Structure-Aware Lower Bounds and Broadening the Horizon of Tractability for QBF

TL;DR

This paper advances the understanding of the computational complexity of QSAT by establishing tight lower bounds and developing new algorithms based on structural graph parameters like treedepth and feedback vertex number.

Contribution

It introduces structure-aware reductions and algorithms that provide a nearly complete complexity classification of QSAT under various graph-theoretic parameters.

Findings

Established tight lower bounds for QSAT with bounded treedepth and feedback vertex number.

Developed new algorithms that complement the lower bounds, clarifying the problem's complexity landscape.

Extended results to various graph representations, including incidence and primal graphs.

Abstract

The QSAT problem, which asks to evaluate a quantified Boolean formula (QBF), is of fundamental interest in approximation, counting, decision, and probabilistic complexity and is also considered the prototypical PSPACEcomplete problem. As such, it has previously been studied under various structural restrictions (parameters), most notably parameterizations of the primal graph representation of instances. Indeed, it is known that QSAT remains PSPACE-complete even when restricted to instances with constant treewidth of the primal graph, but the problem admits a double-exponential fixed-parameter algorithm parameterized by the vertex cover number (primal graph). However, prior works have left a gap in our understanding of the complexity of QSAT when viewed from the perspective of other natural representations of instances, most notably via incidence graphs. In this paper, we develop structure-aware reductions which allow us to obtain essentially tight lower bounds for highly restricted instances of QSAT, including instances whose incidence graphs have bounded treedepth or feedback vertex number. We complement these lower bounds with novel algorithms for QSAT which establish a nearly-complete picture of the problem's complexity under standard graph-theoretic parameterizations. We also show implications for other natural graph representations, and obtain novel upper as well as lower bounds for QSAT under more fine-grained parameterizations of the primal graph.