2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS) (2016)

New Brunswick, New Jersey, USA

Oct. 9, 2016 to Oct. 11, 2016

ISSN: 0272-5428

ISBN: 978-1-5090-3933-3

pp: 704-713

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2016.80

ABSTRACT

We study the hard-core (gas) model defined on independent sets of an input graph where the independent sets are weighted by a parameter (aka fugacity) λ > 0. For constant Δ, previous work of Weitz (2006) established an FPTAS for the partition function for graphs of maximum degree Δ when λλc(Δ). The threshold λc(Δ) is the critical point for the statistical physics phase transition for uniqueness/non-uniqueness on the infinite Δ-regular tree. The running time of Weitz's algorithm is exponential in logΔ. Here we present an FPRAS for the partition function whose running time is O*(n2). We analyze the simple single-site Markov chain known as the Glauber dynamics for sampling from the associated Gibbs distribution. We prove there exists a constant Δ0 such that for all graphs with maximum degree Δ ≥ Δ0 and girth ≥ 7 (i.e., no cycles of length ≤ 6), the mixing time of the Glauber dynamics is O(n log n) when λ

INDEX TERMS

Partitioning algorithms, Approximation algorithms, Algorithm design and analysis, Heuristic algorithms, Markov processes, Couplings, Electronic mail

CITATION

C. Efthymiou, T. P. Hayes, D. Stefankovic, E. Vigoda and Y. Yin, "Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model,"

*2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)*, New Brunswick, New Jersey, USA, 2016, pp. 704-713.

doi:10.1109/FOCS.2016.80

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