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Polynomial-Time Algorithms for Contiguous Art Gallery and Related Problems

Authors Ahmad Biniaz , Anil Maheshwari , Magnus Christian Ring Merrild , Joseph S. B. Mitchell , Saeed Odak , Valentin Polishchuk , Eliot W. Robson , Casper Moldrup Rysgaard , Jens Kristian Refsgaard Schou , Thomas Shermer , Jack Spalding-Jamieson , Rolf Svenning , Da Wei Zheng

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Art Gallery Problem
  • Computational Geometry
  • Combinatorics
  • Discrete Algorithms

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Abstract

We introduce the contiguous art gallery problem which is to guard the boundary of a simple polygon with a minimum number of guards such that each guard covers exactly one contiguous portion of the boundary. Art gallery problems are often NP-hard. In particular, it is NP-hard to minimize the number of guards to see the boundary of a simple polygon, without the contiguity constraint.
This paper is a merge of three concurrent works [Ahmad Biniaz et al., 2024; Magnus Christian Ring Merrild et al., 2024; Eliot W. Robson et al., 2024] each showing that (surprisingly) the contiguous art gallery problem is solvable in polynomial time. The common idea of all three approaches is developing a greedy function that maps a point on the boundary to the furthest point on the boundary so that the contiguous interval along the boundary between them could be guarded by one guard. Repeatedly applying this function immediately leads to an OPT+1 approximation. By studying this greedy algorithm, we present three different approaches that achieve an optimal solution. The first and second approach apply this greedy algorithm from different points on the boundary that could be found in advance or on the fly while traversing along the boundary (respectively). The third approach represents this function as a piecewise linear rational function, which can be reduced to an abstract arc cover problem involving infinite families of arcs. We identify other problems that can be represented by similar functions, and solve them via the third approach.
From the combinatorial point of view, we show that any n-vertex polygon can be guarded by at most ⌊(n-2)/2⌋ guards. This bound is tight because there are polygons that require this many guards.

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Ahmad Biniaz, Anil Maheshwari, Magnus Christian Ring Merrild, Joseph S. B. Mitchell, Saeed Odak, Valentin Polishchuk, Eliot W. Robson, Casper Moldrup Rysgaard, Jens Kristian Refsgaard Schou, Thomas Shermer, Jack Spalding-Jamieson, Rolf Svenning, and Da Wei Zheng. Polynomial-Time Algorithms for Contiguous Art Gallery and Related Problems. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 20:1-20:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.20

Author Details

Ahmad Biniaz
  • School of Computer Science, University of Windsor, Canada
Anil Maheshwari
  • School of Computer Science, Carleton University, Canada
Magnus Christian Ring Merrild
  • Department of Computer Science, Aarhus University, Denmark
Joseph S. B. Mitchell
  • Department of Applied Mathematics and Statistics, Stony Brook University, NY, USA
Saeed Odak
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
Valentin Polishchuk
  • Communications and Transport Systems, Linköping Univeristy, Sweden
Eliot W. Robson
  • Department of Computer Science, University of Illinois Urbana-Champaign, IL, USA
Casper Moldrup Rysgaard
  • Department of Computer Science, Aarhus University, Denmark
Jens Kristian Refsgaard Schou
  • Department of Computer Science, Aarhus University, Denmark
Thomas Shermer
  • School of Computing Science, Simon Fraser University, Burnaby, Canada
Jack Spalding-Jamieson
  • Independent, Vancouver, Canada
Rolf Svenning
  • Department of Computer Science, Aarhus University, Denmark
Da Wei Zheng
  • Department of Computer Science, University of Illinois Urbana-Champaign, IL, USA

Funding

  • Biniaz, Ahmad: Research supported in part by NSERC.
  • Maheshwari, Anil: Research supported in part by NSERC.
  • Merrild, Magnus Christian Ring: Research supported by Independent Research Fund Denmark (DFF), grant 10.46540/3103-00334B.
  • Mitchell, Joseph S. B.: Partially supported by the National Science Foundation (CCF-2007275).
  • Odak, Saeed: Research supported by NSERC.
  • Polishchuk, Valentin: Partially supported by the Swedish Transport Administration and the Swedish Research Council.
  • Rysgaard, Casper Moldrup: Research supported by Independent Research Fund Denmark (DFF), grant 9131-00113B.
  • Schou, Jens Kristian Refsgaard: Research supported by Independent Research Fund Denmark (DFF), grant 9131-00113B.
  • Shermer, Thomas: Research supported by NSERC.
  • Svenning, Rolf: Research supported by Independent Research Fund Denmark (DFF), grant 9131-00113B.
  • Zheng, Da Wei: Research supported in part by an NSERC PGSD.

Acknowledgements

We are grateful to the anonymous reviewers for helpful suggestions, and we thank Joseph O'Rourke for organizing the open problem session at the Canadian Conference on Computational Geometry 2024 (CCCG24), where Thomas C. Shermer proposed the contiguous art gallery problem. The authors further thank the participants of CCCG24 for their lively discussions of the problem at the conference. We thank the members of the Stony Brook CG Group open problem seminar (2022), where Joe Mitchell posed questions regarding a greedy algorithm for the "CBG" (Contiguous Boundary Guarding) problem and its variants.

Supplementary Materials

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