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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.40

On Optimization and Counting of Non-Broken Bases of Matroids

Authors: Dorna Abdolazimi, Kasper Lindberg, and Shayan Oveis Gharan

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract

Given a matroid M = (E,I), and a total ordering over the elements E, a broken circuit is a circuit where the smallest element is removed and an NBC independent set is an independent set in I with no broken circuit. The set of NBC independent sets of any matroid M define a simplicial complex called the broken circuit complex which has been the subject of intense study in combinatorics. Recently, Adiprasito, Huh and Katz showed that the face of numbers of any broken circuit complex form a log-concave sequence, proving a long-standing conjecture of Rota. We study counting and optimization problems on NBC bases of a generic matroid. We find several fundamental differences with the independent set complex: for example, we show that it is NP-hard to find the max-weight NBC base of a matroid or that the convex hull of NBC bases of a matroid has edges of arbitrary large length. We also give evidence that the natural down-up walk on the space of NBC bases of a matroid may not mix rapidly by showing that for some family of matroids it is NP-hard to count the number of NBC bases after certain conditionings.

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Dorna Abdolazimi, Kasper Lindberg, and Shayan Oveis Gharan. On Optimization and Counting of Non-Broken Bases of Matroids. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abdolazimi_et_al:LIPIcs.APPROX/RANDOM.2023.40,
  author =	{Abdolazimi, Dorna and Lindberg, Kasper and Gharan, Shayan Oveis},
  title =	{{On Optimization and Counting of Non-Broken Bases of Matroids}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{40:1--40:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.40},
  URN =		{urn:nbn:de:0030-drops-188653},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.40},
  annote =	{Keywords: Complexity, Hardness, Optimization, Counting, Random walk, Local to Global, Matroids}
}

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DOI: 10.4230/LIPIcs.ITCS.2022.61

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

Authors: Farzam Ebrahimnejad, Ansh Nagda, and Shayan Oveis Gharan

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

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.

Cite as

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)

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@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/entities/document/10.4230/LIPIcs.ITCS.2022.61},
  URN =		{urn:nbn:de:0030-drops-156579},
  doi =		{10.4230/LIPIcs.ITCS.2022.61},
  annote =	{Keywords: perfect matchings, approximate sampling, approximate counting, expanders}
}

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Documents authored by Gharan, Shayan Oveis

On Optimization and Counting of Non-Broken Bases of Matroids

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

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