Counting perfect matchings in graphs of degree 3

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

    1 Scopus citations

    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

    All Science Journal Classification (ASJC) codes

    • Theoretical Computer Science
    • Computer Science(all)

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  • 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