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Author Maneva, Elitza ♦ Mossel, Elchanan ♦ Wainwright, Martin J.
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©2007
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword k-SAT ♦ Gibbs sampling ♦ Markov random field ♦ Satisfiability problems ♦ Belief propagation ♦ Factor graph ♦ Message-passing ♦ Sum-product ♦ Survey propagation
Abstract This article provides a new conceptual perspective on survey propagation, which is an iterative algorithm recently introduced by the statistical physics community that is very effective in solving random $\textit{k}-SAT$ problems even with densities close to the satisfiability threshold. We first describe how any SAT formula can be associated with a novel family of Markov random fields (MRFs), parameterized by a real number ρ ∈ [0, 1]. We then show that applying belief propagation---a well-known “message-passing” technique for estimating marginal probabilities---to this family of MRFs recovers a known family of algorithms, ranging from pure survey propagation at one extreme (ρ = 1) to standard belief propagation on the uniform distribution over SAT assignments at the other extreme (ρ = 0). Configurations in these MRFs have a natural interpretation as partial satisfiability assignments, on which a partial order can be defined. We isolate $\textit{cores}$ as minimal elements in this partial ordering, which are also fixed points of survey propagation and the only assignments with positive probability in the MRF for ρ = 1. Our experimental results for $\textit{k}$ = 3 suggest that solutions of random formulas typically do not possess non-trivial cores. This makes it necessary to study the structure of the space of partial assignments for ρ < 1 and investigate the role of assignments that are very close to being cores. To that end, we investigate the associated lattice structure, and prove a weight-preserving identity that shows how any MRF with ρ > 0 can be viewed as a “smoothed” version of the uniform distribution over satisfying assignments (ρ = 0). Finally, we isolate properties of Gibbs sampling and message-passing algorithms that are typical for an ensemble of $\textit{k}-SAT$ problems.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2007-07-01
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 54
Issue Number 4

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Source: ACM Digital Library