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# Irrational Guards are Sometimes Needed

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LIPIcs.SoCG.2017.3.pdf
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## Cite As

Mikkel Abrahamsen, Anna Adamaszek, and Tillmann Miltzow. Irrational Guards are Sometimes Needed. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.SoCG.2017.3

## Abstract

In this paper we study the art gallery problem, which is one of the fundamental problems in computational geometry. The objective is to place a minimum number of guards inside a simple polygon so that the guards together can see the whole polygon. We say that a guard at position x sees a point y if the line segment xy is contained in the polygon. Despite an extensive study of the art gallery problem, it remained an open question whether there are polygons given by integer coordinates that require guard positions with irrational coordinates in any optimal solution. We give a positive answer to this question by constructing a monotone polygon with integer coordinates that can be guarded by three guards only when we allow to place the guards at points with irrational coordinates. Otherwise, four guards are needed. By extending this example, we show that for every n, there is a polygon which can be guarded by 3n guards with irrational coordinates but needs 4n guards if the coordinates have to be rational. Subsequently, we show that there are rectilinear polygons given by integer coordinates that require guards with irrational coordinates in any optimal solution.
##### Keywords
• art gallery problem
• computational geometry
• irrational numbers

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## References

1. Pankaj Kumar Agarwal, Kurt Mehlhorn, and Monique Teillaud. Dagstuhl Seminar 11111, Computational Geometry, March 13-18 , 2011.
2. Saugata Basu, Richard Pollack, and Marie-Françoise Roy. Algorithms in real algebraic geometry. Springer-Verlag Berlin Heidelberg, 2006.
3. Patrice Belleville. Computing two-covers of simple polygons. Master’s thesis, McGill University, 1991.
4. Édouard Bonnet and Tillmann Miltzow. An approximation algorithm for the art gallery problem. CoRR, abs/1607.05527, 2016.
5. Édouard Bonnet and Tillmann Miltzow. Parameterized hardness of art gallery problems. In Proceedings of the 24th Annual European Symposium on Algorithms (ESA), pages 19:1-19:17, 2016.
6. Björn Brodén, Mikael Hammar, and Bengt J. Nilsson. Guarding lines and 2-link polygons is APX-hard. In Proceedings of the 13th Canadian Conference on Computational Geometry (CCCG), pages 45-48, 2001.
7. John Canny. Some algebraic and geometric computations in PSPACE. In Proceedings of the twentieth annual ACM symposium on Theory of computing (STOC), pages 460-467. ACM, 1988.
8. Jean Cardinal. Computational geometry column 62. SIGACT News, 46(4):69-78, December 2015. URL: http://dx.doi.org/10.1145/2852040.2852053.
9. Vasek Chvátal. A combinatorial theorem in plane geometry. Journal of Combinatorial Theory, Series B, 18(1):39-41, 1975.
10. Pedro Jussieu de Rezende, Cid C. de Souza, Stephan Friedrichs, Michael Hemmer, Alexander Kröller, and Davi C. Tozoni. Engineering art galleries. In Algorithm Engineering: Selected Results and Surveys, LNCS, pages 379-417. Springer, 2016.
11. The Sage Developers. SageMath, the Sage Mathematics Software System (Version 7.4), 2016. http://www.sagemath.org.
12. Alon Efrat and Sariel Har-Peled. Guarding galleries and terrains. Inf. Process. Lett., 100(6):238-245, 2006.
13. Stephan Eidenbenz, Christoph Stamm, and Peter Widmayer. Inapproximability results for guarding polygons and terrains. Algorithmica, 31(1):79-113, 2001.
14. Sándor Fekete. Private communication.
15. Steve Fisk. A short proof of Chvátal’s watchman theorem. J. Comb. Theory, Ser. B, 24(3):374, 1978.
16. Stephan Friedrichs, Michael Hemmer, James King, and Christiane Schmidt. The continuous 1.5D terrain guarding problem: Discretization, optimal solutions, and PTAS. Journal of Computational Geometry, 7(1):256-284, 2016.
17. Erik Krohn and Bengt J. Nilsson. Approximate guarding of monotone and rectilinear polygons. Algorithmica, 66(3):564-594, 2013.
18. Der-Tsai Lee and Arthur K. Lin. Computational complexity of art gallery problems. IEEE Transactions on Information Theory, 32(2):276-282, 1986.
19. Jiří Matoušek. Intersection graphs of segments and ∃ ℝ. CoRR, abs/1406.2636, 2014.
20. Joseph O'Rourke. Art Gallery Theorems and Algorithms. Oxford University Press, 1987.
21. Joseph O'Rourke and Kenneth Supowit. Some NP-hard polygon decomposition problems. IEEE Transactions on Information Theory, 29(2):181-190, 1983.
22. Joseph O’Rourke. Visibility. In Jacob E. Goodman and Joseph O’Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 28. Chapman &Hall/CRC, second edition, 2004.
23. Günter Rote. EuroCG open problem session, 2011. See the personal webpage of Günter Rote: URL: http://page.mi.fu-berlin.de/rote/Papers/slides/Open-Problem_artgallery-Morschach-EuroCG-2011.pdf.
24. Marcus Schaefer. Complexity of some geometric and topological problems. In International Symposium on Graph Drawing, pages 334-344. Springer, 2009.
25. Dietmar Schuchardt and Hans-Dietrich Hecker. Two NP-hard art-gallery problems for ortho-polygons. Math. Log. Q., 41:261-267, 1995.
26. Thomas C. Shermer. Recent results in art galleries. Proceedings of the IEEE, 80(9):1384-1399, 1992.
27. Ana Paula Tomás. Guarding thin orthogonal polygons is hard. In Fundamentals of Computation Theory, pages 305-316. Springer, 2013.
28. Jorge Urrutia. Art gallery and illumination problems. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 973-1027. Elsevier, 2000.
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