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.

翻译：约束满足问题旨在判定关系结构 $\mathcal{A}$ 上的一组约束是否可满足（CSP$(\mathcal{A})$）。我们研究 CSP$(\mathcal{A} \cup \mathcal{B})$，其中 $\mathcal{A}$ 是一个结构而 $\mathcal{B}$ 是一个异类结构，并分析当最多允许 $k$ 个异类约束时其（参数化）复杂性。我们建立了与多个被广泛研究但先前难以分类的问题之间的联系，并获得了可迁移的复杂性结果。我们采用逻辑与代数方法的新颖途径，为任意有限结构得出了 FPT 与 pNP 的二分性，为布尔结构和 $(\mathbb{N},=)$ 的一阶归约（等式 CSP）提供了更精细的二分性，同时为一般的 $\omega$-范畴结构获得了许多部分结果。