Efficient Algorithms for the max k -vertex cover Problem

We first devise moderately exponential exact algorithms for max k-vertex cover, with time-complexity exponential in n but with polynomial space-complexity by developing a branch and reduce method based upon the measure-and-conquer technique. We then prove that, there exists an exact algorithm for max k-vertex cover with complexity bounded above by the maximum among c and γ , for some γ < 2, where τ is the cardinality of a minimum vertex cover of G (note that max k-vertex cover / ∈ FPT with respect to parameter k unless FPT = W[1]), using polynomial space. 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.