Valuated Matroid Intersection I: Optimality Criteria

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.

Gross substitutability: An algorithmic survey

“…Nisan and Segal [36] propose a solution that explores properties of gross substitutes to build a suitable linear program. Finally, Murota [32,33] gives a strongly polynomial time algorithm for this problem based on a cycle-canceling approach.…”

“…Lemma 9.3 (Murota [32]). If the allocation is optimal, the exchange graph has no negative cycles and therefore, the shortest-path distance is well defined.…”

“…Cycle-canceling algorithms. Finally we describe a purely combinatorial approach proposed by Murota [32,33] based on the Klein's cycle-canceling technique [24] for the minimum-cost flow problem. This approach has the advantage that it leads to a strongly polynomial time algorithm.…”

Computing Walrasian Equilibria: Fast Algorithms and Structural Properties

“…The version of this theorem for gross substitutes is due to Murota [Mur96b,Mur96c] and it was originally proved in the context of valuated matroids, which are known to be equivalent to gross substitutes under a certain transformation. We refer the reader to Lemma 10.1 in [PL] for a proof of this lemma in the language of gross substitute valuations: b 1 ), (a 2 , b 2 ), .…”

“…In a sequence of two foundational papers [Mur96b,Mur96c], Murota shows that the assigment problem for valuated matroids, a class of functions introduced by Dress and Wenzel [DW90] can be solved in strongly polynomial time. We show how this algorithm can be used to obtain anÕ(nm + n 3 ) strongly polynomial time algorithm for problem of computing a Walrasian equilibrium for gross substitute valuations.…”