New complexity results for the k-covers problem

Abstract

The k-covers problem (kCP) asks us to compute a minimum cardinality set of stringsof given length k > 1 that covers a given string. It was shown in a recent paper, by reduction to 3-SAT, that the k-covers problem is NP-complete. In this paper we introduce a new problem, that we call the k-Bounded Relaxed Vertex Cover Problem (RVCPk), which we show is equivalent to k-Bounded Set Cover (SCPk). We show further that kCP is a special case of RVCPk restricted to certain classes Gx,k of graphs that represent all strings x. Thus a minimum k-cover can be approximated to within a factor k in polynomial time. We discuss approximate solutions of kCP, and we state a number of conjectures and open problems related to kCP and Gx,k.