A Topological Version of Schaefer's Dichotomy Theorem

TL;DR

This paper explores the topological complexity of boolean CSPs, establishing a dichotomy that parallels Schaefer's computational classification by linking NP-completeness to topological universality.

Contribution

It introduces a topological perspective on boolean CSPs, defining projection-universality and proving it aligns with NP-completeness, extending Schaefer's dichotomy into topology.

Findings

Projection-universality characterizes NP-complete CSPs

Topological dichotomy mirrors computational complexity classification

Homotopy-universality applies to SAT variants

Abstract

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.