Schaefer's dichotomy theorem [Schaefer, STOC'78] states that a boolean constraint satisfaction problem (CSP) is polynomial-time solvable if one of six given conditions holds for every type of constraint allowed in its instances. Otherwise, it is NP-complete. In this paper, we analyze boolean CSPs in terms of their topological complexity, instead of their computational complexity. We attach a natural topological space to the set of solutions of a boolean CSP and introduce the notion of projection-universality. We prove that a boolean CSP is projection-universal if and only if it is categorized as NP-complete by Schaefer's dichotomy theorem, showing that the dichotomy translates exactly from computational to topological complexity. We show a similar dichotomy for SAT variants and homotopy-universality.

翻译：Schaefer两分性定理[Schaefer, STOC'78]指出：若布尔约束满足问题(CSP)的实例中允许的每一类约束满足六种给定条件之一，则该问题可在多项式时间内求解；否则即为NP完全问题。本文从拓扑复杂度而非计算复杂度角度分析布尔CSP。我们为布尔CSP的解集关联一个自然拓扑空间，并引入投影泛性质的概念。证明布尔CSP具有投影泛性质当且仅当它被Schaefer两分性定理归类为NP完全问题，揭示该两分性从计算复杂度到拓扑复杂度的精确转化。此外，我们展示了SAT变体与同伦泛性质之间的类似两分性结果。