License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.MFCS.2016.64
URN: urn:nbn:de:0030-drops-64769
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6476/
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Lagerkvist, Victor ; Roy, Biman

A Preliminary Investigation of Satisfiability Problems Not Harder than 1-in-3-SAT

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Abstract

The parameterized satisfiability problem over a set of Boolean
relations Gamma (SAT(Gamma)) is the problem of determining
whether a conjunctive formula over Gamma has at least one
model. Due to Schaefer's dichotomy theorem the computational
complexity of SAT(Gamma), modulo polynomial-time reductions, has
been completely determined: SAT(Gamma) is always either tractable
or NP-complete. More recently, the problem of studying the
relationship between the complexity of the NP-complete cases of
SAT(Gamma) with restricted notions of reductions has attracted
attention. For example, Impagliazzo et al. studied the complexity of
k-SAT and proved that the worst-case time complexity increases
infinitely often for larger values of k, unless 3-SAT is solvable in
subexponential time. In a similar line of research Jonsson et al.
studied the complexity of SAT(Gamma) with algebraic tools borrowed
from clone theory and proved that there exists an NP-complete problem
SAT(R^{neq,neq,neq,01}_{1/3}) such that there cannot exist any NP-complete SAT(Gamma) problem with strictly lower worst-case time complexity: the easiest NP-complete SAT(Gamma) problem. In this paper
we are interested in classifying the NP-complete SAT(Gamma)
problems whose worst-case time complexity is lower than 1-in-3-SAT but
higher than the easiest problem SAT(R^{neq,neq,neq,01}_{1/3}). Recently it was conjectured that there only exists three satisfiability problems of this form. We prove that this conjecture does not hold and that there is an infinite number of such SAT(Gamma) problems. In the process we determine several algebraic properties of 1-in-3-SAT and related problems, which could be of independent interest for constructing exponential-time algorithms.

BibTeX - Entry

@InProceedings{lagerkvist_et_al:LIPIcs:2016:6476,
  author =	{Victor Lagerkvist and Biman Roy},
  title =	{{A Preliminary Investigation of Satisfiability Problems Not Harder than 1-in-3-SAT}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{64:1--64:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Piotr Faliszewski and Anca Muscholl and Rolf Niedermeier},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6476},
  URN =		{urn:nbn:de:0030-drops-64769},
  doi =		{10.4230/LIPIcs.MFCS.2016.64},
  annote =	{Keywords: clone theory, universal algebra, satisfiability problems}
}

Keywords: clone theory, universal algebra, satisfiability problems
Collection: 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)
Issue Date: 2016
Date of publication: 19.08.2016

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