Susceptibility Propagation for Constraint Satisfaction Problems

We study the susceptibility propagation, a message-passing algorithm to compute correlation functions. It is applied to constraint satisfaction problems and its accuracy is examined. As a heuristic method to find a satisfying assignment, we propose susceptibility-guided decimation where correlations among the variables play an important role. We apply this novel decimation to locked occupation problems, a class of hard constraint satisfaction problems exhibited recently. It is shown that the present method performs better than the standard belief-guided decimation.

💡 Research Summary

The paper introduces Susceptibility Propagation (SP), a message‑passing scheme derived as the linear response of Belief Propagation (BP) to infinitesimal external fields, to compute two‑point connected correlation functions in discrete constraint satisfaction problems (CSPs). By differentiating the BP update equations with respect to the fields, the authors obtain new messages—cavity susceptibilities—that obey simple linear update rules (equations 8–9). In the binary case these are expressed in log‑likelihood form as η and \hat η, leading to a compact linear system y = My + b whose fixed point, when it exists, yields exact pairwise correlations. The existence and convergence of the fixed point depend on the spectral radius of the matrix M; on tree‑like factor graphs the method reproduces the exact marginals and correlations, while on loopy graphs (e.g., a ring for the 1‑in‑2 SAT problem) the system may lack a fixed point and the messages diverge.

Building on SP, the authors propose a “susceptibility‑guided decimation” heuristic. Traditional BP‑decimation fixes the most polarized variable according to its marginal entropy. In contrast, the new approach identifies strongly correlated variable pairs via the η‑messages and fixes their relative orientation (e.g., x_i = x_j or x_i = 1 − x_j). This is particularly advantageous for Locked Occupation Problems (LOPs), a class of CSPs where solutions are isolated and local marginals carry little information. Experiments on several LOPs—including 1‑in‑4 SAT, 1‑in‑K SAT, and parity‑check constraints—show that susceptibility‑guided decimation outperforms standard BP‑guided decimation, achieving higher success rates and solving instances that were previously intractable.

The paper also discusses practical issues: on loopy graphs SP may diverge, but introducing a finite temperature (softening the BP updates) can restore convergence. Small‑scale tests comparing SP‑estimated correlations with exact enumeration reveal modest errors that diminish as the problem size grows and the factor graph becomes locally tree‑like.

In conclusion, the work demonstrates that incorporating pairwise correlation information via susceptibility propagation enriches the BP framework and yields a more powerful heuristic for hard CSPs. The authors acknowledge the need for further theoretical analysis of the spectral conditions for convergence, extensions to higher‑order correlations, and efficient online updating of susceptibility messages during decimation.