Weighted Matroid Intersection Problem Research Articles

Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity Problems

Spanning trees are a representative example of linear matroid bases that are efficiently countable. Perfect matchings of Pfaffian bipartite graphs are a countable example of common bases of two matrices. Generalizing these two, Webb Counting Bases, Ph.D. thesis, University of Waterloo, 2004 introduced the notion of Pfaffian pairs as a pair of matrices for which counting of their common bases is tractable via the Cauchy--Binet formula. This paper studies counting on linear matroid problems extending Webb's work. We first introduce “Pfaffian parities” as an extension of Pfaffian pairs to the linear matroid parity problem, which is a common generalization of the linear matroid intersection problem and the matching problem. We show that a large number of efficiently countable discrete structures are interpretable as special cases of Pfaffian pairs and parities. Our study then turns to algorithmic aspects. We observe that the fastest randomized algorithms for the linear matroid intersection and parity problems by Harvey SIAM J. Comput., 39 (2009), pp. 679--702 and Cheung, Lau, and Leung ACM Trans. Algorithms, 10 (2014), pp. 1--26 can be derandomized for Pfaffian pairs and parities. We further present polynomial-time algorithms to count the number of minimum-weight solutions on weighted Pfaffian pairs and parities. Our algorithms make use of Frank's weight-splitting lemma for the weighted matroid intersection problem and the algebraic optimality criterion of the weighted linear matroid parity problem given by Iwata and Kobayashi SIAM J. Comput., 51 (2002) pp. STOC17-238--STOC17-280.

Exact and approximation algorithms for weighted matroid intersection

In this paper, we propose new exact and approximation algorithms for the weighted matroid intersection problem. Our exact algorithm is faster than previous algorithms when the largest weight is relatively small. Our approximation algorithm delivers a $$(1-\epsilon )$$-approximate solution with a running time significantly faster than most known exact algorithms. The core of our algorithms is a decomposition technique: we decompose an instance of the weighted matroid intersection problem into a set of instances of the unweighted matroid intersection problem. The computational advantage of this approach is that we can make use of fast unweighted matroid intersection algorithms as a black box for designing algorithms. More precisely, we show that we can solve the weighted matroid intersection problem via solving W instances of the unweighted matroid intersection problem, where W is the largest given weight, assuming that all given weights are integral. Furthermore, we can find a $$(1-\epsilon )$$-approximate solution via solving $$O(\epsilon ^{-1} \log r)$$ instances of the unweighted matroid intersection problem, where r is the smaller rank of the two given matroids. Our algorithms make use of the weight-splitting approach of Frank (J Algorithms 2(4):328–336, 1981) and the geometric scaling scheme of Duan and Pettie (J ACM 61(1):1, 2014). Our algorithms are simple and flexible: they can be adapted to special cases of the weighted matroid intersection problem, using specialized unweighted matroid intersection algorithms. In addition, we give a further application of our decomposition technique: we solve efficiently the rank-maximal matroid intersection problem, a problem motivated by matching problems under preferences.

Shortest bibranchings and valuated matroid intersection

For a digraph D = (V, A) and a partition {S, T} of V, an arc set \({B \subseteq A}\) is called an S−T if each vertex in T is reachable from S and each vertex in S reaches T in the subgraph (V, B). Bibranchings commonly generalize bipartite edge covers and arborescences. A totally dual integral linear system determining the S−T polytope is provided by Schrijver, and the shortest S−T problem, whose objective is to find an S−T of minimum total arc weight, can be solved in polynomial time by the ellipsoid method or a faster combinatorial algorithm due to Keijsper and Pendavingh. The valuated matroid intersection problem, introduced by Murota, is a weighted generalization of the independent matching problem, including the independent assignment problem and the weighted matroid intersection problem. The valuated matroid intersection problem can be solved efficiently with polynomially many value oracles by extending classical combinatorial algorithms for the weighted matroid intersection problem. In this paper, we show that the shortest S−T problem is polynomially reducible to the valuated matroid intersection problem. This reduction suggests one answer to why the shortest S−T problem is tractable, and implies new combinatorial algorithms for the shortest S−T problem based on the valuated matroid intersection algorithm, where a value oracle corresponds to computing a minimum-weight arborescence.

Covering directed graphs by in-trees

Given a directed graph D=(V,A) with a set of d specified vertices S={s 1,?,s d }?V and a function f : S?? where ? denotes the set of positive integers, we consider the problem which asks whether there exist ? i=1 d f(s i ) in-trees denoted by $T_{i,1},T_{i,2},\ldots,T_{i,f(s_{i})}$ for every i=1,?,d such that $T_{i,1},\ldots,T_{i,f(s_{i})}$ are rooted at s i , each T i,j spans vertices from which s i is reachable and the union of all arc sets of T i,j for i=1,?,d and j=1,?,f(s i ) covers A. In this paper, we prove that such set of in-trees covering A can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in ? i=1 d f(s i ) and the size of D. Furthermore, for the case where D is acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.

Facets of the independent path-matching polytope

Cunningham and Geelen introduced the independent path-matching problem as a common generalization of the weighted matching problem and the weighted matroid intersection problem. Associated with an independent path-matching is an independent path-matching vector. The independent path-matching polytope of an instance of the independent path-matching problem is the convex hull of all the independent path-matching vectors. Cunningham and Geelen described a system of linear inequalities defining the independent path-matching polytope. In this paper, we characterize which inequalities in this system induce facets of the independent path-matching polytope, generalizing previous results on the matching polytope and the common independent set polytope.

