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Robustly Guarding Polygons

Authors Rathish Das , Omrit Filtser , Matthew J. Katz , Joseph S.B. Mitchell

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Author Details

Rathish Das
  • University of Houston, TX, USA
Omrit Filtser
  • The Open University of Israel, Israel
Matthew J. Katz
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
Joseph S.B. Mitchell
  • Stony Brook University, NY, USA

Funding

  • Katz, Matthew J.: Partially supported by Grant 495/23 from the Israel Science Foundation and Grant 2019715/CCF-20-08551 from the US-Israel Binational Science Foundation/US National Science Foundation.
  • Mitchell, Joseph S.B.: Partially supported by the National Science Foundation (CCF-2007275).

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Rathish Das, Omrit Filtser, Matthew J. Katz, and Joseph S.B. Mitchell. Robustly Guarding Polygons. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.47

Abstract

We propose precise notions of what it means to guard a domain "robustly", under a variety of models. While approximation algorithms for minimizing the number of (precise) point guards in a polygon is a notoriously challenging area of investigation, we show that imposing various degrees of robustness on the notion of visibility coverage leads to a more tractable (and realistic) problem for which we can provide approximation algorithms with constant factor guarantees.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • geometric optimization
  • approximation algorithms
  • guarding

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References

  1. Mikkel Abrahamsen, Anna Adamaszek, and Tillmann Miltzow. Irrational guards are sometimes needed. In 33rd International Symposium on Computational Geometry (SoCG 2017), volume 77, page 3. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  2. Mikkel Abrahamsen, Anna Adamaszek, and Tillmann Miltzow. The art gallery problem is ∃ ℝ-complete. ACM Journal of the ACM (JACM), 69(1):1-70, 2021. Google Scholar
  3. Akanksha Agrawal, Kristine VK Knudsen, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. The parameterized complexity of guarding almost convex polygons. In 36th International Symposium on Computational Geometry, SoCG 2020, pages LIPIcs-SoCG. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  4. Akanksha Agrawal and Meirav Zehavi. Parameterized analysis of art gallery and terrain guarding. In Computer Science-Theory and Applications: 15th International Computer Science Symposium in Russia, CSR 2020, Yekaterinburg, Russia, June 29-July 3, 2020, Proceedings 15, pages 16-29. Springer, 2020. Google Scholar
  5. Greg Aloupis, Prosenjit Bose, Vida Dujmovic, Chris Gray, Stefan Langerman, and Bettina Speckmann. Triangulating and guarding realistic polygons. Comput. Geom., 47(2):296-306, 2014. Google Scholar
  6. Yoav Amit, Joseph S. B. Mitchell, and Eli Packer. Locating guards for visibility coverage of polygons. International Journal of Computational Geometry & Applications, 20(05):601-630, 2010. Google Scholar
  7. Rom Aschner, Matthew J. Katz, Gila Morgenstern, and Yelena Yuditsky. Approximation schemes for covering and packing. In Subir Kumar Ghosh and Takeshi Tokuyama, editors, WALCOM: Algorithms and Computation, 7th International Workshop, WALCOM 2013, Kharagpur, India, February 14-16, 2013. Proceedings, volume 7748 of Lecture Notes in Computer Science, pages 89-100, 2013. Google Scholar
  8. Pradeesha Ashok and Meghana M Reddy. Efficient guarding of polygons and terrains. In Frontiers in Algorithmics: 13th International Workshop, FAW 2019, Sanya, China, April 29-May 3, 2019, Proceedings 13, pages 26-37. Springer, 2019. Google Scholar
  9. Stav Ashur, Omrit Filtser, and Matthew J. Katz. A constant-factor approximation algorithm for vertex guarding a wv-polygon. J. Comput. Geom., 12(1):128-144, 2021. Google Scholar
  10. Stav Ashur, Omrit Filtser, Matthew J. Katz, and Rachel Saban. Terrain-like graphs: Ptass for guarding weakly-visible polygons and terrains. Comput. Geom., 101:101832, 2022. Google Scholar
  11. Boaz Ben-Moshe, Matthew J. Katz, and Joseph S. B. Mitchell. A constant-factor approximation algorithm for optimal 1.5d terrain guarding. SIAM J. Comput., 36(6):1631-1647, 2007. Google Scholar
  12. Pritam Bhattacharya, Subir Kumar Ghosh, and Bodhayan Roy. Approximability of guarding weak visibility polygons. Discrete Applied Mathematics, 228:109-129, 2017. Google Scholar
  13. Édouard Bonnet and Tillmann Miltzow. An approximation algorithm for the art gallery problem. In 33rd Symposium on Computational Geometry (SoCG 2017), volume 77, page 20, 2017. Google Scholar
  14. Édouard Bonnet and Tillmann Miltzow. Parameterized hardness of art gallery problems. ACM Transactions on Algorithms (TALG), 16(4):1-23, 2020. Google Scholar
