A Characterisation of First-Order Constraint Satisfaction Problems

TL;DR

This paper provides algebraic and combinatorial characterisations of certain finite relational structures, leading to decidability results and algorithms for identifying and solving first-order definable constraint satisfaction problems.

Contribution

It introduces new algebraic and combinatorial criteria for first-order CSPs, along with algorithms for their recognition and solution, advancing understanding of their computational complexity.

Findings

Decidable to determine if a CSP is first-order definable

NP-complete general problem, polynomial-time for cores

Algorithm to produce solutions for first-order definable CSPs

Abstract

We describe simple algebraic and combinatorial characterisations of finite relational core structures admitting finitely many obstructions. As a consequence, we show that it is decidable to determine whether a constraint satisfaction problem is first-order definable: we show the general problem to be NP-complete, and give a polynomial-time algorithm in the case of cores. A slight modification of this algorithm provides, for first-order definable CSP's, a simple poly-time algorithm to produce a solution when one exists. As an application of our algebraic characterisation of first order CSP's, we describe a large family of L-complete CSP's.