Approximation of Self-stabilizing Vertex Cover Less Than 2

A vertex cover of a graph is a subset of vertices such that each edge has at least one endpoint in the subset. Determining the minimum vertex cover is a well-known NP-complete problem in a sequential setting. Several techniques, e.g., depth-first search, a local ratio theorem, and semidefinite relaxation, have given good approximation algorithms. However, some of them cannot be applied to a distributed setting, in particular self-stabilizing algorithms. Thus only a 2-approximation solution based on a self-stabilizing maximal matching has been obviously known until now. In this paper we propose a new self-stabilizing vertex cover algorithm that achieves (2–1/Δ)-approximation ratio, where Δ is the maximum degree of a given network. We first introduce a sequential (2–1/Δ)-approximation algorithm that uses a maximal matching with the high-degree-first order of vertices. Then we present a self-stabilizing algorithm based on the same idea, and show that the output of the algorithm is the same as that of the sequential one.