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Counting and Sampling Perfect Matchings in Regular Expanding Non-Bipartite Graphs

Authors Farzam Ebrahimnejad, Ansh Nagda, Shayan Oveis Gharan



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Author Details

Farzam Ebrahimnejad
  • University of Washington, Seattle, WA, USA
Ansh Nagda
  • University of Washington, Seattle, WA, USA
Shayan Oveis Gharan
  • University of Washington, Seattle, WA, USA

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Farzam Ebrahimnejad, Ansh Nagda, and Shayan Oveis Gharan. Counting and Sampling Perfect Matchings in Regular Expanding Non-Bipartite Graphs. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 61:1-61:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITCS.2022.61

Abstract

We show that the ratio of the number of near perfect matchings to the number of perfect matchings in d-regular strong expander (non-bipartite) graphs, with 2n vertices, is a polynomial in n, thus the Jerrum and Sinclair Markov chain [Jerrum and Sinclair, 1989] mixes in polynomial time and generates an (almost) uniformly random perfect matching. Furthermore, we prove that such graphs have at least Ω(d)ⁿ many perfect matchings, thus proving the Lovasz-Plummer conjecture [L. Lovász and M.D. Plummer, 1986] for this family of graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Expander graphs and randomness extractors
  • Mathematics of computing → Combinatoric problems
  • Mathematics of computing → Markov-chain Monte Carlo methods
Keywords
  • perfect matchings
  • approximate sampling
  • approximate counting
  • expanders

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References

  1. N. Anari, S. Oveis Gharan, and C. Vinzant. Log-concave polynomials i: Entropy and a deterministic approximation algorithm for counting bases of matroids. Duke Mathematical Journal, 2021. to appear. Google Scholar
  2. Alexander Barvinok. A bound for the number of vertices of a polytope with applications. Combinatorica, 33(1):1-10, 2013. Google Scholar
  3. Alexander I. Barvinok. Polynomial time algorithms to approximate permanents and mixed discriminants within a simply exponential factor. Random Struct. Algorithms, 14(1):29-61, 1999. Google Scholar
  4. Ivona Bezáková, Daniel Stefankovic, Vijay V. Vazirani, and Eric Vigoda. Accelerating simulated annealing for the permanent and combinatorial counting problems. In SODA, pages 900-907. ACM Press, 2006. Google Scholar
  5. Béla Bollobás and Brendan D McKay. The number of matchings in random regular graphs and bipartite graphs. Journal of Combinatorial Theory, Series B, 41(1):80-91, 1986. Google Scholar
  6. Charles Bordenave. A new proof of Friedman’s second eigenvalue Theorem and its extension to random lifts. Annales scientifiques de l'Ecole normale supérieure, 2019. Google Scholar
  7. M. Chudnovsky and P Seymour. Perfect matchings in planar cubic graphs. Combinatorica, 32:403-424, 2012. Google Scholar
  8. G.P. Egorychev. The solution of van der waerden’s problem for permanents. Advances in Math., 42:299-305, 1981. Google Scholar
  9. Louis Esperet, František Kardoš, Andrew D. King, Daniel Král, and Serguei Norine. Exponentially many perfect matchings in cubic graphs. Advances in Mathematics, 227(4):1646-1664, July 2011. Google Scholar
  10. D. I. Falikman. Proof of the van der waerden’s conjecture on the permanent of a doubly stochastic matrix. Mat. Zametki, 29(6):931-938, 1981. (in Russian). Google Scholar
  11. J. Friedman. A proof of alon’s second eigenvalue conjecture and related problems. Mem. Amer. Math. Soc., 195(910), 2008. Google Scholar
  12. David Gamarnik and Dmitriy Katz. A deterministic approximation algorithm for computing the permanent of a 0,1 matrix. Journal of Computer and System Sciences, 76(8):879-883, 2010. Google Scholar
  13. Leonid Gurvits. Hyperbolic polynomials approach to van der waerden/schrijver-valiant like conjectures: sharper bounds, simpler proofs and algorithmic applications. In Jon M. Kleinberg, editor, STOC, pages 417-426. ACM, 2006. Google Scholar
  14. Mark Jerrum and Alistair Sinclair. Approximating the permanent. SIAM journal on computing, 18(6):1149-1178, 1989. Google Scholar
  15. Mark Jerrum, Alistair Sinclair, and Eric Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. Journal of the ACM (JACM), 51(4):671-697, 2004. Google Scholar
  16. Mark Jerrum and Umesh V. Vazirani. A mildly exponential approximation algorithm for the permanent. Algorithmica, 16(4/5):392-401, 1996. Google Scholar
  17. N. Linial, A. Samorodnitsky, and A. Wigderson. A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents. Combinatorica, 20:545-568, 2000. Google Scholar
  18. L. Lovász and M.D. Plummer. Matching Theory. Elsevier Science, 1986. Google Scholar
  19. Robert W. Robinson and Nicholas C. Wormald. Almost all regular graphs are hamiltonian. Random Structures & Algorithms, 5(2):363-374, 1994. Google Scholar
  20. Mark Rudelson, Alex Samorodnitsky, and Ofer Zeitouni. Hafnians, perfect matchings and gaussian matrices. Ann. Probab., 44(4):2858-2888, July 2016. Google Scholar
  21. A. Schrijver. Counting 1-factors in regular bipartite graphs. Journal of Combinatorial Theory, B 72:122-135, 1998. Google Scholar
  22. R Michael Tanner. Explicit concentrators from generalized n-gons. SIAM Journal on Algebraic Discrete Methods, 5(3):287-293, 1984. Google Scholar
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