关系结构的图：受限类型
Graphs of relational structures: restricted types
摘要：
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，我们进一步发展的方法之一，代数方法可以实现，并将其应用到一些类型的CSP。这种方法在我们的LICS 2004论文中介绍，涉及有限代数和关系结构的局部结构的研究。它将一个代数A或一个关系结构S与一个图联系起来，图的顶点是A（或S）的元素，边表示A的子集，使得A的某些项运算的限制在子集上是“好的”，即充当三种类型之一的运算：半格、多数或仿射。在本文中，我们使用这一理论，并考虑代数的边缘从一个限制的类型集。我们证明了标准代数结构下的类型限制。然后，我们表明，如果在一个关系结构中的类型边是受限制的，那么相应的CSP可以解决在多项式时间通过特定的算法。特别是，我们给出了一个新的，有点更直观的证明有界宽度定理：CSP代数A有界宽度当且仅当A不包含仿射边。实际上，这个结果表明有界宽度意味着宽度（2，3）。最后，证明了无半格边代数的幂子代数很少，即这类代数上的CSP也是多项式时间的。本文所得到的方法和结果是作者2017年证明二分法猜想的重要组成部分。二分法猜想也被朱克独立证明。
英文摘要：
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.
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