### 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 (2-1^{χ′}(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 |
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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

*Information Processing Letters*,

*111*(10), 465-468. https://doi.org/10.1016/j.ipl.2011.02.006

**An approximation algorithm dependent on edge-coloring number for minimum maximal matching problem.** / Matsumoto, Yusuke; Kamiyama, Naoyuki; Imai, Keiko.

Research output: Contribution to journal › Article

*Information Processing Letters*, vol. 111, no. 10, pp. 465-468. https://doi.org/10.1016/j.ipl.2011.02.006

}

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 -