(M) − γ(Q ∩ (A ∪ M)).

Addition of these inequalities yields Ω(M) ≥ Ω(M) − γ(Q).The above lemma shows “(OPT) ⇒ (NNC)”.(NNC) ⇒ (POT): By the well-known fact in graph theory, (NNC) implies the

existence of a function p : V + ∪ V − → R such that

γ(a) + p(∂+a) − p(∂−a) ≥ 0 (a ∈ A).

This condition for a ∈ A ∪M is equivalent to the condition (i) in Theorem 4.1. Fora = (u, v) ∈ A+, where u ∈ C+(B+, v), it means

ω+(B+, u, v) − p(u) + p(v) ≤ 0.

18 K. MUROTA

Namely,

ω+[p+](B+, u, v) ≤ 0 (u ∈ C+(B+, v)),

which in turn implies the condition (ii) by Lemma 3.1. Similarly, the above conditionfor a ∈ A− implies the condition (iii). Thus “(NNC) ⇒ (POT)” has been shown.

(POT) ⇒ (OPT): For any independent assignment M and any function p : V + ∪V − → R we see

Ω(M) = ω+(∂+M) + ω−(∂−M) + w(M)

=

[ω+(∂+M) +

∑a∈M

p(∂+a)

]+

[ω−(∂−M) −

∑a∈M

p(∂−a)

]+

∑a∈M

[w(a) − p(∂+a) + p(∂−a)]

= ω+[p+](∂+M) + ω−[−p−](∂−M) +∑a∈M

wp(a),

where wp(a) = w(a) − p(∂+a) + p(∂−a).Suppose M and p satisfy (i)–(iii) of Theorem 4.1, and take an arbitrary indepen-

dent assignment M ′. Then we have

Ω(M ′) = ω+[p+](∂+M ′) + ω−[−p−](∂−M ′) +∑

a∈M ′

wp(a)

≤ ω+[p+](∂+M) + ω−[−p−](∂−M)

= Ω(M).

This shows that M is optimal, establishing “(POT) ⇒ (OPT)”.Finally for the second half of Theorem 4.1 we note in the above inequality that

Ω(M ′) = Ω(M) if and only if ω+[p+](∂+M ′) = ω+[p+](∂+M), ω−[−p−](∂−M ′) =ω−[−p−](∂−M), wp(a) = 0 for a ∈ M ′.

We have completed the proofs of Theorem 4.1 and Theorem 4.3.

VALUATED MATROID INTERSECTION, I 19

5. Extension to VIAP(k).

5.1. Theorems. In this section the optimality criteria for VIAP(k) introducedat the end of Section 2 are derived from Theorem 4.1 and Theorem 4.3. The proofsare given in Section 5.2.

Theorem 5.1. (1) A feasible solution (M,B+, B−) for VIAP(k) is optimal ifand only if there exists a “potential” function p : V + ∪ V − → R such that

(i) w(a) − p(∂+a) + p(∂−a)

≤ 0 (a ∈ A)= 0 (a ∈ M)

(ii) B+ is a maximum-weight base of M+ with respect to ω+[p+],(iii) B− is a maximum-weight base of M− with respect to ω−[−p−],(iv) p(u) ≥ p(v) (u ∈ V +, v ∈ B+ − ∂+M),(v) p(u) ≤ p(v) (u ∈ V −, v ∈ B− − ∂−M).(2) Let p be a potential that satisfies (i)–(v) above for some (optimal) (M0, B

+0 , B−

0 ).Then (M,B+, B−) is optimal if and only if it satisfies (i)–(v).

To express the optimality in terms of negative cycles we need to introduce anauxiliary graph G(M,B+,B−) = (V , A) associated with (M,B+, B−), which is a slightmodification of the one used in Section 4. The vertex set V of G(M,B+,B−) is given by

V = V + ∪ V − ∪ s+, s−,

where s+ and s− are new vertices referred to as the source vertex and the sink vertexrespectively. The arc set A consists of eight disjoint parts:

A = (A ∪ M) ∪ (A+ ∪ F+ ∪ S+) ∪ (A− ∪ F− ∪ S−),

where

A = a | a ∈ A (copy of A),

M = a | a ∈ M (a: reorientation of a),

A+ = (u, v) | u ∈ B+, v ∈ V + − B+, u ∈ C+(B+, v),

F+ = (u, s+) | u ∈ V +,

S+ = (s+, v) | v ∈ B+ − ∂+M,

A− = (v, u) | u ∈ B−, v ∈ V − − B−, u ∈ C−(B−, v),

F− = (s−, u) | u ∈ V −,

S− = (v, s−) | v ∈ B− − ∂−M.

