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

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

An application of Farkas' lemma to finite-valued constraint satisfaction problems over infinite domains

“…Remark 4.3. We remark that Theorem 4.2 is consistent with the characterisation of Imp(wPol(Γ)) for finite-domain finite-valued languages [19,20]. Indeed, for a finite-domain finite-valued language Γ it holds that…”

“…. , k} : ρ(x i ) − Combined with (19) above and the fact that e ∈ F, this implies (20). This proves that ρ ∈ Expr(Γ), as desired.…”

“…where the first inequality follows from [19,Theorem 3.3], and the second inequality follows from Proposition 3.3 applied to the specific case of a finite-domain finite-valued language.…”

The Complexity of General-Valued CSPs

“…We remark that, in some papers (e.g., in [13]), fractional polymorphisms (and closely related objects called weighted polymorphisms) are defined as rational-valued functions, which is sufficient for analysing the complexity of VCSPs with finite constraint languages. However, real-valued fractional polymorphisms are necessary to analyse infinite constraint languages [24,38,49].…”