### Abstract

Counting perfect matchings is an interesting and challenging combinatorial task. It has important applications in statistical physics. As the general problem is #P complete, it is usually tackled by randomized heuristics and approximation schemes. The trivial running times for exact algorithms are O*((n-1)!!)=O *(n!!)=O *((n/2)! 2 ^{n/2}) for general graphs and O *((n/2)!) for bipartite graphs. Ryser's old algorithm uses the inclusion exclusion principle to handle the bipartite case in time O*(2 ^{n/2}). It is still the fastest known algorithm handling arbitrary bipartite graphs. For graphs with n vertices and m edges, we present a very simple argument for an algorithm running in time O *(1.4656 ^{m-n} ). For graphs of average degree 3 this is O*(1.2106 ^{n} ), improving on the previously fastest algorithm of Björklund and Husfeldt. We also present an algorithm running in time O*(1.4205 ^{m-n} ) or O*(1.1918 ^{n} ) for average degree 3 graphs. The purpose of these simple algorithms is to exhibit the power of the m-n measure. Here, we don't investigate the further improvements possible for larger average degrees by applying the measure-and-conquer method.

Original language | English (US) |
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Title of host publication | Fun with Algorithms - 6th International Conference, FUN 2012, Proceedings |

Pages | 189-197 |

Number of pages | 9 |

DOIs | |

State | Published - Jun 13 2012 |

Event | 6th International Conference on Fun with Algorithms, FUN 2012 - Venice, Italy Duration: Jun 4 2012 → Jun 6 2012 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7288 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 6th International Conference on Fun with Algorithms, FUN 2012 |
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Country | Italy |

City | Venice |

Period | 6/4/12 → 6/6/12 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Fun with Algorithms - 6th International Conference, FUN 2012, Proceedings*(pp. 189-197). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7288 LNCS). https://doi.org/10.1007/978-3-642-30347-0_20

}

*Fun with Algorithms - 6th International Conference, FUN 2012, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7288 LNCS, pp. 189-197, 6th International Conference on Fun with Algorithms, FUN 2012, Venice, Italy, 6/4/12. https://doi.org/10.1007/978-3-642-30347-0_20

**Counting perfect matchings in graphs of degree 3.** / Furer, Martin.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Counting perfect matchings in graphs of degree 3

AU - Furer, Martin

PY - 2012/6/13

Y1 - 2012/6/13

N2 - Counting perfect matchings is an interesting and challenging combinatorial task. It has important applications in statistical physics. As the general problem is #P complete, it is usually tackled by randomized heuristics and approximation schemes. The trivial running times for exact algorithms are O*((n-1)!!)=O *(n!!)=O *((n/2)! 2 n/2) for general graphs and O *((n/2)!) for bipartite graphs. Ryser's old algorithm uses the inclusion exclusion principle to handle the bipartite case in time O*(2 n/2). It is still the fastest known algorithm handling arbitrary bipartite graphs. For graphs with n vertices and m edges, we present a very simple argument for an algorithm running in time O *(1.4656 m-n ). For graphs of average degree 3 this is O*(1.2106 n ), improving on the previously fastest algorithm of Björklund and Husfeldt. We also present an algorithm running in time O*(1.4205 m-n ) or O*(1.1918 n ) for average degree 3 graphs. The purpose of these simple algorithms is to exhibit the power of the m-n measure. Here, we don't investigate the further improvements possible for larger average degrees by applying the measure-and-conquer method.

AB - Counting perfect matchings is an interesting and challenging combinatorial task. It has important applications in statistical physics. As the general problem is #P complete, it is usually tackled by randomized heuristics and approximation schemes. The trivial running times for exact algorithms are O*((n-1)!!)=O *(n!!)=O *((n/2)! 2 n/2) for general graphs and O *((n/2)!) for bipartite graphs. Ryser's old algorithm uses the inclusion exclusion principle to handle the bipartite case in time O*(2 n/2). It is still the fastest known algorithm handling arbitrary bipartite graphs. For graphs with n vertices and m edges, we present a very simple argument for an algorithm running in time O *(1.4656 m-n ). For graphs of average degree 3 this is O*(1.2106 n ), improving on the previously fastest algorithm of Björklund and Husfeldt. We also present an algorithm running in time O*(1.4205 m-n ) or O*(1.1918 n ) for average degree 3 graphs. The purpose of these simple algorithms is to exhibit the power of the m-n measure. Here, we don't investigate the further improvements possible for larger average degrees by applying the measure-and-conquer method.

UR - http://www.scopus.com/inward/record.url?scp=84861969856&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861969856&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-30347-0_20

DO - 10.1007/978-3-642-30347-0_20

M3 - Conference contribution

SN - 9783642303463

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 189

EP - 197

BT - Fun with Algorithms - 6th International Conference, FUN 2012, Proceedings

ER -