A New Empirical Study of Weak Backdoors

Work by Kilby, Slaney, Thiebaux andWalsh [1] showed that the backdoors and backbones of unstructured Random 3SAT instances are largely disjoint. In this work we extend this study to the consideration of backdoors in SAT encodings of structured problems. We show that the results of Kilby et al. also apply to structured problems. Further, we analyse the frequency with which individual variables appear in backdoors for specific problem instances. In all problem classes there are variables with particularly high frequencies of backdoor membership. Backbone variables that do appear in backdoors typically appear in very few. It has previously been hypothesised that modern SAT solvers implicitly exploit small backdoors [2], for example, through the use of restart policies. Modern SAT solvers use clause learning, and we show how clause learning interacts with the backdoor structure of problems to enhance the opportunities for SAT solvers to exploit small backdoors in this way. We show that clause learning reduces the size of backdoors in the augmented problem, whilst simultaneously reducing the number of variables that appear in backdoors.