Focused local search for random 3-satisfiability

Abstract

A local search algorithm solving an NP-complete optimization problem can be viewed as astochastic process moving in an ‘energy landscape’ towards eventually finding an optimalsolution. For the random 3-satisfiability problem, the heuristic of focusing the local moves onthe currently unsatisfied clauses is known to be very effective: the time to solution has beenobserved to grow only linearly in the number of variables, for a given clauses-to-variables ratioα sufficiently far below the critical satisfiability thresholdαc≈4.27. We present numerical results on the behaviour of three focused local search algorithms forthis problem, considering in particular the characteristics of a focused variant of simpleMetropolis dynamics. We estimate the optimal value for the ‘temperature’ parameterη for this algorithm, such that its linear time regime extends as close toαc as possible. Similar parameter optimization is performed also for the well-known WalkSATalgorithm and for the less studied, but very well performing focused record-to-recordtravel method. We observe that with an appropriate choice of parameters, thelinear time regime for each of these algorithms seems to extend well into ratiosα>4.2—much further than has so far been generally assumed. We discuss the statistics ofsolution times for the algorithms, relate their performance to the process of ‘whitening’,and present some conjectures on the shape of their computational phase diagrams.