An approximation algorithm dependent on edge-coloring number for minimum maximal matching problem

摘要

We consider the minimum maximal matching problem, which is NP-hard (Yannakakis and Gavril (1980) [18]). Given an unweighted simple graph G = (V, E), the problem seeks to find a maximal matching of minimum cardinality. It was unknown whether there exists a non-trivial approximation algorithm whose approximation ratio is less than 2 for any simple graph. Recently, Z. Gotthilf et al. (2008) [5] presented a (2-C(log|V|/|V|)approximation algorithm, where c is an arbitrary constant. In this paper, we present a (2 —1/x'(G)approximation algorithm based on an LP relaxation, where x'(G) is the edge-coloring number of G. Our algorithm is the first non-trivial approximation algorithm whose approximation ratio is independent of |V|. Moreover, it is known that the minimum maximal matching problem is equivalent to the edge dominating set problem. Therefore, the edge dominating set problem is also (2 -1/x'(G))-approximable. From edge-coloring theory, the approximation ratio of our algorithm is 2 - 1/Δ(G)+1, where Δ(G) represents the maximum degree of G. In our algorithm, an LP formulation for the edge dominating set problem is used. Fujito and Nagamochi (2002) [4] showed the integrality gap of the LP formulation for bipartite graphs is at least 2 -1/Δ(G). Moreover, x'(C) is Δ(G) for bipartite graphs. Thus, as far as an approximation algorithm for the minimum maximal matching problem uses the LP formulation, we believe our result is the best possible.