Necessary Conditions for Tractability 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 andŽivný 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.

Sherali-Adams Relaxations for Valued CSPs

“…A special case of Lemma 3 for finite-valued constraint languages was proved by the authors in [27]. Lemma 3 has also been observed in [22] and in another recent paper of the authors [29].…”

Representing Fitness Landscapes by Valued Constraints to Understand the Complexity of Local Search

“…An approach that has not yet been explored in any detail, is to extend the modelling and analysis techniques recently developed for valued constraint satisfaction problems [4,12,21,22,2,8] to analyse the computational difficulty of local search. In this paper we begin the development of a novel approach to understanding fitness landscapes based on representing those landscapes as valued constraint satisfaction problems (VCSPs), and studying the properties of the associated constraint graphs.…”

A Galois Connection for Weighted (Relational) Clones of Infinite Size

“…Cohen et al have introduced an algebraic theory of weighted clones [10], further extended in [28,23], for classifying the computational complexity of valued constraint languages. This theory establishes a one-to-one correspondence between valued constraint languages closed under expressibility (which does not change the complexity of the associated class of optimisation problems), called weighted relational clones, and weighted clones [10].…”

“…This is an extension of (a part of) the algebraic approach to CSPs which relies on a one-to-one correspondence between constraint languages closed under pp-definability (which does not change the complexity of the associated class of decision problems), called relational clones, and clones [7], thus making it possible to use deep results from universal algebra. This theory has been developed primarily as an aid for studying the computational complexity of valued CSPS (and indeed, recent progress on valued CSPs [26,27,21,30,20] and on special cases of valued CSPs [31] heavily rely on the theory introduced in [10]), but the theory is interesting in its own right [28,23,33,32].…”