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**Published in:** LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

The inverse satisfiability problem over a set of Boolean relations Gamma (Inv-SAT(Gamma)) is the computational decision problem of, given a set of models R, deciding whether there exists a SAT(Gamma) instance with R as its set of models. This problem is co-NP-complete in general and a dichotomy theorem for finite Γ containing the constant Boolean relations was obtained by Kavvadias and Sideri. In this paper we remove the latter condition and prove that Inv-SAT(Gamma) is always either tractable or co-NP-complete for all finite sets of relations Gamma, thus solving a problem open since 1998. Very little of the techniques used by Kavvadias and Sideri are applicable and we have to turn to more recently developed algebraic approaches based on partial polymorphisms. We also consider the case when Γ is infinite, where the situation differs markedly from the case of SAT. More precisely, we show that there exists infinite Gamma such that Inv-SAT(Gamma) is tractable even though there exists finite Delta is subset of Gamma such that Inv-SAT(Delta) is co-NP-complete.

Victor Lagerkvist and Biman Roy. A Dichotomy Theorem for the Inverse Satisfiability Problem. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{lagerkvist_et_al:LIPIcs.FSTTCS.2017.39, author = {Lagerkvist, Victor and Roy, Biman}, title = {{A Dichotomy Theorem for the Inverse Satisfiability Problem}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {39:1--39:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.39}, URN = {urn:nbn:de:0030-drops-83745}, doi = {10.4230/LIPIcs.FSTTCS.2017.39}, annote = {Keywords: Complexity Theory, Inverse Satisfiability, Clone Theory} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

The exponential-time hypothesis (ETH) states that 3-SAT is not solvable in subexponential time, i.e. not solvable in O(c^n) time for arbitrary c > 1, where n denotes the number of variables. Problems like k-SAT can be viewed as special cases of the constraint satisfaction problem (CSP), which is the problem of determining whether a set of constraints is satisfiable. In this paper we study the worst-case time complexity of NP-complete CSPs. Our main interest is in the CSP problem parameterized by a constraint language Gamma (CSP(Gamma)), and how the choice of Gamma affects the time complexity. It is believed that CSP(Gamma) is either tractable or NP-complete, and the algebraic CSP dichotomy conjecture gives a sharp delineation of these two classes based on algebraic properties of constraint languages. Under this conjecture and the ETH, we first rule out the existence of subexponential algorithms for finite domain NP-complete CSP(Gamma) problems. This result also extends to certain infinite-domain CSPs and structurally restricted CSP(Gamma) problems. We then begin a study of the complexity of NP-complete CSPs where one is allowed to arbitrarily restrict the values of individual variables, which is a very well-studied subclass of CSPs. For such CSPs with finite domain D, we identify a relation SD such that (1) CSP({SD}) is NP-complete and (2) if CSP(Gamma) over D is NP-complete and solvable in O(c^n) time, then CSP({SD}) is solvable in O(c^n) time, too. Hence, the time complexity of CSP({SD}) is a lower bound for all CSPs of this particular kind. We also prove that the complexity of CSP({SD}) is decreasing when |D| increases, unless the ETH is false. This implies, for instance, that for every c>1 there exists a finite-domain Gamma such that CSP(Gamma) is NP complete and solvable in O(c^n) time.

Peter Jonsson, Victor Lagerkvist, and Biman Roy. Time Complexity of Constraint Satisfaction via Universal Algebra. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{jonsson_et_al:LIPIcs.MFCS.2017.17, author = {Jonsson, Peter and Lagerkvist, Victor and Roy, Biman}, title = {{Time Complexity of Constraint Satisfaction via Universal Algebra}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {17:1--17:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.17}, URN = {urn:nbn:de:0030-drops-80710}, doi = {10.4230/LIPIcs.MFCS.2017.17}, annote = {Keywords: Clone Theory, Universal Algebra, Constraint Satisfaction Problems} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

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.

Victor Lagerkvist and Biman Roy. A Preliminary Investigation of Satisfiability Problems Not Harder than 1-in-3-SAT. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 64:1-64:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{lagerkvist_et_al:LIPIcs.MFCS.2016.64, author = {Lagerkvist, Victor and Roy, Biman}, 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 = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.64}, URN = {urn:nbn:de:0030-drops-64769}, doi = {10.4230/LIPIcs.MFCS.2016.64}, annote = {Keywords: clone theory, universal algebra, satisfiability problems} }

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