A mixture-model for the behaviour of SLS algorithms for SAT

Abstract

Stochastic Local Search (SLS) algorithms are amongst the most effective approaches for solving hard and large propositional satisfiability (SAT) problems. Prominent and successful SLS algorithms for SAT, including many members of the WalkSAT and GSAT families of algorithms, tend to show highly regular behaviour when applied to hard SAT instances: The run-time distributions (RTDs) of these algorithms are closely approximated by exponential distributions. The deeper reasons for this regular behaviour are, however, essentially unknown. In this study we show that there are hard problem instances, e.g., from the phase transition region of the widely studied class of Uniform Random 3-SAT instances, for which the RTDs for well-known SLS algorithms such as GWSAT or WalkSAT/SKC deviate substantially from exponential distributions. We investigate these irregular instances and show that the respective RTDs can be modelled using mixtures of exponential distributions. We present evidence that such mixture distributions reflect stagnation behaviour in the search process caused by traps in the underlying search spaces. This leads to the formulation of a new model of SLS behaviour as a simple Markov process. This model subsumes and extends earlier characterisations of SLS behaviour and provides plausible explanations for many empirical observations.