Survey Propagation as local equilibrium equations

TL;DR

This paper derives Survey Propagation equations as local equilibrium conditions in an extended variable space, linking them to solution clustering in combinatorial problems like K-SAT and graph coloring.

Contribution

It introduces a new derivation of SP equations as sum-product marginals in an extended space, clarifying the geometric clustering interpretation.

Findings

SP equations correspond to local equilibrium conditions in an extended variable space.

The entropy of the local equilibrium function matches the logarithm of solution clusters.

Results clarify the geometric interpretation of clustering in random combinatorial problems.

Abstract

It has been shown experimentally that a decimation algorithm based on Survey Propagation (SP) equations allows to solve efficiently some combinatorial problems over random graphs. We show that these equations can be derived as sum-product equations for the computation of marginals in an extended space where the variables are allowed to take an additional value -- $*$ -- when they are not forced by the combinatorial constraints. An appropriate ``local equilibrium condition'' cost/energy function is introduced and its entropy is shown to coincide with the expected logarithm of the number of clusters of solutions as computed by SP. These results may help to clarify the geometrical notion of clusters assumed by SP for the random K-SAT or random graph coloring (where it is conjectured to be exact) and helps to explain which kind of clustering operation or approximation is enforced in general/small sized models in which it is known to be inexact.