Rank Vertex Cover as a Natural Problem for Algebraic Compression

The question of the existence of a polynomial kernelization of the VERTEX COVER ABOVE LP problem was a long-standing, notorious open problem in parameterized complexity. Some years ago, the breakthrough work by Kratsch and Wahlstriim on representative sets finally answered this question in the affirmative [FOCS 2012]. In this paper, we present an alternative, algebraic compression of the VERTEX COVER ABOVE LP problem into the RANK VERTEX COVER problem. Here, the input consists of a graph G, a parameter k, and a bijection between V(G) and the set of columns of a representation of a matroid M, and the objective is to find a vertex cover whose rank is upper bounded by k.