Tradeoffs in Backdoors: Inconsistency Detection, Dynamic Simplification, and Preprocessing

There has been considerable interest in the identification of structural properties of combinatorial problems that lead to efficient algorithms for solving them. The notion of backdoors captures hidden structure exploited by state-of-the-art constraint solvers. We prove that strong backdoor identification with respect to the tractable classes Horn and 2CNF becomes harder than NP (unless NP=coNP) as soon as the inconsequential sounding feature of empty clause detection (present in all modern SAT solvers) is added. More interestingly, in practice such a feature as well as polynomial time constraint propagation mechanisms often lead to much smaller backdoor sets. We show experimentally that instances from real-world domains such as car configuration and logistics that have thousands of variables often have backdoors of only a few variables. We evaluate the effect of different preprocessors on the structure of the instances and hence on the backdoor size. Finally, strong backdoors have been mostly studied for unsatisfiable instances. In this work, we also look into strong backdoors of satisfiable instances and their relationship to solution counting.