Backdoor Sets of Quantified Boolean Formulas

Abstract

We generalize the notion of backdoor sets from propositional formulas to quantified Boolean formulas in conjunctive normal form (QCNF). We develop parameterized algorithms that admit uniform polynomial time QCNF evaluation parameterized by the size of smallest strong backdoor sets. For our algorithms we develop a theory of variable dependency which is of independent interest. As a result, we obtain hierarchies of classes of tractable QCNF formulas with the classes of quantified Horn and quantified 2CNF formulas, respectively, at their first level, thus gradually generalizing these two prominent tractable classes. In contrast to known tractable classes based on bounded treewidth, the number of quantifier alternations of our classes is unbounded.