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DOI: 10.4230/LIPIcs.SAT.2025.10
Fine-Grained Complexity Analysis of Dependency Quantified Boolean Formulas

Authors: Che Cheng, Long-Hin Fung, Jie-Hong Roland Jiang, Friedrich Slivovsky, and Tony Tan

Published in: LIPIcs, Volume 341, 28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025)

Abstract
Dependency Quantified Boolean Formulas (DQBF) extend Quantified Boolean Formulas by allowing each existential variable to depend on an explicitly specified subset of the universal variables. The satisfiability problem for DQBF is NEXP-complete in general, with only a few tractable fragments known to date. We investigate the complexity of DQBF with k existential variables (k-DQBF) under structural restrictions on the matrix - specifically, when it is in Conjunctive Normal Form (CNF) or Disjunctive Normal Form (DNF) - as well as under constraints on the dependency sets. For DNF matrices, we obtain a clear classification: 2-DQBF is PSPACE-complete, while 3-DQBF is NEXP-hard, even with disjoint dependencies. For CNF matrices, the picture is more nuanced: we show that the complexity of k-DQBF ranges from NL-complete for 2-DQBF with disjoint dependencies to NEXP-complete for 6-DQBF with arbitrary dependencies.
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Che Cheng, Long-Hin Fung, Jie-Hong Roland Jiang, Friedrich Slivovsky, and Tony Tan. Fine-Grained Complexity Analysis of Dependency Quantified Boolean Formulas. In 28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 341, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cheng_et_al:LIPIcs.SAT.2025.10,
  author =	{Cheng, Che and Fung, Long-Hin and Jiang, Jie-Hong Roland and Slivovsky, Friedrich and Tan, Tony},
  title =	{{Fine-Grained Complexity Analysis of Dependency Quantified Boolean Formulas}},
  booktitle =	{28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-381-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{341},
  editor =	{Berg, Jeremias and Nordstr\"{o}m, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2025.10},
  URN =		{urn:nbn:de:0030-drops-237441},
  doi =		{10.4230/LIPIcs.SAT.2025.10},
  annote =	{Keywords: Dependency quantified Boolean formulas, complexity, completeness, conjunctive normal form, disjunctive normal form}
}
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