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International Symposium on Mathematical Foundations of Computer Science

MFCS 2011: Mathematical Foundations of Computer Science 2011 pp 231–242Cite as

An Algebraic Theory of Complexity for Valued Constraints: Establishing a Galois Connection

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6907))

Abstract

The complexity of any optimisation problem depends critically on the form of the objective function. Valued constraint satisfaction problems are discrete optimisation problems where the function to be minimised is given as a sum of cost functions defined on specified subsets of variables. These cost functions are chosen from some fixed set of available cost functions, known as a valued constraint language. We show in this paper that when the costs are non-negative rational numbers or infinite, then the complexity of a valued constraint problem is determined by certain algebraic properties of this valued constraint language, which we call weighted polymorphisms. We define a Galois connection between valued constraint languages and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterised. These results provide a new approach in the search for tractable valued constraint languages.

This research was supported by EPSRC grant EP/F01161X/1. Stanislav Živný is supported by a Junior Research Fellowship at University College, Oxford.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Royal Holloway, University of London, UK

    David A. Cohen & Páidí Creed

  2. Department of Computer Science, University of Oxford, UK

    Peter G. Jeavons

  3. University College, University of Oxford, UK

    Stanislav Živný

Authors
  1. David A. Cohen
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  2. Páidí Creed
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  3. Peter G. Jeavons
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  4. Stanislav Živný
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Editor information

Editors and Affiliations

  1. Institute of Informatics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland

    Filip Murlak  & Piotr Sankowski  & 

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Cohen, D.A., Creed, P., Jeavons, P.G., Živný, S. (2011). An Algebraic Theory of Complexity for Valued Constraints: Establishing a Galois Connection. In: Murlak, F., Sankowski, P. (eds) Mathematical Foundations of Computer Science 2011. MFCS 2011. Lecture Notes in Computer Science, vol 6907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22993-0_23

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  • DOI: https://doi.org/10.1007/978-3-642-22993-0_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22992-3

  • Online ISBN: 978-3-642-22993-0

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