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Parameter Analysis for Guarding Terrains

Authors Akanksha Agrawal, Sudeshna Kolay, Meirav Zehavi

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Author Details

Akanksha Agrawal
  • Ben-Gurion University of the Negev, Beersheba, Israel
Sudeshna Kolay
  • Ben-Gurion University of the Negev, Beersheba, Israel
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beersheba, Israel

Acknowledgements

The second author would like to thank Prof. Mark de Berg for very insightful preliminary discussions for the second problem. The first and third authors are thankful to Prof. Saket Saurabh for helpful discussions.

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Akanksha Agrawal, Sudeshna Kolay, and Meirav Zehavi. Parameter Analysis for Guarding Terrains. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SWAT.2020.4

Abstract

The Terrain Guarding problem is a well-known variant of the famous Art Gallery problem. Only second to Art Gallery, it is the most well-studied visibility problem in Discrete and Computational Geometry, which has also attracted attention from the viewpoint of Parameterized complexity. In this paper, we focus on the parameterized complexity of Terrain Guarding (both discrete and continuous) with respect to two natural parameters. First we show that, when parameterized by the number r of reflex vertices in the input terrain, the problem has a polynomial kernel. We also show that, when parameterized by the number c of minima in the terrain, Discrete Orthogonal Terrain Guarding has an XP algorithm.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Terrain Guarding
  • Reflex Vertices
  • Terrain Minima
  • FPT Algorithm
  • XP Algorithm
  • Kernelization

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References

  1. Pradeesha Ashok, Fedor V. Fomin, Sudeshna Kolay, Saket Saurabh, and Meirav Zehavi. Exact algorithms for terrain guarding. ACM Trans. Algorithms, 14(2):25:1-25:20, 2018. Google Scholar
  2. 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
  3. Édouard Bonnet and Panos Giannopoulos. Orthogonal terrain guarding is np-complete. In Symposium on Computational Geometry, SoCG 2018, pages 11:1-11:15, 2018. Google Scholar
  4. Édouard Bonnet and Panos Giannopoulos. Orthogonal terrain guarding is np-complete. JoCG, 10(2):21-44, 2019. Google Scholar
  5. Danny Z. Chen, Vladimir Estivill-Castro, and Jorge Urrutia. Optimal guarding of polygons and monotone chains. In Proceedings of the 7th Canadian Conference on Computational Geometry, CCCG, pages 133-138, 1995. Google Scholar
  6. K. L. Clarkson and K. R. Varadarajan. Improved approximation algorithms for geometric set cover. Discrete and Computational Geometry, 37(1):43-58, 2007. Google Scholar
  7. M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized algorithms. Springer, 2015. Google Scholar
  8. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  9. R. Downey and M. R. Fellows. Fundamentals of parameterized complexity. Springer, 2013. Google Scholar
  10. Stephane Durocher, Pak Ching Li, and Saeed Mehrabi. Guarding orthogonal terrains. In Proceedings of the 27th Canadian Conference on Computational Geometry, CCCG, 2015. Google Scholar
  11. M. K. Elbassioni, E. Krohn, D. Matijevic, J. Mestre, and D. Severdija. Improved approximations for guarding 1.5-dimensional terrains. Algorithmica, 60(2):451-463, 2011. Google Scholar
  12. S. Friedrichs, M. Hemmer, J. King, and C. Schmidt. The continuous 1.5D terrain guarding problem: Discretization, optimal solutions, and PTAS. JoCG, 7(1):256-284, 2016. Google Scholar
  13. M. R. Garey and D. S. Johnson. Computers and intractability: a guide to the theory of NP-completeness. W.H. Freeman, New York, 1979. Google Scholar
  14. M. Gibson, G. Kanade, E. Krohn, and K. Varadarajan. Guarding terrains via local search. JoCG, 5(1):168-178, 2014. Google Scholar
  15. M. J. Katz and G. S. Roisman. On guarding the vertices of rectilinear domains. Computational Geometry, 39(3):219 - -228, 2008. Google Scholar
  16. F. Khodakarami, F. Didehvar, and A. Mohades. A fixed-parameter algorithm for guarding 1.5D terrains. Theoretical Computer Science, 595:130-142, 2015. Google Scholar
  17. James King. A 4-approximation algorithm for guarding 1.5-dimensional terrains. In Proceedings of the 7th Latin American Symposium on Theoretical Informatics, LATIN, volume 3887, pages 629-640, 2006. Google Scholar
  18. James King and Erik Krohn. Terrain guarding is np-hard. SIAM J. Comput., 40(5):1316-1339, 2011. Google Scholar
  19. Yangdi Lyu and Alper Üngör. A fast 2-approximation algorithm for guarding orthogonal terrains. In Proceedings of the 28th Canadian Conference on Computational Geometry, CCCG, pages 161-167, 2016. Google Scholar
  20. S. Mehrabi. Guarding the vertices of an orthogonal terrain using vertex guards. arXiv:1512.08292, 2015. Google Scholar
  21. Joseph O'Rourke. Art Gallery Theorems and Algorithms. Oxford University Press, Inc., 1987. Google Scholar
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