Clustering of solutions in hard satisfiability problems

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.Comment: 8 pages, 9 figure

Scaling of running time of quantum adiabatic algorithm for propositional satisfiability

We numerically study quantum adiabatic algorithm for the propositional satisfiability. A new class of previously unknown hard instances is identified among random problems. We numerically find that the running time for such instances grows exponentially with their size. Worst case complexity of quantum adiabatic algorithm therefore seems to be exponential.Comment: 7 page