New complexity results for the k-covers problem

The k-covers problem (kCP) asks us to compute a minimum cardinality set of strings of 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 (RVCP"k), which we show is equivalent to k-bounded set cover (SCP"k). We show further that kCP is a special case of RVCP"k restricted to certain classes G"x","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 G"x","k.

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