Geometric Embeddability of Complexes Is ∃ℝ-Complete

Authors Mikkel Abrahamsen , Linda Kleist , Tillmann Miltzow

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Mikkel Abrahamsen
  • University of Copenhagen, Denmark
Linda Kleist
  • Technische Universität Braunschweig, Germany
Tillmann Miltzow
  • Utrecht University, The Netherlands


We thank Arkadiy Skopenkov for his kind and swift help with acquiring literature and Martin Tancer for pointing out a mistake in a previous version of this manuscript.

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Mikkel Abrahamsen, Linda Kleist, and Tillmann Miltzow. Geometric Embeddability of Complexes Is ∃ℝ-Complete. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in ℝ^d is complete for the Existential Theory of the Reals for all d ≥ 3 and k ∈ {d-1,d}. Consequently, the problem is polynomial time equivalent to determining whether a polynomial equation system has a real solution and other important problems from various fields related to packing, Nash equilibria, minimum convex covers, the Art Gallery Problem, continuous constraint satisfaction problems, and training neural networks. Moreover, this implies NP-hardness and constitutes the first hardness result for the algorithmic problem of geometric embedding (abstract simplicial) complexes. This complements recent breakthroughs for the computational complexity of piece-wise linear embeddability.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • simplicial complex
  • geometric embedding
  • linear embedding
  • hypergraph
  • recognition
  • existential theory of the reals


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