Valuated Matroid Intersection I: Optimality Criteria

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.