We consider the minimum maximal matching problem, which is NP-hard (Yannakakis and Gavril (1980) ). 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)  presented a (2-clog|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 (2-1χ′(G))-approximable. From edge-coloring theory, the approximation ratio of our algorithm is 2-1Δ(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)  showed the integrality gap of the LP formulation for bipartite graphs is at least 2-1Δ(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.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Signal Processing
- Information Systems
- Computer Science Applications