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-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.
| Original language | English |
|---|---|
| Pages (from-to) | 465-468 |
| Number of pages | 4 |
| Journal | Information Processing Letters |
| Volume | 111 |
| Issue number | 10 |
| DOIs | |
| Publication status | Published - Apr 30 2011 |
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All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Signal Processing
- Information Systems
- Computer Science Applications
Cite this
An approximation algorithm dependent on edge-coloring number for minimum maximal matching problem. / Matsumoto, Yusuke; Kamiyama, Naoyuki; Imai, Keiko.
In: Information Processing Letters, Vol. 111, No. 10, 30.04.2011, p. 465-468.Research output: Contribution to journal › Article
}
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
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.
UR - http://www.scopus.com/inward/record.url?scp=79951575134&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=79951575134&partnerID=8YFLogxK
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 -