  15. Dorit Borrmann, Pedro J de Rezende, Cid C de Souza, Sándor P Fekete, Stephan Friedrichs, Alexander Kröller, Andreas Nüchter, Christiane Schmidt, and Davi C Tozoni. Point guards and point clouds: Solving general art gallery problems. In Proceedings of the 29th Symposium on Computational Geometry, pages 347-348, 2013. Google Scholar
  16. Björn Brodén, Mikael Hammar, and Bengt J Nilsson. Guarding lines and 2-link polygons is apx-hard. In The Thirteenth Canadian Conference on Computational Geometry-CCCG'01, University of Waterloo, ON (13-15 August 2001), pages 45-48. Proc. 13th Canadian Conference on Computational Geometry, 2001. Google Scholar
  17. Hervé Brönnimann and Michael T. Goodrich. Almost optimal set covers in finite vc-dimension. Discret. Comput. Geom., 14(4):463-479, 1995. Google Scholar
  18. Ajay Deshpande, Taejung Kim, Erik D. Demaine, and Sanjay E. Sarma. A pseudopolynomial time O(log n)-approximation algorithm for art gallery problems. In Algorithms and Data Structures, 10th International Workshop, WADS'07, Halifax, Canada, August 15-17, 2007, Proceedings, pages 163-174, 2007. Google Scholar
  19. Michael Gene Dobbins, Andreas Holmsen, and Tillmann Miltzow. Smoothed analysis of the art gallery problem. arXiv preprint, 2018. URL: https://arxiv.org/abs/1811.01177.
  20. Alon Efrat. The complexity of the union of (alpha, beta)-covered objects. SIAM J. Comput., 34(4):775-787, 2005. Google Scholar
  21. Alon Efrat and Sariel Har-Peled. Guarding galleries and terrains. Inf. Process. Lett., 100(6):238-245, 2006. URL: https://doi.org/10.1016/j.ipl.2006.05.014.
  22. Alon Efrat, Sariel Har-Peled, and Joseph S. B. Mitchell. Approximation algorithms for location problems in sensor networks. In 2nd International Conference on Broadband Networks (BROADNETS 2005), 3-7 October 2005, Boston, Massachusetts, USA, pages 767-776, 2005. URL: https://doi.org/10.1109/ICBN.2005.1589677.
  23. Stephan Eidenbenz, Christoph Stamm, and Peter Widmayer. Inapproximability results for guarding polygons and terrains. Algorithmica, 31(1):79-113, 2001. Google Scholar
  24. Jeff Erickson, Ivor Van Der Hoog, and Tillmann Miltzow. Smoothing the gap between np and er. SIAM Journal on Computing, pages FOCS20-102 - FOCS20-138, 2022. Google Scholar
  25. Stephan Friedrichs, Michael Hemmer, James King, and Christiane Schmidt. The continuous 1.5D terrain guarding problem: Discretization, optimal solutions, and PTAS. JoCG, 7(1):256-284, 2016. Google Scholar
  26. Matt Gibson, Gaurav Kanade, Erik Krohn, and Kasturi Varadarajan. Guarding terrains via local search. Journal of Computational Geometry, 5(1):168-178, 2014. Google Scholar
  27. Simon B Hengeveld and Tillmann Miltzow. A practical algorithm with performance guarantees for the art gallery problem. In 37th International Symposium on Computational Geometry, 2021. Google Scholar
  28. Gil Kalai and Jiří Matoušek. Guarding galleries where every point sees a large area. Israel Journal of Mathematics, 101(1):125-139, 1997. Google Scholar
  29. James King and David G. Kirkpatrick. Improved approximation for guarding simple galleries from the perimeter. Discrete & Computational Geometry, 46(2):252-269, 2011. URL: https://doi.org/10.1007/s00454-011-9352-x.
  30. David Kirkpatrick. Guarding galleries with no nooks. In Proceedings of the 12th Canadian Conference on Computational Geometry, pages 43-46. Citeseer, 2000. Google Scholar
  31. David Kirkpatrick. An O(lg lg OPT)-approximation algorithm for multi-guarding galleries. Discrete & Computational Geometry, 53:327-343, 2015. Google Scholar
  32. Erik A Krohn and Bengt J Nilsson. Approximate guarding of monotone and rectilinear polygons. Algorithmica, 66:564-594, 2013. Google Scholar
  33. D Lee and Arthurk Lin. Computational complexity of art gallery problems. IEEE Transactions on Information Theory, 32(2):276-282, 1986. Google Scholar
  34. Lucas Meijer and Tillmann Miltzow. Sometimes two irrational guards are needed. arXiv preprint, 2022. URL: https://arxiv.org/abs/2212.01211.
  35. Joseph O’Rourke. Visibility. In Handbook of discrete and computational geometry, pages 875-896. Chapman and Hall/CRC, 2017. Google Scholar
  36. Thomas C Shermer. Recent results in art galleries (geometry). Proceedings of the IEEE, 80(9):1384-1399, 1992. Google Scholar
  37. Davi C Tozoni, Pedro J De Rezende, and Cid C De Souza. Algorithm 966: A practical iterative algorithm for the art gallery problem using integer linear programming. ACM Transactions on Mathematical Software (TOMS), 43(2):1-27, 2016. Google Scholar
  38. Pavel Valtr. Guarding galleries where no point sees a small area. Israel Journal of Mathematics, 104:1-16, 1998. Google Scholar
  39. Pavel Valtr. On galleries with no bad points. Discrete & Computational Geometry, 21:193-200, 1999. Google Scholar
  40. A. Frank van der Stappen and Mark H. Overmars. Motion planning amidst fat obstacles (extended abstract). In Kurt Mehlhorn, editor, Proceedings of the Tenth Annual Symposium on Computational Geometry, Stony Brook, New York, USA, June 6-8, 1994, pages 31-40. ACM, 1994. Google Scholar
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