On Counting Perfect Matchings in General Graphs

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


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.



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.


  1. 1.
    Broder, A.Z.: How hard is it to marry at random? (On the approximation of the permanent). In: Proceedings of the 18th Annual ACM Symposium on Theory of Computing (STOC), pp. 50–58 (1986). Erratum in Proceedings of the 20th Annual ACM Symposium on Theory of Computing, p. 551 (1988)Google Scholar
  2. 2.
    Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gallai, T.: Kritische Graphen II. Magyar Tud. Akad. Mat. Kutató Int. Kőzl. 8, 273–395 (1963)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Gallai, T.: Maximale systeme unabhängiger kanten. Magyar Tud. Akad. Mat. Kutató Int. Kőzl 9, 401–413 (1964)zbMATHGoogle Scholar
  5. 5.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)zbMATHGoogle Scholar
  6. 6.
    Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. Comput. 18(6), 1149–1178 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. J. ACM 51(4), 671–697 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jerrum, M.R., Valiant, L.G., Vazirani, V.V.: Random generation of combinatorial structures from a uniform distribution. Theoret. Comput. Sci. 43(2–3), 169–188 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kasteleyn, P.W.: Graph theory and crystal physics. In: Graph Theory and Theoretical Physics, pp. 43–110, Academic Press, London (1967)Google Scholar
  10. 10.
    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  11. 11.
    Lovász, L.: A note on factor-critical graphs. Stud. Sci. Math. Hungar 7(11), pp. 279–280 (1972)Google Scholar
  12. 12.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Hoboken (1998)zbMATHGoogle Scholar
  13. 13.
    Sinclair, A.J.: Algorithms for Random Generation and Counting: A Markov Chain Approach. Birkhäuser, Basel (1988)zbMATHGoogle Scholar
  14. 14.
    Valiant, L.G.: The complexity of computing the permanent. Theoret. Comput. Sci. 8(2), 189–201 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

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

Personalised recommendations