Fenchel-type duality for matroid valuations

The weighted matroid intersection problem has recently been extended to the valuated matroid intersection problem: Given a pair of valuated matroidsMi = (V, ℬi, ωi) (i = 1,2) defined on a common ground setV, find a common baseB ∈ ℬ1 ∩ ℬ2 that maximizesω1(B) + ω2(B). This paper develops a Fenchel-type duality theory related to this problem with a view to establishing a convexity framework for nonlinear integer programming. A Fenchel-type min max theorem and a discrete separation theorem are given. Furthermore, the subdifferentials of matroid valuations are investigated. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

The optimal path-matching problem

We describe a common generalization of the weighted matching problem and the weighted matroid intersection problem. In this context we establish common generalizations of the main results on those two problems—polynomial-time solvability, min-max theorems, and totally dual integral polyhedral descriptions. New application of these results include a strongly polynomial separation algorithm for the convex hull of matchable sets of a graph, and a polynomial-time algorithm to compute the rank of a certain matrix of indeterminates.

Valuated Matroid Intersection I: Optimality Criteria

The independent assignment problem (or the weighted matroid intersection problem) is extended using Dress and 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 as follows: given a bipartite graph $G = (V^ + ,V^ - ;A)$ with arc weight $w:A \to \mathbf{R}$ and matroid valuations $\omega ^ + $ and $\omega^ - $ on $V^ + $ and $V^ - $ respectively, find a matching $M( \subseteq A)$ that maximizes $\sum \{ \omega (a) \mid a \in M\} + \omega^ + (\partial ^ + M) + \omega^ - (\partial ^ - M)$, where $\partial ^ + M$ and $\partial ^ - M$ 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.

Valuated Matroid Intersection II: Algorithms

Based on the optimality criteria established in part I [SIAM J. Discrete Math., 9 (1996), pp. 545–561] we show a primal-type cycle-canceling algorithm and a primal–dual-type augmenting algorithm for the valuated independent assignment problem: given a bipartite graph $G = (V^ + ,V^ - ;A)$ with arc weight $w:A \to \mathbf{R}$ and matroid valuations $\omega^ + $ and $\omega ^ - $ on $V^ + $ and $V^ - $, respectively; find a matching $M( \subseteq A)$ that maximizes $\sum \{ w(a)\mid a \in M\} + \omega^ + (\partial ^ + M) + \omega^ - (\partial ^ - M)$, where $\partial ^ + M$ and $\partial ^ - M$ denote the sets of vertices in $V^ + $ and $V^ - $ incident to M. The proposed algorithms generalize the previous algorithms for the independent assignment problem as well as for the weighted matroid intersection problem, including those due to Lawler [Math. Prog., 9 (1975), pp. 31–56], Ini and Tomizawa [J. Oper. Res. Soc. Japan, 19 (1976), pp. 32–57], Fujishige [J. Oper. Res. Soc. Japan, 20 (1977), pp. 1–15], Frank [J. Algorithms, 2 (1981), pp. 328–336], and Zimmermann [Discrete Appl. Math., 36 (1992), pp. 179–189].

AN EFFICIENT COST SCALING ALGORITHM FOR THE INDEPENDENT ASSIGNMENT PROBLEM

An efficient cost scaling algorithm is presented for the independent assignment problem of Iri and Tomizawa, which is equivalent to the weighted matroid intersection problem of Edmonds. Our algorithm in general can be viewed as a generalization of Orlin and Ahuja's scaling algorithm for the ordinary assignment problem. On a bipartite graph with n vertices and integer arc costs bounded by C, an optimal r-independent assignment can be found in O(√<rgt:n^2 log(rC)) time by our algorithm under an independence oracle for matorids.

Two algorithms for weighted matroid intersection

Consider a finite setE, a weight functionw:EźR, and two matroidsM1 andM2 defined onE. The weighted matroid intersection problem consists of finding a setI⊆E, independent in both matroids, that maximizes Σ{w(e):e inI}. We present an algorithm of complexity O(nr(r+c+logn)) for this problem, wheren=|E|,r=min(rank(M1), rank (M2)),c=max (c1,c2) and, fori=1,2,ci is the complexity of finding the circuit ofIź{e} inMi (or show that none exists) wheree is inE andI⊆E is independent inM1 andM2. A related problem is to find a maximum weight set, independent in both matroids, and of given cardinalityk (if one exists). Our algorithm also solves this problem. In addition, we present a second algorithm that, given a feasible solution of cardinalityk, finds an optimal one of the same cardinality. A sensitivity analysis on the weights is easy to perform using this approach. Our two algorithms are related to existing algorithms. In fact, our framework provides new simple proofs of their validity. Other contributions of this paper are the existence of nonnegative reduced weights (Theorem 6), allowing the improved complexity bound, and the introduction of artificial elements, allowing an improved start and flexibility in the implementation of the algorithms.

Matroid optimization with the interleaving of two ordered sets

We study the structure of the minimum weight base of a matroid M = (E, I) the order of whose element set E is determined by the interleaving of two ordered subsets of E, R and W. The results imply an interesting application in economics, and are useful for the rapid recomputation of the minimum weight base when the order of E is successively modified by changing the interleaving of R and W. As a special case of the main result, the following parametric problem is efficiently solved: For M = (E, I) a matroid with weighted element set E, and R a subset of E, find for all feasible values of q, the minimum weight base of M containing exactly q elements of R. This parametric problem is a weighted matroid intersection problem and hence can be solved by known matroid intersection algorithms. The approach in this paper is different, and vastly improves the efficiency of the solution, as well as determining structural information about the bases.