On the MAX k-VERTEX COVER problem

Federico Della Croce (1) , Vangelis Paschos (2)

1 DAUIN - Department of Computer Engineering
2 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision

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 complexity exponential to n (note that the known results concerned time bounds of the form $n^(O(k))$) 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 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.

Domaines: Algorithme et structure de données [cs.DS]; Complexité [cs.CC]; Recherche opérationnelle [math.OC]

Federico Della Croce, Vangelis Paschos. On the MAX k-VERTEX COVER problem. 2011. ⟨hal-00875629⟩