Abstract. We study numerically the solution space structure of random 3-SAT problems close to the SAT/UNSAT transition. This is done by considering chains of satisfiability problems, where clauses are added sequentially to a problem instance. Using the overlap measure of similarity between different solutions found on the same problem instance, we examine geometrical changes as a function of α. In each chain, the overlap distribution is first smooth, but then develops a tiered structure, indicating that the solutions are found in well separated clusters. On chains of not too large instances, all remaining solutions are eventually observed to be found in only one small cluster before vanishing. This condensation transition point is estimated by finite size scaling to be α_c = 4.26 with an apparent critical exponent of about 1.7. The average overlap value is also observed to increase with α up to the transition, indicating a reduction in solutions space size, in accordance with theoretical predictions. The solutions are generated by a local heuristic, ASAT, and compared to those found by the Survey Propagation algorithm up to α_c.

Key words: finite-size scaling; energy landscapes (experiment); network dynamics; random graphs, networks

E-print number: cond-mat/0702672
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Received 21 June 2007, accepted for publication 4 October 2007
Published 24 October 2007