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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.46

Counting Hypergraph Matchings up to Uniqueness Threshold

Authors: Renjie Song, Yitong Yin, and Jinman Zhao

Published in: LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

Abstract

We study the problem of approximately counting matchings in hypergraphs of bounded maximum degree and maximum size of hyperedges. With an activity parameter lambda, each matching M is assigned a weight lambda^{|M|}. The counting problem is formulated as computing a partition function that gives the sum of the weights of all matchings in a hypergraph. This problem unifies two extensively studied statistical physics models in approximate counting: the hardcore model (graph independent sets) and the monomer-dimer model (graph matchings). For this model, the critical activity lambda_c= (d^d)/(k (d-1)^{d+1}) is the threshold for the uniqueness of Gibbs measures on the infinite (d+1)-uniform (k+1)-regular hypertree. Consider hypergraphs of maximum degree at most k+1 and maximum size of hyperedges at most d+1. We show that when lambda < lambda_c, there is an FPTAS for computing the partition function; and when lambda = lambda_c, there is a PTAS for computing the log-partition function. These algorithms are based on the decay of correlation (strong spatial mixing) property of Gibbs distributions. When lambda > 2lambda_c, there is no PRAS for the partition function or the log-partition function unless NP=RP. Towards obtaining a sharp transition of computational complexity of approximate counting, we study the local convergence from a sequence of finite hypergraphs to the infinite lattice with specified symmetry. We show a surprising connection between the local convergence and the reversibility of a natural random walk. This leads us to a barrier for the hardness result: The non-uniqueness of infinite Gibbs measure is not realizable by any finite gadgets.

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Renjie Song, Yitong Yin, and Jinman Zhao. Counting Hypergraph Matchings up to Uniqueness Threshold. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 46:1-46:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{song_et_al:LIPIcs.APPROX-RANDOM.2016.46,
  author =	{Song, Renjie and Yin, Yitong and Zhao, Jinman},
  title =	{{Counting Hypergraph Matchings up to Uniqueness Threshold}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
  pages =	{46:1--46:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-018-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{60},
  editor =	{Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.46},
  URN =		{urn:nbn:de:0030-drops-66690},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2016.46},
  annote =	{Keywords: approximate counting; phase transition; spatial mixing}
}

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Counting Hypergraph Matchings up to Uniqueness Threshold

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