Threshold behaviour of WalkSAT and Focused Metropolis search on random 3-satisfiability

Abstract. An important heuristic in local search algorithms for Satisfiability is focusing, i.e. restricting the selection of flipped variables to those appearing in presently unsatisfied clauses. We consider the behaviour on large randomly generated 3-SAT instances of two focused solution methods: the well-known WalkSAT algorithm, and the straightforward algorithm obtained by imposing the focusing constraint on basic Metropolis dynamics. We observe that both WalkSAT and our Focused Metropolis Search method are quite sensitive to the proper choice of their “noise” and “temperature ” parameters, and attempt to estimate ideal values for these, such that the algorithms exhibit generically linear solution times as close to the satisfiability transition threshold αc ≈ 4.267 as possible. With an appropriate choice of parameters, the linear time regime of both algorithms seems to extend well into clauses-to-variables ratios α> 4.2, which is much further than has generally been assumed in the literature.