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DOI: 10.4230/LIPIcs.CCC.2019.31
URN: urn:nbn:de:0030-drops-108533
URL: http://drops.dagstuhl.de/opus/volltexte/2019/10853/
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Bringmann, Karl ; Fischer, Nick ; Künnemann, Marvin

A Fine-Grained Analogue of Schaefer's Theorem in P: Dichotomy of Exists^k-Forall-Quantified First-Order Graph Properties

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Abstract

An important class of problems in logics and database theory is given by fixing a first-order property psi over a relational structure, and considering the model-checking problem for psi. Recently, Gao, Impagliazzo, Kolokolova, and Williams (SODA 2017) identified this class as fundamental for the theory of fine-grained complexity in P, by showing that the (Sparse) Orthogonal Vectors problem is complete for this class under fine-grained reductions. This raises the question whether fine-grained complexity can yield a precise understanding of all first-order model-checking problems. Specifically, can we determine, for any fixed first-order property psi, the exponent of the optimal running time O(m^{c_psi}), where m denotes the number of tuples in the relational structure? Towards answering this question, in this work we give a dichotomy for the class of exists^k-forall-quantified graph properties. For every such property psi, we either give a polynomial-time improvement over the baseline O(m^k)-time algorithm or show that it requires time m^{k-o(1)} under the hypothesis that MAX-3-SAT has no O((2-epsilon)^n)-time algorithm. More precisely, we define a hardness parameter h = H(psi) such that psi can be decided in time O(m^{k-epsilon}) if h <=2 and requires time m^{k-o(1)} for h >= 3 unless the h-uniform HyperClique hypothesis fails. This unveils a natural hardness hierarchy within first-order properties: for any h >= 3, we show that there exists a exists^k-forall-quantified graph property psi with hardness H(psi)=h that is solvable in time O(m^{k-epsilon}) if and only if the h-uniform HyperClique hypothesis fails. Finally, we give more precise upper and lower bounds for an exemplary class of formulas with k=3 and extend our classification to a counting dichotomy.

BibTeX - Entry

@InProceedings{bringmann_et_al:LIPIcs:2019:10853,
  author =	{Karl Bringmann and Nick Fischer and Marvin K{\"u}nnemann},
  title =	{{A Fine-Grained Analogue of Schaefer's Theorem in P: Dichotomy of Exists^k-Forall-Quantified First-Order Graph Properties}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{31:1--31:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Amir Shpilka},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10853},
  URN =		{urn:nbn:de:0030-drops-108533},
  doi =		{10.4230/LIPIcs.CCC.2019.31},
  annote =	{Keywords: Fine-grained Complexity, Hardness in P, Hyperclique Conjecture, Constrained Triangle Detection}
}

Keywords: Fine-grained Complexity, Hardness in P, Hyperclique Conjecture, Constrained Triangle Detection
Seminar: 34th Computational Complexity Conference (CCC 2019)
Issue Date: 2019
Date of publication: 16.07.2019


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