License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2022.61
URN: urn:nbn:de:0030-drops-156579
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15657/
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Ebrahimnejad, Farzam ; Nagda, Ansh ; Gharan, Shayan Oveis

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

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LIPIcs-ITCS-2022-61.pdf (0.5 MB)

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.

BibTeX - Entry

@InProceedings{ebrahimnejad_et_al:LIPIcs.ITCS.2022.61,
  author =	{Ebrahimnejad, Farzam and Nagda, Ansh and Gharan, Shayan Oveis},
  title =	{{Counting and Sampling Perfect Matchings in Regular Expanding Non-Bipartite Graphs}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{61:1--61:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15657},
  URN =		{urn:nbn:de:0030-drops-156579},
  doi =		{10.4230/LIPIcs.ITCS.2022.61},
  annote =	{Keywords: perfect matchings, approximate sampling, approximate counting, expanders}
}

Keywords: perfect matchings, approximate sampling, approximate counting, expanders
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

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