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

TL;DR

This paper extends the Galois connection between weighted clones and relational clones to infinite sizes with real weights, demonstrating the preservation of complexity in valued CSPs.

Contribution

It proves the Galois connection holds for infinite weighted clones with real weights, generalizing previous finite results and maintaining complexity equivalences.

Findings

Galois connection established for infinite weighted clones

Complexity of valued CSPs is preserved under this extension

Answers an open question from Cohen et al. (2013)

Abstract

A Galois connection between clones and relational clones on a fixed finite domain is one of the cornerstones of the so-called algebraic approach to the computational complexity of non-uniform Constraint Satisfaction Problems (CSPs). Cohen et al. established a Galois connection between finitely-generated weighted clones and finitely-generated weighted relational clones [SICOMP'13], and asked whether this connection holds in general. We answer this question in the affirmative for weighted (relational) clones with real weights and show that the complexity of the corresponding valued CSPs is preserved.