  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 NP-hard 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.
