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Documents authored by Fung, Long-Hin

Document
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.SAT.2023.10
On the Complexity of k-DQBF

Authors: Long-Hin Fung and Tony Tan

Published in: LIPIcs, Volume 271, 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)

Abstract
Recently Dependency Quantified Boolean Formula (DQBF) has attracted a lot of attention in the SAT community. Intuitively, a DQBF is a natural extension of quantified boolean formula where for each existential variable, one can specify the set of universal variables it depends on. It has been observed that a DQBF with k existential variables - henceforth denoted by k-DQBF - is essentially a k-CNF formula in succinct representation. However, beside this and the fact that the satisfiability problem is NEXP-complete, not much is known about DQBF. In this paper we take a closer look at k-DQBF and show that a number of well known classical results on k-SAT can indeed be lifted to k-DQBF, which shows a strong resemblance between k-SAT and k-DQBF. More precisely, we show the following. a) The satisfiability problem for 2- and 3-DQBF is PSPACE- and NEXP-complete, respectively. b) There is a parsimonious polynomial time reduction from arbitrary DQBF to 3-DQBF. c) Many polynomial time projections from SAT to languages in NP can be lifted to polynomial time reductions from the satisfiability of DQBF to languages in NEXP. d) Languages in the class NSPACE[s(n)] can be reduced to the satisfiability of 2-DQBF with O(s(n)) universal variables. e) Languages in the class NTIME[t(n)] can be reduced to the satisfiability of 3-DQBF with O(log t(n)) universal variables. The first result parallels the well known classical results that 2-SAT and 3-SAT are NL- and NP-complete, respectively.
Cite as

Long-Hin Fung and Tony Tan. On the Complexity of k-DQBF. In 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 271, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fung_et_al:LIPIcs.SAT.2023.10,
  author =	{Fung, Long-Hin and Tan, Tony},
  title =	{{On the Complexity of k-DQBF}},
  booktitle =	{26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)},
  pages =	{10:1--10:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-286-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{271},
  editor =	{Mahajan, Meena and Slivovsky, Friedrich},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2023.10},
  URN =		{urn:nbn:de:0030-drops-184729},
  doi =		{10.4230/LIPIcs.SAT.2023.10},
  annote =	{Keywords: Dependency quantified boolean formulas, existential variables, complexity}
}
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