The connection between the complexity of constraint languages and clone theory, discovered by Cohen and Jeavons in a series of papers, has been a fruitful line of research on the complexity of CSPs. In a recent result, Cohen et al. [14] have established a Galois connection between the complexity of valued constraint languages and so-called weighted clones. In this paper, we initiate the study of weighted clones. Firstly, we prove an analogue of Rosenberg’s classification of minimal clones for weighted clones. Secondly, we show minimality of several weighted clones whose support clone is generated by a single minimal operation. Finally, we classify all Boolean weighted clones. This classification implies a complexity classification of Boolean valued constraint languages obtained by Cohen et al. [13]


Cost Function Constraint Satisfaction Problem Galois Connection Constraint Language Suitable Scaling 
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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Páidí Creed
    • 1
  • Stanislav Živný
    • 2
  1. 1.Department of Computer ScienceRoyal Holloway, University of LondonUK
  2. 2.University CollegeUniversity of OxfordUK

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