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

We consider the minimum maximal matching problem, which is NPNP-hard (Yannakakis and Gavril (1980) [18]). Given an unweighted simple graph G=(V,E)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−clog|V||V|)-approximation algorithm, where c is an arbitrary constant.In this paper, we present a (2−1χ′(G))-approximation algorithm based on an LP relaxation, where χ′(G)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||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 (2−1χ′(G))-approximable. From edge-coloring theory, the approximation ratio of our algorithm is 2−1Δ(G)+1, where Δ(G)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 2−1Δ(G). Moreover, χ′(G)χ′(G) is Δ(G)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.

Research highlights
► We consider the minimum maximal matching problem.
► The linear programming relaxation technique is used in our approximation algorithm.
► The approximation ratio of our algorithm is dependent on the edge-coloring number.