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 are of interest because they capture key aspects of state-of-the-art constraint solvers. In particular, it was recently shown that the problem of identifying a strong Horn- or 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. In fact, despite the worst-case complexity results for strong backdoor detection, we show that Satz-Rand is remarkably good at finding small strong backdoors on a range of experimental domains. Our results suggest that structural notions explored for designing efficient algorithms for combinatorial problems should capture both statically and dynamically identifiable properties.
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