Predicting Satisfiability at the Phase Transition

Abstract

Uniform random 3-SAT at the solubility phase transition is one of the most widely studied and empirically hardest distributions of SAT instances. For 20 years, this distribution has been used extensively for evaluating and comparing algorithms. In this work, we demonstrate that simple rules can predict the solubility of these instances with surprisingly high accuracy. Specifically, we show how classification accuracies of about 70% can be obtained based on cheaply (polynomial-time) computable features on a wide range of instance sizes. We argue in two ways that classification accuracy does not decrease with instance size: first, we show that our models' predictive accuracy remains roughly constant across a wide range of problem sizes; second, we show that a classifier trained on small instances is sufficient to achieve very accurate predictions across the entire range of instance sizes currently solvable by complete methods. Finally, we demonstrate that a simple decision tree based on only two features, and again trained only on the smallest instances, achieves predictive accuracies close to those of our most complex model. We conjecture that this two-feature model outperforms random guessing asymptotically; due to the model's extreme simplicity, we believe that this conjecture is a worthwhile direction for future theoretical work.