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A Topological Version of Schaefer’s Dichotomy Theorem

Authors Patrick Schnider , Simon Weber

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ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Complexity theory and logic
Keywords
  • Computational topology
  • Boolean CSP
  • satisfiability
  • computational complexity
  • solution space
  • homotopy universality
  • homological connectivity

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Abstract

Schaefer’s dichotomy theorem states that a Boolean constraint satisfaction problem (CSP) is polynomial-time solvable if one of four 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. Motivated by complexity and topological universality results in computational geometry, 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 and Simon Weber. A Topological Version of Schaefer’s Dichotomy Theorem. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 77:1-77:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.77

Author Details

Patrick Schnider
  • Department of Computer Science, ETH Zürich, Switzerland
Simon Weber
  • Department of Computer Science, ETH Zürich, Switzerland

Funding

  • Weber, Simon: Swiss National Science Foundation under project no. 204320.

Acknowledgements

We thank Tillmann Miltzow and Sebastian Meyer for the helpful discussions. We further thank the SoCG reviewers for their helpful comments and pointing out related work.

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