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On Counting Perfect Matchings in General Graphs

  • Daniel Štefankovič
  • Eric Vigoda
  • John Wilmes
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

Counting perfect matchings has played a central role in the theory of counting problems. The permanent, corresponding to bipartite graphs, was shown to be #P-complete to compute exactly by Valiant (1979), and a fully polynomial randomized approximation scheme (FPRAS) was presented by Jerrum, Sinclair, and Vigoda (2004) using a Markov chain Monte Carlo (MCMC) approach. However, it has remained an open question whether there exists an FPRAS for counting perfect matchings in general graphs. In fact, it was unresolved whether the same Markov chain defined by JSV is rapidly mixing in general. In this paper, we show that it is not. We prove torpid mixing for any weighting scheme on hole patterns in the JSV chain. As a first step toward overcoming this obstacle, we introduce a new algorithm for counting matchings based on the Gallai−Edmonds decomposition of a graph, and give an FPRAS for counting matchings in graphs that are sufficiently close to bipartite. In particular, we obtain a fixed-parameter tractable algorithm for counting matchings in general graphs, parameterized by the greatest “order” of a factor-critical subgraph.

Notes

Acknowledgements

This research was supported in part by NSF grants CCF-1617306, CCF-1563838, CCF-1318374, and CCF-1717349. The authors are grateful to Santosh Vempala for many illuminating conversations about Markov chains and the structure of factor-critical graphs.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Daniel Štefankovič
    • 1
  • Eric Vigoda
    • 2
  • John Wilmes
    • 2
  1. 1.University of RochesterRochesterUSA
  2. 2.Georgia Institute of TechnologyAtlantaUSA

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