Finite Algebras with Hom-Sets of Polynomial Size

TL;DR

This paper characterizes finite algebras with polynomially bounded hom-set sizes, providing internal criteria, decision algorithms, and applications to tractable constraint satisfaction problems in computational complexity.

Contribution

It offers a new internal characterization of such algebras, along with polynomial-time decision procedures and applications to CSP tractability.

Findings

Algebras with no nontrivial strongly abelian congruences have polynomially bounded hom-sets.

Deciding this property is polynomial-time feasible for finite signature algebras.

Homomorphisms to such algebras can be computed efficiently, impacting CSP complexity classifications.

Abstract

We provide an internal characterization of those finite algebras (i.e., algebraic structures) $\mathbf A$ such that the number of homomorphisms from any finite algebra $\mathbf X$ to $\mathbf A$ is bounded from above by a polynomial in the size of $\mathbf X$. Namely, an algebra $\mathbf A$ has this property if, and only if, no subalgebra of $\mathbf A$ has a nontrivial strongly abelian congruence. We also show that the property can be decided in polynomial time for algebras in finite signatures. Moreover, if $\mathbf A$ is such an algebra, the set of all homomorphisms from $\mathbf X$ to $\mathbf A$ can be computed in polynomial time given $\mathbf X$ as input. As an application of our results to the field of computational complexity, we characterize inherently tractable constraint satisfaction problems over fixed finite structures, i.e., those that are tractable and remain tractable after expanding the fixed structure by arbitrary relations or functions.