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Precise Upper and Lower Bounds for the Monotone Constraint Satisfaction Problem

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Mathematical Foundations of Computer Science 2015 (MFCS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9234))

Abstract

The monotone constraint satisfaction problem (MCSP) is the problem of, given an existentially quantified positive formula, decide whether this formula has a model. This problem is a natural generalization of the constraint satisfaction problem, which can be seen as the problem of determining whether a conjunctive formula has a model. In this paper we study the worst-case time complexity, measured with respect to the number of variables, n, of the MCSP problem parameterized by a constraint language \(\varGamma \) (MCSP\((\varGamma )\)). We prove that the complexity of the NP-complete MCSP\((\varGamma )\) problems on a given finite domain D falls into exactly \(|D| - 1\) cases and ranges from \(O(2^{n})\) to \(O(|D|^n)\). We give strong lower bounds and prove that MCSP\((\varGamma )\), for any constraint language \(\varGamma \) over any finite domain, is solvable in \(O(|D'|^n)\) time, where \(D'\) is the domain of the core of \(\varGamma \), but not solvable in \(O(|D'|^{\delta n})\) time for any \(\delta < 1\), unless the strong exponential-time hypothesis fails. Hence, we obtain a complete understanding of the worst-case time complexity of MCSP\((\varGamma )\) for constraint languages over arbitrary finite domains.

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Acknowledgments

The author is grateful towards Peter Jonsson for several helpful discussions regarding the content of this paper, and to the anonymous reviewers for many suggestions for improvement.

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Correspondence to Victor Lagerkvist .

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Lagerkvist, V. (2015). Precise Upper and Lower Bounds for the Monotone Constraint Satisfaction Problem. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9234. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48057-1_28

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  • DOI: https://doi.org/10.1007/978-3-662-48057-1_28

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