On the max k-vertex cover problem ∗

Given a graph G(V,E) of order n and a constant k 6 n, the max k-vertex cover problem consists of determining k vertices that cover the maximum number of edges in G. In its (standard) parameterized version, max k-vertex cover can be stated as follows: “given G, k and parameter l, does G contain k vertices that cover at least l edges?”. We first devise moderately exponential exact algorithms for max k-vertex cover, with complexity exponential to n (note that the known results concerned time bounds of the form n) by developing a branch and reduce method based upon the measure-and-conquer technique. We then prove that, interestingly enough, although max k-vertex cover is non fixed parameter tractable with respect to l, it is fixed parameter tractable with respect to the size τ of a minimum vertex cover of G. We also point out that the same happens for a lot of well-known problems quite different from max k-vertex cover. We finally study approximation of max k-vertex cover by moderately exponential algorithms. The general goal of the issue of moderately exponential approximation is to catch-up on polynomial inapproximability, by providing algorithms achieving, with worst-case running times importantly smaller than those needed for exact computation, approximation ratios unachievable in polynomial time.