New complexity results for the kcovers problem
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The kcovers 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 3SAT, that the kcovers problem is NPcomplete. In this paper we introduce a new problem, that we call the kbounded relaxed vertex cover problem (RVCPk), which we show is equivalent to kbounded 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 kcover 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.
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