Fixed-Parameter Tractability of Dependency QBF with Structural Parameters

We study dependency quantified Boolean formulas (DQBF), an extension of QBF in which dependencies of existential variables are listed explicitly rather than being implicit in the order of quantifiers. DQBF evaluation is a canonical NEXPTIME-complete problem, a complexity class containing many prominent problems that arise in Knowledge Representation and Reasoning. One approach for solving such hard problems is to identify and exploit structural properties captured by numerical parameters such that bounding these parameters gives rise to an efficient algorithm. This idea is captured by the notion of fixed-parameter tractability (FPT). We initiate the study of DQBF through the lens of fixed-parameter tractability and show that the evaluation problem becomes FPT under two natural parameterizations: the treewidth of the primal graph of the DQBF instance combined with a restriction on the interactions between the dependency sets, and also the treedepth of the primal graph augmented by edges representing dependency sets.