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Inapproximability of the Independent Set Polynomial Below the Shearer Threshold

Authors Andreas Galanis, Leslie Ann Goldberg, Daniel Stefankovic

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  • approximate counting
  • independent set polynomial
  • 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).

Cite As Get BibTex

Andreas Galanis, Leslie Ann Goldberg, and Daniel Stefankovic. Inapproximability of the Independent Set Polynomial Below the Shearer Threshold. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ICALP.2017.28

Author Details

Andreas Galanis
Leslie Ann Goldberg
Daniel Stefankovic

References

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