Clustering of solutions in hard satisfiability problems

We study numerically the solution space structure of random SAT problems close to the SAT/UNSAT transition. This is done by considering chains of satis. ability problems, where clauses are addedsequentially 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 a. 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 alpha(c) = 4.26 with an apparent critical exponent of about 1.7. The average overlap value is also observed to increase with a 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 alpha(.)(c)