Necessary conditions for tractability of valued CSPs

TL;DR

This paper explores the structure of weighted clones in valued constraint satisfaction problems, providing necessary conditions for their tractability and analyzing when these conditions are also sufficient.

Contribution

It extends previous work on weighted clones, offering new insights into the conditions that determine the computational complexity of valued CSPs.

Findings

Some necessary conditions are also sufficient for tractability.

Other necessary conditions are proven not to be sufficient.

The results have implications for classifying the complexity of valued CSPs.

Abstract

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. [SICOMP'13] have characterised 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 Zivny from [CP'11/SICOMP'13] 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.