The algebraic approach to the Constraint Satisfaction Problem (CSP) uses high order symmetries of relational structures -- polymorphisms -- to study the complexity of the CSP. In this paper we further develop one of the methods the algebraic approach can be implemented, and apply it to some kinds of the CSP. This method was introduced in our LICS 2004 paper and involves the study of the local structure of finite algebras and relational structures. It associates with an algebra A or a relational structure S a graph, whose vertices are the elements of A (or S), the edges represent subsets of A such that the restriction of some term operation of A is `good' on the subset, that is, act as an operation of one of the 3 types: semilattice, majority, or affine. In this paper we use this theory and consider algebras with edges from a restricted set of types. We prove type restrictions are preserved under the standard algebraic constructions. Then we show that if the types edges in a relational structure are restricted, then the corresponding CSP can be solved in polynomial time by specific algorithms. In particular, we give a new, somewhat more intuitive proof of the Bounded Width Theorem: the CSP over algebra A has bounded width if and only if A does not contain affine edges. Actually, this result shows that bounded width implies width (2,3). Finally, we prove that algebras without semilattice edges have few subalgebras of powers, that is, the CSP over such algebras is also polynomial time. The methods and results obtained in this paper are important ingredients of the 2017 proof of the Dichotomy Conjecture by the author. The Dichotomy Conjecture was also proved independently by Zhuk.

中文翻译：

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关系结构图：受限类型

约束满足问题 (CSP) 的代数方法使用关系结构的高阶对称性——多态性——来研究 CSP 的复杂性。在本文中，我们进一步开发了一种可以实现代数方法的方法，并将其应用于某些类型的 CSP。这种方法是在我们的 LICS 2004 论文中介绍的，涉及有限代数和关系结构的局部结构的研究。它将代数 A 或关系结构 S 与一个图相关联，其顶点是 A（或 S）的元素，边代表 A 的子集，使得 A 的某些项操作的限制对子集是“好”的，也就是说，作为以下 3 种类型之一的运算：半格、多数或仿射。在本文中，我们使用该理论并考虑具有来自一组受限类型的边的代数。我们证明在标准代数构造下保留了类型限制。然后我们证明，如果关系结构中的类型边受到限制，则可以通过特定算法在多项式时间内求解相应的 CSP。特别是，我们给出了有界宽度定理的一个新的、更直观的证明：当且仅当 A 不包含仿射边时，代数 A 上的 CSP 具有有界宽度。实际上，这个结果表明有界宽度意味着宽度 (2,3)。最后，我们证明了没有半格边的代数几乎没有幂的子代数，也就是说，这些代数上的 CSP 也是多项式时间。本文获得的方法和结果是作者2017年二分猜想证明的重要组成部分。朱克也独立证明了二分法猜想。