Clustering of solutions in hard satisfiability problems

TL;DR

This paper investigates the structure of solution spaces in random 3-SAT problems near the phase transition, revealing clustering behavior and a condensation transition point around alpha_c=4.26, with implications for local search algorithms.

Contribution

It introduces a detailed analysis of solution space clustering and identifies the condensation transition point in random 3-SAT problems, comparing local search and survey propagation methods.

Findings

Solution space shrinks as clause-to-variable ratio increases.

Solutions form well-separated clusters near the transition.

Condensation transition occurs at alpha_c ≈ 4.26.

Abstract

We study the structure of the solution space and behavior of local search methods on random 3-SAT problems close to the SAT/UNSAT transition. Using the overlap measure of similarity between different solutions found on the same problem instance we show that the solution space is shrinking as a function of alpha. We consider chains of satisfiability problems, where clauses are added sequentially. In each such chain, the overlap distribution is first smooth, and then develops a tiered structure, indicating that the solutions are found in well separated clusters. On chains of not too large instances, all solutions are eventually observed to be in only one small cluster before vanishing. This condensation transition point is estimated to be alpha_c = 4.26. The transition approximately obeys finite-size scaling with an apparent critical exponent of about 1.7. We compare the solutions found by a local heuristic, ASAT, and the Survey Propagation algorithm up to alpha_c.