The arc length γ(a) = γ(M,B+,B−)(a) (a ∈ A) is defined by

γ(a) =

−w(a) (a ∈ A)w(a) (a = (u, v) ∈ M, a = (v, u) ∈ M)−ω+(B+, u, v) (a = (u, v) ∈ A+)−ω−(B−, u, v) (a = (v, u) ∈ A−)0 (a ∈ F+ ∪ S+ ∪ F− ∪ S−)

(5.1)

20 K. MUROTA

Theorem 5.2. A feasible solution (M,B+, B−) for VIAP(k) is optimal if andonly if there exists in G(M,B+,B−) no negative cycle with respect to the arc lengthγ(M,B+,B−).

Remark 5.1. The definition of F± could be replaced by

F+ = (u, s+) | u ∈ ∂+M ∪ (V + − B+), F− = (s−, u) | u ∈ ∂−M ∪ (V − − B−)

without affecting the above theorem. The present definition is more convenient forthe algorithm to be developed in Part II. 2

Remark 5.2. When k = r+ = r−, the auxiliary graph G(M,B+,B−) contains theauxiliary graph GM of Section 4 as a subgraph. 2

5.2. Proofs. We formulate VIAP(k) as a valuated independent assignment prob-lem on Gk = (V +

k , V −k ; Ak) with valuated matroids M+

k = (V +k ,B+

k , ω+k ) and M−

k =(V −

k ,B−k , ω−

k ) having a common rank r+ + r− − k. The graph Gk = (V +k , V −

k ;Ak) isdefined as follows:

V +k = V + ∪ U+

k , U+k ≡ u+

i | 1 ≤ i ≤ r− − k,

V −k = V − ∪ U−

k , U−k ≡ u−

i | 1 ≤ i ≤ r+ − k,

Ak = A ∪ (u, u−i ) | u ∈ V +, u−

i ∈ U−k ∪ (u+

i , u) | u ∈ V −, u+i ∈ U+

k .

The valuated matroid M+k is the direct sum of M+ and the free matroid on U+

k withtrivial valuation (which is zero), i.e., B+

k = B ∪ U+k | B ∈ B+ and ω+

k (B ∪ U+k ) =

ω+(B) for B ∈ B+. Similarly for M−k . The weight wk : Ak → R is defined by

wk(a) =

w(a) (a ∈ A)0 (a ∈ Ak − A)

With an independent assignment Mk in Gk we can associate a feasible solution(M,B+, B−) for VIAP(k) by defining M = Mk ∩ A, B+ = ∂+Mk − U+

k and B− =∂−Mk − U−

k . Conversely, from (M,B+, B−) feasible for VIAP(k) we can constructan independent assignment Mk in Gk. Moreover, we have Ω(M,B+, B−) = Ωk(Mk),where Ωk(Mk) ≡ wk(Mk) + ω+

k (∂+Mk) + ω−k (∂−Mk).

Through this reduction of VIAP(k) to the valuated independent assignment prob-lem Theorem 4.1 and Theorem 4.3 translate into Theorem 5.1 and Theorem 5.2, re-spectively.

Acknowledgements. The author is grateful to Andreas Dress for a stimulatingcomment on [23] on the occasion of the 15th International Symposium on Mathemat-ical Programming at Ann Arbor, August 1994, which motivated the present work. Healso thanks Satoru Iwata for careful reading of the manuscript and for fruitful discus-sion, which resulted in the extension given in Section 5. The discussion with AndrasSebo was also fruitful, leading to Remark 3.3.

VALUATED MATROID INTERSECTION, I 21

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