Efficient algorithms for the max $$k$$ -vertex cover problem - Della Croce, Federico; Paschos, Vangelis Th (Journal of combinatorial optimization, 2014)
Given a graph G(V,E) of order n and a constant k⩽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 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 ck and γτ, for some γ<2, where τ is the cardinality of a minimum vertex cover of G (note that \textscmax k \textsc−vertexcover∉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.