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Topological Universality of the Art Gallery Problem

Authors Jack Stade, Jamie Tucker-Foltz

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Author Details

Jack Stade
  • Department of Mathematics, University of Cambridge, UK
Jamie Tucker-Foltz
  • School of Engineering and Applied Sciences, Harvard University, Boston, MA, USA

Funding

  • Tucker-Foltz, Jamie: This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE1745303. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Acknowledgements

We are very grateful to Simon Weber for his helpful, detailed feedback on an earlier version of this paper. We are also grateful to Tillmann Miltzow for numerous helpful suggestions.

Cite AsGet BibTex

Jack Stade and Jamie Tucker-Foltz. Topological Universality of the Art Gallery Problem. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 58:1-58:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.58

Abstract

We prove that any compact semi-algebraic set is homeomorphic to the solution space of some art gallery problem. Previous works have established similar universality theorems, but holding only up to homotopy equivalence, rather than homeomorphism, and prior to this work, the existence of art galleries even for simple spaces such as the Möbius strip or the three-holed torus were unknown. Our construction relies on an elegant and versatile gadget to copy guard positions with minimal overhead. It is simpler than previous constructions, consisting of a single rectangular room with convex slits cut out from the edges. We show that both the orientable and non-orientable surfaces of genus n admit galleries with only O(n) vertices.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Art gallery
  • Homeomorphism
  • Exists-R
  • ETR
  • Semi-algebraic sets
  • Universality

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References

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