Counting perfect matchings in graphs of degree 3

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    1 Citation (Scopus)

    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 languageEnglish (US)
    Title of host publicationFun with Algorithms - 6th International Conference, FUN 2012, Proceedings
    Pages189-197
    Number of pages9
    DOIs
    StatePublished - Jun 13 2012
    Event6th International Conference on Fun with Algorithms, FUN 2012 - Venice, Italy
    Duration: Jun 4 2012Jun 6 2012

    Publication series

    NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume7288 LNCS
    ISSN (Print)0302-9743
    ISSN (Electronic)1611-3349

    Other

    Other6th International Conference on Fun with Algorithms, FUN 2012
    CountryItaly
    CityVenice
    Period6/4/126/6/12

    Fingerprint

    Perfect Matching
    Counting
    Graph in graph theory
    Bipartite Graph
    Fast Algorithm
    Inclusion-Exclusion principle
    Statistical Physics
    Exact Algorithms
    Approximation Scheme
    Trivial
    Heuristics
    Arbitrary
    Physics

    All Science Journal Classification (ASJC) codes

    • Theoretical Computer Science
    • Computer Science(all)

    Cite this

    Furer, M. (2012). Counting perfect matchings in graphs of degree 3. In 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
    Furer, Martin. / Counting perfect matchings in graphs of degree 3. Fun with Algorithms - 6th International Conference, FUN 2012, Proceedings. 2012. pp. 189-197 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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    title = "Counting perfect matchings in graphs of degree 3",
    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 Bj{\"o}rklund 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.",
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    Furer, M 2012, Counting perfect matchings in graphs of degree 3. in 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.

    Fun with Algorithms - 6th International Conference, FUN 2012, Proceedings. 2012. p. 189-197 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7288 LNCS).

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

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    Furer M. Counting perfect matchings in graphs of degree 3. In Fun with Algorithms - 6th International Conference, FUN 2012, Proceedings. 2012. p. 189-197. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-30347-0_20