License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.58
URN: urn:nbn:de:0030-drops-179082
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17908/
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Stade, Jack ; Tucker-Foltz, Jamie

Topological Universality of the Art Gallery Problem

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LIPIcs-SoCG-2023-58.pdf (1 MB)

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.

BibTeX - Entry

@InProceedings{stade_et_al:LIPIcs.SoCG.2023.58,
  author =	{Stade, Jack and Tucker-Foltz, Jamie},
  title =	{{Topological Universality of the Art Gallery Problem}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{58:1--58:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17908},
  URN =		{urn:nbn:de:0030-drops-179082},
  doi =		{10.4230/LIPIcs.SoCG.2023.58},
  annote =	{Keywords: Art gallery, Homeomorphism, Exists-R, ETR, Semi-algebraic sets, Universality}
}

Keywords: Art gallery, Homeomorphism, Exists-R, ETR, Semi-algebraic sets, Universality
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023

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