The Constraint Satisfaction Problem (CSP) is a problem of computing a homomorphism $\mathbf{R}\to \mathbf{\Gamma}$ between two relational structures, where $\mathbf{R}$ is defined over a domain $V$ and $\mathbf{\Gamma}$ is defined over a domain $D$. In a fixed template CSP, denoted $\rm{CSP}(\mathbf{\Gamma})$, the right side structure $\mathbf{\Gamma}$ is fixed and the left side structure $\mathbf{R}$ is unconstrained. In the last two decades it was discovered that the reasons that make fixed template CSPs polynomially solvable are of algebraic nature, namely, templates that are tractable should be preserved under certain polymorphisms. From this perspective the following problem looks natural: given a prespecified finite set of algebras ${\mathcal B}$ whose domain is $D$, is it possible to present the solution set of a given instance of $\rm{CSP}(\mathbf{\Gamma})$ as a subalgebra of ${\mathbb A}_1\times ... \times {\mathbb A}_{|V|}$ where ${\mathbb A}_i\in {\mathcal B}$? We study this problem and show that it can be reformulated as an instance of a certain fixed-template CSP over another template $\mathbf{\Gamma}^{\mathcal B}$. We study conditions under which $\rm{CSP}(\mathbf{\Gamma})$ can be reduced to $\rm{CSP}(\mathbf{\Gamma}^{\mathcal B})$. This issue is connected with the so-called CSP with an input prototype, formulated in the following way: given a homomorphism from $\mathbf{R}$ to $\mathbf{\Gamma}^{\mathcal B}$ find a homomorphism from $\mathbf{R}$ to $\mathbf{\Gamma}$. We prove that if ${\mathcal B}$ contains only tractable algebras, then the latter CSP with an input prototype is tractable. We also prove that $\rm{CSP}(\mathbf{\Gamma}^{\mathcal B})$ can be reduced to $\rm{CSP}(\mathbf{\Gamma})$ if the set ${\mathcal B}$, treated as a relation over $D$, can be expressed as a primitive positive formula over $\mathbf{\Gamma}$.

翻译：约束满足问题（CSP）是计算两个关系结构之间的同态 $\mathbf{R}\to \mathbf{\Gamma}$ 的问题，其中 $\mathbf{R}$ 定义于论域 $V$，$\mathbf{\Gamma}$ 定义于论域 $D$。在固定模板CSP（记作 $\rm{CSP}(\mathbf{\Gamma})$）中，右侧结构 $\mathbf{\Gamma}$ 固定，而左侧结构 $\mathbf{R}$ 不受约束。近二十年来，研究发现固定模板CSP具有多项式可解性的原因在于其代数性质，即可解模板必须被某种多态性所保持。基于此视角，以下问题自然成立：给定一个预先指定的、定义于论域 $D$ 上的有限代数集合 ${\mathcal B}$，能否将 $\rm{CSP}(\mathbf{\Gamma})$ 的给定实例的解集表示为 ${\mathbb A}_1\times ... \times {\mathbb A}_{|V|}$（其中 ${\mathbb A}_i\in {\mathcal B}$）的子代数？我们研究该问题，并证明其可重新表述为另一个模板 $\mathbf{\Gamma}^{\mathcal B}$ 上的固定模板CSP实例。我们探讨了 $\rm{CSP}(\mathbf{\Gamma})$ 可约化为 $\rm{CSP}(\mathbf{\Gamma}^{\mathcal B})$ 的条件。该问题与所谓的带输入原型的CSP相关，其形式化表述为：给定从 $\mathbf{R}$ 到 $\mathbf{\Gamma}^{\mathcal B}$ 的同态，求从 $\mathbf{R}$ 到 $\mathbf{\Gamma}$ 的同态。我们证明：若 ${\mathcal B}$ 仅包含可解代数，则上述带输入原型的CSP是可解的。此外，若将 ${\mathcal B}$ 视为 $D$ 上的关系，且该关系可表示为 $\mathbf{\Gamma}$ 上的原始正公式，则 $\rm{CSP}(\mathbf{\Gamma}^{\mathcal B})$ 可约化为 $\rm{CSP}(\mathbf{\Gamma})$。