CSPs with Few Alien Constraints

TL;DR

This paper investigates the complexity of constraint satisfaction problems with a limited number of alien constraints, establishing a dichotomy and transferring results across various structures using logical and algebraic methods.

Contribution

It introduces a novel approach to analyze CSPs with alien constraints, providing a comprehensive FPT versus pNP dichotomy and sharper classifications for specific structures.

Findings

Established an FPT versus pNP dichotomy for finite structures.

Derived sharper dichotomies for Boolean structures and equality CSPs.

Provided partial results for general ω-categorical structures.

Abstract

The constraint satisfaction problem asks to decide if a set of constraints over a relational structure $\mathcal{A}$ is satisfiable (CSP$(\mathcal{A})$). We consider CSP$(\mathcal{A} \cup \mathcal{B})$ where $\mathcal{A}$ is a structure and $\mathcal{B}$ is an alien structure, and analyse its (parameterized) complexity when at most $k$ alien constraints are allowed. We establish connections and obtain transferable complexity results to several well-studied problems that previously escaped classification attempts. Our novel approach, utilizing logical and algebraic methods, yields an FPT versus pNP dichotomy for arbitrary finite structures and sharper dichotomies for Boolean structures and first-order reducts of $(\mathbb{N},=)$ (equality CSPs), together with many partial results for general $\omega$-categorical structures.