License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2019.31
URN: urn:nbn:de:0030-drops-108533
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10853/
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
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}
}
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Keywords: |
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Fine-grained Complexity, Hardness in P, Hyperclique Conjecture, Constrained Triangle Detection |
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Collection: |
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34th Computational Complexity Conference (CCC 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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16.07.2019 |
