TY - JOUR
T1 - An approximation algorithm dependent on edge-coloring number for minimum maximal matching problem
AU - Matsumoto, Yusuke
AU - Kamiyama, Naoyuki
AU - Imai, Keiko
N1 - Funding Information:
The work of the second and third authors was supported in part by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science. The third author was also supported in part by Chuo University Grant for Special Research.
PY - 2011/4/30
Y1 - 2011/4/30
N2 - 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-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) [4] 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.
AB - 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-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) [4] 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.
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U2 - 10.1016/j.ipl.2011.02.006
DO - 10.1016/j.ipl.2011.02.006
M3 - Article
AN - SCOPUS:79951575134
VL - 111
SP - 465
EP - 468
JO - Information Processing Letters
JF - Information Processing Letters
SN - 0020-0190
IS - 10
ER -