Equivariant algorithms for constraint satisfaction problems over coset templates

TL;DR

This paper introduces a new, concise proof method linking equivariant algorithms solving CSPs over coset templates to the property of being 2-Helly, revealing connections between CSP complexity and descriptive logic.

Contribution

The paper presents a novel, self-contained proof technique that connects equivariant CSP algorithms with the 2-Helly property, simplifying existing proofs and highlighting new theoretical links.

Findings

Equivariant algorithms imply the template is 2-Helly.

Bounded width and fixed-point definability coincide with 2-Helly.

New connections between CSP theory and descriptive complexity are established.

Abstract

We investigate the Constraint Satisfaction Problem (CSP) over templates with a group structure, and algorithms solving CSP that are equivariant, i.e. invariant under a natural group action induced by a template. Our main result is a method of proving the implication: if CSP over a coset template T is solvable by an equivariant algorithm then T is 2-Helly (or equivalently, has a majority polymorphism). Therefore bounded width, and definability in fixed-point logics, coincide with 2-Helly. Even if these facts may be derived from already known results, our new proof method has two advantages. First, the proof is short, self-contained, and completely avoids referring to the omitting-types theorems. Second, it brings to light some new connections between CSP theory and descriptive complexity theory, via a construction similar to CFI graphs.