Reducing the rank of a matroid

Abstract

We consider the rank reduction problem for matroids: Given a matroid M and an integer k, find a minimum size subset of elements of M whose removal reduces the rank of M by at least k. When M is a graphical matroid this problem is the minimum k-cut problem, which admits a 2-approximation algorithm. In this paper we show that the rank reduction problem for transversal matroids is essentially at least as hard to approximate as the densest k-subgraph problem. We also prove that, while the problem is easily solvable in polynomial time for partition matroids, it is NPhard when considering the intersection of two partition matroids. Our proof shows, in particular, that the maximum vertex cover problem is NP-hard on bipartite graphs, which answers an open problem of B. Simeone.