Reducing the rank of a matroid

TL;DR

This paper investigates the computational complexity of the rank reduction problem in various matroid classes, revealing its hardness in transversal and intersection of partition matroids, and providing polynomial solutions for partition matroids.

Contribution

It establishes the hardness of rank reduction for transversal and intersecting partition matroids, and offers polynomial algorithms for partition matroids, connecting matroid problems to classical NP-hard problems.

Findings

Rank reduction in transversal matroids is as hard as densest k-subgraph.

NP-hardness of maximum vertex cover on bipartite graphs.

Polynomial-time solution for partition matroids.

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 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.