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DOI: 10.4230/LIPIcs.ICALP.2017.28
URN: urn:nbn:de:0030-drops-73962
URL: http://drops.dagstuhl.de/opus/volltexte/2017/7396/
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Galanis, Andreas ; Goldberg, Leslie Ann ; Stefankovic, Daniel

Inapproximability of the Independent Set Polynomial Below the Shearer Threshold

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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. For 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|<lambda*(D). Our main result establishes for the first time a computational transition at the Shearer threshold. We show that for all D>=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. 1. For -lambda*(D)<lambda<lambda_c(D), the problem admits an FPTAS. 2. For lambda<-lambda*(D) or lambda>lambda_c(D), the problem is NP-hard. Rather 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).

BibTeX - Entry

@InProceedings{galanis_et_al:LIPIcs:2017:7396,
  author =	{Andreas Galanis and Leslie Ann Goldberg and Daniel Stefankovic},
  title =	{{Inapproximability of the Independent Set Polynomial Below the Shearer Threshold}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{28:1--28:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7396},
  URN =		{urn:nbn:de:0030-drops-73962},
  doi =		{10.4230/LIPIcs.ICALP.2017.28},
  annote =	{Keywords: approximate counting, independent set polynomial, Shearer threshold}
}

Keywords: approximate counting, independent set polynomial, Shearer threshold
Seminar: 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)
Issue Date: 2017
Date of publication: 06.07.2017


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