The complexity of almost perfect matchings in uniform hypergraphs with high codegree

Abstract

In this paper we prove that the problem APM(k, r, c) of deciding whether a given k-uniform hypergraph H, with minimum (k - 1)-wise vertex degree at least c|V (H)|, contains a matching missing exactly r vertices, that is, a set of disjoint edges of size at least (|V (H)|- r)/k, is NP-complete for c < 1/k , while for c > 1/k , and r > 0 we provide a polynomial time algorithm for the corresponding search problem. © Springer-Verlag Berlin Heidelberg 2009.