On Efficient Fixed Parameter Algorithms for WEIGHTED VERTEX COVER

We investigate the fixed parameter complexity of one of the most popular problems in combinatorial optimization, Weighted Vertex Cover. Given a graph G = (V, E), a weight function ω : V → R, and k ∈ R, Weighted Vertex Cover (WVC for short) asks for a subset C of vertices in V of weight at most k such that every edge of G has at least one endpoint in C. WVC and its variants have all been shown to be NP-complete. We show that, when restricting the range of ω to positive integers, the so-called Integer-WVC can be solved as fast as unweighted Vertex Cover (time O(1.271 + k|V |)). Our main result is that if the range of ω is restricted to positive reals ≥ 1, then so-called Real-WVC can be solved in time O(1.3954 + k|V |). If we modify the problem in such a way that k is not the weight of the vertex cover we are looking for, but the number of vertices in a minimum weight vertex cover, then the same running time can be obtained. If the weights are arbitrary (referred to by General-WVC), however, the problem is not fixed parameter tractable unless P = NP .