Bilateral Treewidth for QBF: Where Strategies and Resolution Meet

2026-05-07

AI Summary

Quantified Boolean Formulas (QBFs) remain PSPACE-complete even when restricted to constant treewidth, as the structure of their quantifier prefixes is not captured by traditional treewidth measures. This work introduces a novel parameter called *bilateral treewidth*, which for the first time unifies two major parameterized approaches—strategy construction and Q-resolution—by jointly accounting for both the graph structure of the formula and the alternation pattern of its quantifiers. Building upon a given tree decomposition, we devise a fixed-parameter tractable algorithm that efficiently solves QBF instances with bounded bilateral treewidth. This result substantially broadens the class of tractable instances and overcomes the longstanding divide in parameterized complexity analyses between strategy-based and resolution-based methods.

Abstract

Treewidth is a well-studied decompositional parameter to measure the tree-likeness of a graph. While the propositional satisfiability problem (SAT) is known to be tractable when parameterized by the treewidth of the underlying primal graph, the evaluation of quantified Boolean formulas (QBFs) remains PSPACE-complete even on formulas of constant treewidth. Intuitively, this is because ordinary treewidth does not take into account the prefix of the QBF: it neither distinguishes between existential and universal variables, nor accounts for the order in which they are quantified. In the past, several weaker variants of treewidth have been devised to incorporate prefix-sensitive information. To establish tractability for QBFs under these notions, prior work has employed either strategy- or resolution-based techniques, thereby dividing the parameterized complexity landscape of QBF into two regimes that are incomparable in strength. We establish fixed-parameter tractability with respect to bilateral treewidth, a novel and strictly more powerful decompositional parameter that combines these rivaling approaches by simultaneously allowing for branching on strategies and performing Q-resolution. As in previous works in this direction, our algorithm assumes that a suitable tree decomposition is provided on the input.

Problem

• Quantified Boolean Formulas
• Treewidth
• Parameterized Complexity
• QBF Evaluation
• Prefix Sensitivity

Innovation

• bilateral treewidth
• quantified Boolean formulas
• fixed-parameter tractability
• Q-resolution
• strategy extraction