A Topological Version of Schaefer's Dichotomy Theorem

Patrick Schnider; Simon Weber

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arXiv · 2307.03446

A Topological Version of Schaefer's Dichotomy Theorem

Patrick Schnider·Simon Weber

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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.

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Patrick Schnider, Simon Weber. 2023-07-07. A Topological Version of Schaefer's Dichotomy Theorem. https://arxiv.org/abs/2307.03446

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