Clausal Deletion Backdoors for QBF: a Parameterized Complexity Approach

2026-05-12

AI Summary

Quantified Boolean Formula (QBF) evaluation is PSPACE-complete and lacks a general fixed-parameter tractable (FPT) theory. This work introduces a novel parameter, clause-cover backdoor (CC), which measures the minimum number of clauses containing variables that must be removed to reduce the formula to a tractable class. Leveraging this parameter, the paper establishes an almost complete parameterized complexity classification across three base classes: Horn, 2-CNF, and linear equations. The Horn case is shown to be W[1]-hard, whereas both 2-CNF and linear equation cases admit FPT algorithms. Methodologically, the approach integrates propositional propagation with Gaussian elimination, transcending conventional QBF solving frameworks and offering a new theoretical foundation for the parameterized complexity of QBF.

Abstract

Determining the validity of a quantified Boolean formula (QBF) is a PSPACE-complete problem with rich expressive power. Despite interest in efficient solvers, there is, compared to problems in NP, a lack of positive theoretical results, and in the parameterized complexity setting one often has to restrict the quantifier prefix (e.g., bounding alternations) to obtain fixed parameter tractability (FPT). We propose a new parameter: the number of variables in clauses that has to be removed before reaching a tractable class (a clause covering (CC) backdoor). We are then interested in solving QBF in FPT time given a CC-backdoor of size $k$. We consider the three classical, tractable cases of QBF as base classes: Horn, 2-CNF, and linear equations. We establish W[1]-hardness for Horn but prove FPT for the others, and prove that in a precise, algebraic sense, we are only missing one important case for a full dichotomy. Our algorithms are non-trivial and depend on propagation, and Gaussian elimination, respectively, and are comparably unexplored for QBF.

Problem

• Quantified Boolean Formula
• Parameterized Complexity
• Backdoor
• Clausal Deletion
• Tractable Class

Innovation

• QBF
• parameterized complexity
• backdoor
• fixed-parameter tractability
• Gaussian elimination