Machine Learning-Based Restart Policy for CDCL SAT Solvers

Abstract

Restarts are a critically important heuristic in most modern conflict-driven clause-learning (CDCL) SAT solvers. The precise reason as to why and how restarts enable CDCL solvers to scale efficiently remains obscure. In this paper we address this question, and provide some answers that enabled us to design a new effective machine learning-based restart policy. Specifically, we provide evidence that restarts improve the quality of learnt clauses as measured by one of best known clause quality metrics, namely, literal block distance (LBD). More precisely, we show that more frequent restarts decrease the LBD of learnt clauses, which in turn improves solver performance. We also note that too many restarts can be harmful because of the computational overhead of rebuilding the search tree from scratch too frequently. With this trade-off in mind, between that of learning better clauses vs. the computational overhead of rebuilding the search tree, we introduce a new machine learning-based restart policy that predicts the quality of the next learnt clause based on the history of previously learnt clauses. The restart policy erases the solver’s search tree during its run, if it predicts that the quality of the next learnt clause is below some dynamic threshold that is determined by the solver’s history on the given input. Our machine learning-based restart policy is based on two observations gleaned from our study of LBDs of learnt clauses. First, we discover that high LBD percentiles can be approximated with z-scores of the normal distribution. Second, we find that LBDs, viewed as a sequence, are correlated and hence the LBDs of past learnt clauses can be used to predict the LBD of future ones. With these observations in place, and techniques to exploit them, our new restart policy is shown to be effective over a large benchmark from the SAT Competition 2014 to 2017.