d-QBF with Few Existential Variables Revisited

2026-03-09

AI Summary

This work investigates the parameterized complexity of Quantified Boolean Formulas (QBF) with respect to the number $k$ of existentially quantified variables and the maximum clause size $d$. Under the framework of parameterized complexity and the Exponential Time Hypothesis (ETH), it establishes for the first time that the general $d$-QBF problem inherently requires double-exponential time, specifically ruling out algorithms running in $2^{2^{o(k)}}$ time. Furthermore, for the two-quantifier-block case of $\forall\exists$-QBF, the paper presents a nearly optimal algorithm running in $k^{O_d(k^{d-1})}$ time. These results settle the tight double-exponential lower bound for $d$-QBF parameterized by $k$ and substantially narrow the gap between known upper and lower bounds for $\forall\exists$-QBF.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

Quantified Boolean Formula (QBF) is a notoriously hard generalization of \textsc{SAT}, especially from the point of view of parameterized complexity, where the problem remains intractable for most standard parameters. A recent work by Eriksson et al.~[IJCAI 24] addressed this by considering the case where the propositional part of the formula is in CNF and we parameterize by the number $k$ of existentially quantified variables. One of their main results was that this natural (but so far overlooked) parameter does lead to fixed-parameter tractability, if we also bound the maximum arity $d$ of the clauses of the given CNF. Unfortunately, their algorithm has a \emph{double-exponential} dependence on $k$ ($2^{2^k}$), even when $d$ is an absolute constant. Since the work of Eriksson et al.\ only complemented this with a SETH-based lower bound implying that a $2^{O(k)}$ dependence is impossible, this left a large gap as an open question. Our main result in this paper is to close this gap by showing that the double-exponential dependence is optimal, assuming the ETH: even for CNFs of arity $4$, QBF with $k$ existential variables cannot be solved in time $2^{2^{o(k)}}|φ|^{O(1)}$. Complementing this, we also consider the further restricted case of QBF with only two quantifier blocks ($\forall\exists$-QBF). We show that in this case the situation improves dramatically: for each $d\ge 3$ we show an algorithm with running time $k^{O_d(k ^{d-1})}|φ|^{O(1)}$ and a lower bound under the ETH showing our algorithm is almost optimal.

Problem

• Quantified Boolean Formula
• parameterized complexity
• existential variables
• ETH
• CNF

Innovation

• Quantified Boolean Formula
• parameterized complexity
• existential variables
• ETH lower bound
• fixed-parameter tractability