Reducing the rank of a matroid

Gwenaël Joret (1) , Adrian Vetta (2)

1 ULB - Département d'Informatique [Bruxelles]
2 Department of Mathematics and Statistics [Montréal]

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.

Mots clés: matroid; maximum vertex cover; partial vertex cover

Domaines: Mathématique discrète [cs.DM]

Gwenaël Joret, Adrian Vetta. Reducing the rank of a matroid. Discrete Mathematics and Theoretical Computer Science, 2015, Vol. 17 no.2 (2), pp.143-156. ⟨10.46298/dmtcs.2135⟩. ⟨hal-01349050⟩