Relaxed Survey Propagation for The Weighted Maximum Satisfiability Problem

The survey propagation (SP) algorithm has been shown to work well on large instances of the random 3-SAT problem near its phase transition. It was shown that SP estimates marginals over covers that represent clusters of solutions. The SP-y algorithm generalizes SP to work on the maximum satisfiability (Max-SAT) problem, but the cover interpretation of SP does not generalize to SP-y. In this paper, we formulate the relaxed survey propagation (RSP) algorithm, which extends the SP algorithm to apply to the weighted Max-SAT problem. We show that RSP has an interpretation of estimating marginals over covers violating a set of clauses with minimal weight. This naturally generalizes the cover interpretation of SP. Empirically, we show that RSP outperforms SP-y and other state-of-the-art Max-SAT solvers on random Max-SAT instances. RSP also outperforms state-of-the-art weighted Max-SAT solvers on random weighted Max-SAT instances.

💡 Research Summary

The paper introduces Relaxed Survey Propagation (RSP), a novel message‑passing algorithm that extends the celebrated Survey Propagation (SP) technique from random 3‑SAT to the weighted Maximum Satisfiability (Weighted Max‑SAT) problem. SP has been highly successful on large random 3‑SAT instances near the satisfiability threshold because it exploits a statistical‑physics insight: solutions cluster into “covers,” and SP’s messages estimate the marginal probability that a variable belongs to a particular cluster. However, the direct generalization of SP to optimization‑type SAT problems, known as SP‑y, loses this cover interpretation and suffers from reduced stability when clause weights are introduced.

Core Idea
RSP restores a principled interpretation by defining a weighted‑violation cover: a partial assignment that may violate some clauses, but the total weight of violated clauses is minimal among all such assignments. The algorithm operates on the same factor‑graph representation as SP, but each clause now contributes a penalty proportional to its weight and a temperature‑like parameter (y). Messages are still of three types (0, 1, and “*” for undecided), yet the update rules incorporate the clause weights, effectively biasing the propagation toward assignments that keep the weighted violation cost low.

Theoretical Foundations
The authors show that RSP’s fixed‑point equations are equivalent to the stationary conditions of a variational free‑energy functional
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