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

Abstract

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 χ ′ ( G ) ) -approximation algorithm based on an LP relaxation, where χ ′ ( 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 χ ′ ( 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, χ ′ ( G ) 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.