On the satisfiability threshold and clustering of solutions of random 3-SAT formulas

Abstract

a b s t r a c tWe study the structure of satisfying assignments of a random 3-Sat formula. In particular, we show that a random formula of density α ≥ 4.453 almost surely has no non-trivial ''core'' assignments. Core assignments are certain partial assignments that can be extended to satisfying assignments, and have been studied recently in connection with the Survey Propagation heuristic for random Sat. Their existence implies the presence of clusters of solutions, and they have been shown to exist with high probability below the satisfiability threshold for k-Sat with k ≥ 9 [D. Achlioptas, F. Ricci-Tersenghi, On the solution-space geometry of random constraint satisfaction problems, in: Proc. 38th ACM Symp. Theory of Computing, STOC, 2006, pp. 130-139]. Our result implies that either this does not hold for 3-Sat, or the threshold density for satisfiability in 3-Sat lies below 4.453. The main technical tool that we use is a novel simple application of the first moment method.

Parallel Local Search for the Costas Array Problem

“…However, previous experiments [6] showed that on structured problems such as (non-random) CSPs, independent multiple-walk parallelization does not achieve 2 http://www.g12.cs.mu.oz.au/mzn/costas array/CostasArray.mzn ideal speedups, reaching only a factor 50-70 speedup for 256 cores. It might be because solutions are not uniformly distributed in the search space, and are for instance regrouped in "clusters", as was shown for solutions of SAT problems near the phase transition in [27]. Therefore some sequential computation is needed to get to the vicinity of such clusters.…”

Statistical physics of hard optimization problems

“…In the large K limit they showed that the frozen phase covers a finite fraction (at least 20%) of the satisfiable region. The second study [MS07] gives a rigorous upper bound on the freezing transition in 3-SAT α f < 4.453, which is slightly better than the best known upper bound on the satisfiability transition in 3-SAT [DBM00]. The third study is numerical [ZDEB-10], presented in fig.…”

Large-scale parallelism for constraint-based local search: the costas array case study

“…This aspect of course depends on the energy landscape generated by the cost function. Moreover, this might be theoretically explained by the fact that, as we use structured problem instances and not random instances, solutions may be not uniformly distributed in the search space but are regrouped in clusters, as was shown for solutions of the SAT problems near the phase transition in [51]. We will however see in the following section that there exist other problems like CAP for which linear speedups can be achieved even far beyond 256 cores.…”

“…The classical explanation for an exponential runtime behavior is the fact that the solutions are uniformly distributed in the search space, (and not regrouped in solution clusters [51]) and that the random search algorithm is able to sample the search space in a uniform manner. For the CAP instances, we could thus explain these linear speedups due to the good distribution of solutions over the search space for n > 17 as shown in [65] (although the number of solutions decreases beyond n = 17) and the fact that AS is able to diversify correctly.…”