Necessary Conditions for Tractability of Valued CSPs
Johan Thapper, Stanislav Živný · SIAM Journal on Discrete Mathematics · 2015
The connection between constraint languages and clone theory has been a fruitful line of research on the complexity of constraint satisfaction problems. In a recent result, Cohen et al. [SIAM J. Comput., 42 (2013), pp. 915--1939] have characterized a Galois connection between valued constraint languages and so-called weighted clones. In this paper, we study the structure of weighted clones. We extend the results of Creed and Živný from [Proceedings of the 17th International Conference on Principles and Practice of Constraint Programming, 2011, pp. 210--224] on types of weightings necessarily contained in every nontrivial weighted clone. This result has immediate computational complexity consequences as it provides necessary conditions for tractability of weighted clones and thus valued constraint languages. We demonstrate that some of the necessary conditions are also sufficient for tractability, while others are provably not.