NDLI: Counting Independent Sets Using the Bethe Approximation

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Chandrasekaran, Venkat Shah, Devavrat Shin, Jinwoo Chertkov, Misha Gamarnik, David
Copyright Year	2011
Abstract	We consider the #P-complete problem of counting the number of independent sets in a given graph. Our interest is in understanding the effectiveness of the popular belief propagation (BP) heuristic. BP is a simple iterative algorithm that is known to have at least one fixed point, where each fixed point corresponds to a stationary point of the Bethe free energy (introduced by Yedidia, Freeman, and Weiss IEEE Trans. Inform. Theory, 51 (2004), pp.22822312 in recognition of Bethes earlier work in 1935). The evaluation of the Bethe free energy at such a stationary point (or BP fixed point) leads to the Bethe approximation for the number of independent sets of the given graph. BP is not known to converge in general, nor is an efficient, convergent procedure for finding stationary points of the Bethe free energy known. Furthermore, the effectiveness of the Bethe approximation is not well understood. As the first result of this paper we propose a BP-like algorithm that always converges to a stationary point of the Bethe free energy for any graph for the independent set problem. This procedure finds an ε-approximate stationary point in O(n2d42dε-4log3(nε-1)) iterations for a graph of n nodes with max-degree d. We study the quality of the resulting Bethe approximation using the recently developed loop series framework of Chertkov and Chernyak J. Stat. Mech. Theory Exp., 6 (2006), P06009. As this characterization is applicable only for exact stationary points of the Bethe free energy, we provide a slightly modified characterization that holds for ε-approximate stationary points. We establish that for any graph on n nodes with max-degree d and girth larger than 8dlog2n, the multiplicative error between the number of independent sets and the Bethe approximation decays as 1+O(n-γ) for some γ>0. This provides a deterministic counting algorithm that leads to strictly different results compared to a recent result of Weitz in Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, ACM Press, New York, 2006, pp.140149. Finally, as a consequence of our analysis we prove that the Bethe approximation is exceedingly good for a random 3-regular graph conditioned on the shortest cycle cover conjecture of Alon and Tarsi SIAM J. Algebr. Discrete Methods, 6 (1985), pp.345350 being true.
Starting Page	1012
Ending Page	1034
Page Count	23
File Format	PDF
ISSN	08954801
DOI	10.1137/090767145
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	2
Volume Number	25
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-07-01
Access Restriction	Subscribed
Subject Keyword	belief propagation Probability in computer science Bethe free energy Approximation algorithms Graph theory independent set loop series
Content Type	Text
Resource Type	Article
Subject	Mathematics

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