An Algebraic Characterisation of Complexity for Valued Constraint

  • David A. Cohen
  • Martin C. Cooper
  • Peter G. Jeavons
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4204)


Classical constraint satisfaction is concerned with the feasibility of satisfying a collection of constraints. The extension of this framework to include optimisation is now also being investigated and a theory of so-called soft constraints is being developed. In this extended framework, tuples of values allowed by constraints are given desirability weightings, or costs, and the goal is to find the most desirable (or least cost) assignment.

The complexity of any optimisation problem depends critically on the type of function which has to be minimized. For soft constraint problems this function is a sum of cost functions 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 rational numbers or infinite the complexity of a soft constraint problem is determined by certain algebraic properties of the valued constraint language, which we call feasibility polymorphisms and fractional polymorphisms.

As an immediate application of these results, we show that the existence of a non-trivial fractional polymorphism is a necessary condition for the tractability of a valued constraint language with rational or infinite costs over any finite domain (assuming P ≠ NP).


Cost Function Constraint Satisfaction Problem Algebraic Property Soft Constraint Submodular Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • David A. Cohen
    • 1
  • Martin C. Cooper
    • 2
  • Peter G. Jeavons
    • 3
  1. 1.Department of Computer ScienceRoyal Holloway, University of LondonUK
  2. 2.IRITUniversity of Toulouse IIIFrance
  3. 3.Computing LaboratoryUniversity of OxfordUK

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