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arXiv · 1211.4853

Reducing the rank of a matroid

Gwenaël Joret·Adrian Vetta
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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.

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Gwenaël Joret, Adrian Vetta. 2012-11-20. Reducing the rank of a matroid. https://doi.org/10.46298/dmtcs.2135

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