{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9086","name":"Counting Hypergraph Matchings up to Uniqueness Threshold","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).\r\n\r\nFor 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.\r\n\r\nTowards 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.","keywords":"approximate counting; phase transition; spatial mixing","author":[{"@type":"Person","name":"Song, Renjie","givenName":"Renjie","familyName":"Song"},{"@type":"Person","name":"Yin, Yitong","givenName":"Yitong","familyName":"Yin"},{"@type":"Person","name":"Zhao, Jinman","givenName":"Jinman","familyName":"Zhao"}],"position":46,"pageStart":"46:1","pageEnd":"46:29","dateCreated":"2016-09-06","datePublished":"2016-09-06","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Song, Renjie","givenName":"Renjie","familyName":"Song"},{"@type":"Person","name":"Yin, Yitong","givenName":"Yitong","familyName":"Yin"},{"@type":"Person","name":"Zhao, Jinman","givenName":"Jinman","familyName":"Zhao"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.APPROX-RANDOM.2016.46","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6263","volumeNumber":60,"name":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX\/RANDOM 2016)","dateCreated":"2016-09-06","datePublished":"2016-09-06","editor":[{"@type":"Person","name":"Jansen, Klaus","givenName":"Klaus","familyName":"Jansen"},{"@type":"Person","name":"Mathieu, Claire","givenName":"Claire","familyName":"Mathieu"},{"@type":"Person","name":"Rolim, Jos\u00e9 D. P.","givenName":"Jos\u00e9 D. P.","familyName":"Rolim"},{"@type":"Person","name":"Umans, Chris","givenName":"Chris","familyName":"Umans"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9086","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6263"}}}