{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9808","name":"Inapproximability of the Independent Set Polynomial Below the Shearer Threshold","abstract":"We study the problem of approximately evaluating the independent set polynomial of bounded-degree graphs at a point lambda or, equivalently, the problem of approximating the partition function of the hard-core model with activity lambda on graphs G of max degree D. For lambda>0, breakthrough results of Weitz and Sly established a computational transition from easy to hard at lambda_c(D)=(D-1)^(D-1)\/(D-2)^D, which coincides with the tree uniqueness phase transition from statistical physics. \r\n\r\nFor lambda<0, the evaluation of the independent set polynomial is connected to the conditions of the Lovasz Local Lemma. Shearer identified the threshold lambda*(D)=(D-1)^(D-1)\/D^D as the maximum value p such that every family of events with failure probability at most p and whose dependency graph has max degree D has nonempty intersection. Very recently, Patel and Regts, and Harvey et al. have independently designed FPTASes for approximating the partition function whenever |lambda|=3, for all lambda<-lambda*(D), it is NP-hard to approximate the partition function on graphs of maximum degree D, even within an exponential factor. Thus, our result, combined with the FPTASes for lambda>-lambda*(D), establishes a phase transition for negative activities. In fact, we now have the following picture for the problem of approximating the partition function with activity lambda on graphs G of max degree D. \n1. For -lambda*(D)lambda_c(D), the problem is NP-hard. \r\nRather than the tree uniqueness threshold of the positive case, the phase transition for negative activities corresponds to the existence of zeros for the partition function of the tree below -lambda*(D).","keywords":["approximate counting","independent set polynomial","Shearer threshold"],"author":[{"@type":"Person","name":"Galanis, Andreas","givenName":"Andreas","familyName":"Galanis"},{"@type":"Person","name":"Goldberg, Leslie Ann","givenName":"Leslie Ann","familyName":"Goldberg"},{"@type":"Person","name":"Stefankovic, Daniel","givenName":"Daniel","familyName":"Stefankovic"}],"position":28,"pageStart":"28:1","pageEnd":"28:13","dateCreated":"2017-07-07","datePublished":"2017-07-07","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Galanis, Andreas","givenName":"Andreas","familyName":"Galanis"},{"@type":"Person","name":"Goldberg, Leslie Ann","givenName":"Leslie Ann","familyName":"Goldberg"},{"@type":"Person","name":"Stefankovic, Daniel","givenName":"Daniel","familyName":"Stefankovic"}],"copyrightYear":"2017","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ICALP.2017.28","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/www.its.caltech.edu\/~piyushs\/","isPartOf":{"@type":"PublicationVolume","@id":"#volume6283","volumeNumber":80,"name":"44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)","dateCreated":"2017-07-07","datePublished":"2017-07-07","editor":[{"@type":"Person","name":"Chatzigiannakis, Ioannis","givenName":"Ioannis","familyName":"Chatzigiannakis"},{"@type":"Person","name":"Indyk, Piotr","givenName":"Piotr","familyName":"Indyk"},{"@type":"Person","name":"Kuhn, Fabian","givenName":"Fabian","familyName":"Kuhn"},{"@type":"Person","name":"Muscholl, Anca","givenName":"Anca","familyName":"Muscholl"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9808","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":"#volume6283"}}}