Treewidth with a Quantifier Alternation Revisited

In this paper we take a closer look at the parameterized complexity of ∃∀SAT, the prototypical complete problem of the class Σp2, the second level of the polynomial hierarchy. We provide a number of tight fine-grained bounds on the complexity of this problem and its variants with respect to the most important structural graph parameters. Specifically, we show the following lower bounds (assuming the ETH): It is impossible to decide ∃∀SAT in time less than double-exponential in the input formula’s treewidth. More strongly, we establish the same bound with respect to the formula’s primal vertex cover, a much more restrictive measure. This lower bound, which matches the performance of known algorithms, shows that the degeneration of the performance of treewidth-based algorithms to a tower of exponentials already begins in problems with one quantifier alternation. For the more general ∃∀CSP problem over a non-boolean domain of size B, there is no algorithm running in time 2B , where vc is the input’s primal vertex cover. ∃∀SAT is already NP-hard even when the input formula has constant modular treewidth (or clique-width), indicating that dense graph parameters are less useful for problems in Σp2. For the two weighted versions of ∃∀SAT recently introduced by de Haan and Szeider, called ∃k∀SAT and ∃∀kSAT, we give tight upper and lower bounds parameterized by treewidth (or primal vertex cover) and the weight k. Interestingly, the complexity of these two problems turns out to be quite different: one is double-exponential in treewidth, while the other is double-exponential in k. We complement the above negative results by showing a double-exponential FPT algorithm for QBF parameterized by vertex cover, showing that for this parameter the complexity never goes beyond double-exponential, for any number of quantifier alternations. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes, F.2.2 Nonnumerical Algorithms and Problems, G.2.1 Combinatorics, G.2.2 Graph Theory