Matched Formulas and Backdoor Sets

We study parameterizations of the satisfiability problem for propositional formulas in conjunctive normal form. In particular, we consider two parameters that generalize the notion of matched formulas: (i) the well studied parameter maximum deficiency, and (ii) the size of smallest backdoor sets with respect to certain base classes of bounded maximum deficiency. The simplest base class considered is the class of matched formulas. Our main technical contribution is a hardness result for the detection of weak, strong, and deletion backdoor sets. This result implies, subject to a complexity theoretic assumption, that small backdoor sets with respect to the base classes under consideration cannot be found significantly faster than by exhaustive search.