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

Yusuke Matsumoto, Naoyuki Kamiyama, Keiko Imai

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 languageEnglish
Pages (from-to)465-468
Number of pages4
JournalInformation Processing Letters
Volume111
Issue number10
DOIs
Publication statusPublished - Apr 30 2011

Fingerprint

Edge Coloring
Approximation algorithms
Coloring
Matching Problem
Approximation Algorithms
Dominating Set
Dependent
Simple Graph
Bipartite Graph
Formulation
Approximation
LP Relaxation
Integrality
Maximum Degree
Cardinality
NP-complete problem
Unknown
Arbitrary

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 journalArticle

@article{7953afe96c5042f8a3c9b211cc233c33,
title = "An approximation algorithm dependent on edge-coloring number for minimum maximal matching problem",
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.",
author = "Yusuke Matsumoto and Naoyuki Kamiyama and Keiko Imai",
year = "2011",
month = "4",
day = "30",
doi = "10.1016/j.ipl.2011.02.006",
language = "English",
volume = "111",
pages = "465--468",
journal = "Information Processing Letters",
issn = "0020-0190",
publisher = "Elsevier",
number = "10",

}

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 -