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


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.

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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)


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
  • Art gallery
  • Homeomorphism
  • Exists-R
  • ETR
  • Semi-algebraic sets
  • Universality


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  1. Mikkel Abrahamsen, Anna Adamaszek, and Tillmann Miltzow. The art gallery problem is ∃ℝ-complete. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 65-73, 2018. URL:
  2. Daniel Bertschinger, Nicolas El Maalouly, Tillmann Miltzow, Patrick Schnider, and Simon Weber. Topological art in simple galleries. In Symposium on Simplicity in Algorithms (SOSA), pages 87-116. SIAM, 2022. Google Scholar
  3. Josef Blass and Wlodzimierz Holsztynski. Cubical polyhedra and homotopy, III. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, 53(3-4):275-279, September 1972. URL:
  4. Édouard Bonnet and Tillmann Miltzow. An approximation algorithm for the art gallery problem. In Boris Aronov and Matthew J. Katz, editors, 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, volume 77 of LIPIcs, pages 20:1-20:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL:
  5. Michael G. Dobbins, Andreas Holmsen, and Tillmann Miltzow. A universality theorem for nested polytopes. arXiv preprint arXiv:1908.02213, 2019. Google Scholar
  6. Heisuke Hironaka. Triangulations of algebraic sets. In Algebraic geometry (Proceedings of Symposia in Pure Mathematics, Volume 29, Humboldt State University, Arcata, California, 1974), volume 29, pages 165-185, 1975. Google Scholar
  7. Kyle Roger Hoffman. Triangulation of locally semi-algebraic spaces. PhD thesis, University of Michigan, 2009. Google Scholar
  8. D. T. Lee and Arthur K. Lin. Computational complexity of art gallery problems. IEEE Transactions on Information Theory, 32(2):276-282, 1986. URL:
  9. William S. Massey. A basic course in algebraic topology, volume 127 of Graduate Texts in Mathematics, chapter 1. Springer, 1991. Google Scholar
  10. Lucas Meijer and Tillmann Miltzow. Sometimes two irrational guards are needed. arXiv preprint arXiv:2212.01211, 2022. Google Scholar
  11. Nikolai E Mnëv. The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In Topology and geometry—Rohlin seminar, pages 527-543. Springer, 1988. Google Scholar
  12. Jürgen Richter-Gebert. Mnëv’s universality theorem revisited. Séminaire Lotharingien de Combinatoire [electronic only], 34:15 p.-15 p., 1995. URL:
  13. Peter W. Shor. Stretchability of pseudolines is NP-hard. In Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift, pages 531-554, 1990. URL:
  14. Jack Stade. The point-boundary art gallery problem is ∃ℝ-hard. arXiv preprint arXiv:2210.12817, 2022. URL: