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Quantified Valued Constraint Satisfaction Problem

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Book cover Principles and Practice of Constraint Programming (CP 2018)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11008))

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Abstract

We study the complexity of the quantified and valued extension of the constraint satisfaction problem (QVCSP) for certain classes of languages. This problem is also known as the weighted constraint satisfaction problem with min-max quantifiers [1].

The multimorphisms that preserve a language is the starting point of our analysis. We establish some situations where a QVCSP is solvable in polynomial time by formulating new algorithms or by extending the usage of collapsibility, a property well known for reducing the complexity of the quantified CSP (QCSP) from Pspace to NP. In contrast, we identify some classes of problems for which the VCSP is tractable but the QVCSP is Pspace-hard.

As a main Corollary, we derive an analogue of Shaeffer’s dichotomy between P and Pspace for QCSP on Boolean languages and Cohen et al. dichotomy between P and NP-complete for VCSP on Boolean valued languages: we prove that the QVCSP follows a dichotomy between P and Pspace-complete.

Finally, we exhibit examples of NP-complete QVCSP for domains of size 3 and more, which suggest at best a trichotomy between P, NP-complete and Pspace-complete for the QVCSP.

Supported by Université Clermont Auvergne, CNRS, LIMOS and Université Caen normandie, CNRS, GREYC.

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Notes

  1. 1.

    We deviate marginally from the standard definition, which would require to rescale by the arity.

  2. 2.

    It was anonymous in [23] and the term was coined in [21].

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Acknowledgments

The authors are thankful to the three anonymous reviewers for their valuable comments which have helped us improve the manuscript.

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Authors and Affiliations

  1. Université Paris-Est Créteil, LACL, Créteil, France

    Florent Madelaine

  2. Université Caen Normandie, CNRS, GREYC, Caen, France

    Stéphane Secouard

Authors
  1. Florent Madelaine
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  2. Stéphane Secouard
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Correspondence to Florent Madelaine .

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Editors and Affiliations

  1. Carnegie Mellon University, Pittsburgh, PA, USA

    John Hooker

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Madelaine, F., Secouard, S. (2018). Quantified Valued Constraint Satisfaction Problem. In: Hooker, J. (eds) Principles and Practice of Constraint Programming. CP 2018. Lecture Notes in Computer Science(), vol 11008. Springer, Cham. https://doi.org/10.1007/978-3-319-98334-9_20

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  • DOI: https://doi.org/10.1007/978-3-319-98334-9_20

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