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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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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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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.

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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

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

Funding

  • Ebrahimnejad, Farzam: Research supported in part by NFS grant CCF-1552097.
  • Gharan, Shayan Oveis: Research supported by Air Force Office of Scientific Research grant FA9550-20-1-0212, NSF grants CCF-1552097, CCF-1907845, and a Sloan fellowship.

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