Tradeoffs in the Complexity of Backdoor Detection for Combinatorial Problems

There has been considerable interest in the identification of structural properties of combinatorial problems that lead to efficient algorithms for solving them. Some of these properties are “easily” identifiable, while others such as backdoor sets are of interest because they capture key aspects of state-of-the-art constraint solvers as well as of many real-world problem instances. In particular, it was recently shown that the problem of identifying a strong Hornor 2CNF-backdoor can be solved by exploiting equivalence with deletion backdoors, and is NP-complete. We prove that strong backdoor identification 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 that have thousands of variables often have backdoors of only a few variables. Our results suggest that structural notions explored for designing efficient algorithms for combinatorial problems should capture both statically and dynamically identifiable properties. We also evaluate the effect of different preprocessors on the structure of the instances and hence on the backdoor size. Finally, we also look into strong backdoors for satisfiable instances and their relationship to solution counting.