CSPs with Few Alien Constraints

TL;DR

This work investigates CSPs with alien constraints, i.e., CSP$(\mathcal{A} \cup \mathcal{B})$ where only up to $k$ constraints come from the alien structure $\mathcal{B}$. It establishes an FPT versus para-NP-hard dichotomy for arbitrary finite structures and sharp refinements for Boolean languages and equality/ω-categorical settings by weaving universal algebra, pp-definability, and core/retraction techniques into a cohesive framework. The paper also links alien constraints to model checking, redundancy/implication/equivalence problems, and provides general FPT methods when alien relations are existentially definable over the base structure. These results substantially broaden the landscape of CSP classifications, enabling unified treatments across finite and infinite domains and offering practical criteria for tractability in hybrid constraint settings.

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 $ω$-categorical structures.

Theorems & Definitions (27)

• Theorem 1: Jeavons:tcs98
• Theorem 2: Schaefer:stoc78
• Lemma 3
• Theorem 4
• Theorem 5
• Proposition 6
• Theorem 7
• Theorem 8
• Lemma 9
• Theorem 10