Abstract

. The problem instance of Vertex Cover consists of an undirected graph G = (V; E) and a positive integer k, the question is whether there exists a subset C V of vertices such that each edge in E has at least one of its endpoints in C with jCj k. We improve two recent worst case upper bounds for Vertex Cover. First, Balasubramanian et al. showed that Vertex Cover can be solved in time O(kn + 1:32472 k k 2 ), where n is the number of vertices in G. Afterwards, Downey et al. improved this to O(kn+ 1:31951 k k 2 ). Bringing the exponential base significantly below 1:3, we present the new upper bound O(kn + 1:29175 k k 2 ). 1 Introduction Vertex Cover is a problem of central importance in computer science: { It was among the rst NP-complete problems [7]. { There have been numerous eorts to design ecient approximation algorithms [3], but it is also known to be hard to approximate [1]. { It is of central importance in parameterized complexity theory and has one ...

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