A new look at survey propagation and its generalizations

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 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 wellknown "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 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 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 k-SAT problems.E. MANEVA ET AL. ACM Reference Format:Maneva, E., Mossel, E., and Wainwright, M. J. 2007. A new look at survey propagation and its generalizations.

Phase transitions in the coloring of random graphs

“…Another interesting role of the frozen variables arises within the whitening procedure, introduced in [65] and studied, between others, for the satisfiability problem in [66,67]. This procedure is equivalent to the warning propagation (WP) update (49) which we outlined in sec.…”

“…However, recent works show that if one applies the whitening procedure starting from solutions found by SP on large graphs, whitening converges every time to the trivial fixed point (see detailed studies for K-SAT in [66,67]). A possible solution to this apparent paradox is discussed in sec.…”

“…In the fixed point or all edges are white, then the solution {s i } does not belong to a frozen cluster, or some of the edges stay colored (non-white), then the solution {s i } belongs to a frozen cluster. Note that in the K-SAT problem (but not in general), whitening is equivalent to a more intuitive procedure, where the directed edged are not considered [66,67].…”

“…We wish to offer here an explanation of a paradox observed in [66,67]. The SP algorithm gives information on the frozen variables in the most numerous clusters (m = 0).…”

Leveraging Belief Propagation, Backtrack Search, and Statistics for Model Counting

“…The equations are analogous to standard BP for SAT (see e.g. [13] Figure 4 with ρ = 0 for a full description), differing only in the added κ exponent in the iterative update equation as shown in Figure 1. We use its output as an estimate of the marginals of the variables in BPCount.…”