Optimization Problems with Color-Induced Budget Constraints

Abstract

Gabow and Tarjan [9] provided a very elegant and fast algorithm for the following problem: given a matroid defined on a red and blue colored ground set, determine a basis of minimum cost among those with k red elements, or decide that no such basis exists. In this paper, we investigate possible extensions of this result from ordinary matroids to the more general notion of poset matroids. Poset matroids (also called distributive supermatroids) are defined on the collection of all ideals of an underlying partial order on the ground set. We show that the problem on general poset matroids becomes NP-hard, already if the underlying poset consists of binary trees of height two. On the positive side, we present two polynomial algorithms: one for integer polymatroids, i.e., the case where the poset consists of disjoint chains, and one for the problem to determine a minimum cost ideal of size l with k red elements, i.e., the uniform rank-l poset matroid, on series-parallel posets.