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Parameter Analysis for Guarding Terrains

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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.

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Notes

  1. For the sake of convenience, we use the convention that the end vertices of the terrain are also its reflex vertices, unless otherwise stated.

  2. Recall that in an orthogonal terrain each vertex is adjacent to at most one horizontal edge and at most one vertical edge.

  3. If \(i=1\), then we do not consider the vertex \(v_{i-1}\). Similarly, if \(j=n\), then we do not consider \(v_{j+1}\). Notice that if \(v_{i-1}\) or \(v_{j+1}\) exist, then they are reflex vertices.

  4. The \(1^{st}\) valley contains the vertex \(v_1\).

  5. We use the convention that \(-\infty \) and \(+\infty \) are the smallest and largest elements, respectively, which are added for ease in comparison among vertices of a valley in the dynamic programming routine.

References

  1. Agrawal, A., Knudsen, K.V., Lokshtanov, D., Saurabh, S., Zehavi, M.: The parameterized complexity of guarding almost convex polygons. In: 36th International Symposium on Computational Geometry (SoCG), vol. 164, pp. 3:1–3:16 (2020)

  2. Ashok, P., Fomin, F.V., Kolay, S., Saurabh, S., Zehavi, M.: Exact algorithms for terrain guarding. ACM Trans. Algorithms 14(2), 25:1–25:20 (2018)

  3. Ben-Moshe, B., Katz, M.J., Mitchell, J.S.B.: A constant-factor approximation algorithm for optimal 1.5d terrain guarding. SIAM J. Comput. 36(6), 1631–1647 (2007)

    Article  MathSciNet  Google Scholar 

  4. Bonnet, É., Giannopoulos, P.: Orthogonal terrain guarding is NP-complete. In: Symposium on Computational Geometry, SoCG 2018, pp. 11:1–11:15 (2018)

  5. Bonnet, É., Giannopoulos, P.: Orthogonal terrain guarding is NP-complete. JoCG 10(2), 21–44 (2019)

    MathSciNet  MATH  Google Scholar 

  6. Bonnet, É., Miltzow, T.: Parameterized hardness of art gallery problems. ACM Trans. Algorithms 16(4), 42:1-42:23 (2020)

    Article  MathSciNet  Google Scholar 

  7. Chen, D.Z., Estivill-Castro, V., Urrutia, J.: Optimal guarding of polygons and monotone chains. In: Proceedings of the 7th Canadian Conference on Computational Geometry, CCCG, pp. 133–138 (1995)

  8. Clarkson, K.L., Varadarajan, K.R.: Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37(1), 43–58 (2007)

    Article  MathSciNet  Google Scholar 

  9. Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)

    Book  Google Scholar 

  10. Diestel, R.: Graph Theory, 4th Edition, volume 173 of Graduate Texts in Mathematics. Springer, Berlin (2012)

    Google Scholar 

  11. Downey, R., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Berlin (2013)

    Book  Google Scholar 

  12. Durocher, S., Li, P.C., Mehrabi, S.: Guarding orthogonal terrains. In: Proceedings of the 27th Canadian Conference on Computational Geometry, CCCG (2015)

  13. Elbassioni, M.K., Krohn, E., Matijevic, D., Mestre, J., Severdija, D.: Improved approximations for guarding 1.5-dimensional terrains. Algorithmica 60(2), 451–463 (2011)

    Article  MathSciNet  Google Scholar 

  14. Friedrichs, S., Hemmer, M., King, J., Schmidt, C.: The continuous 1.5D terrain guarding problem: discretization, optimal solutions, and PTAS. JoCG 7(1), 256–284 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)

    MATH  Google Scholar 

  16. Gibson, M., Kanade, G., Krohn, E., Varadarajan, K.: Guarding terrains via local search. JoCG 5(1), 168–178 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Katz, M.J., Roisman, G.S.: On guarding the vertices of rectilinear domains. Comput. Geom. 39(3), 219–228 (2008)

    Article  MathSciNet  Google Scholar 

  18. Khodakarami, F., Didehvar, F., Mohades, A.: A fixed-parameter algorithm for guarding 1.5D terrains. Theor. Comput. Sci. 595, 130–142 (2015)

    Article  MathSciNet  Google Scholar 

  19. King, J.: A 4-approximation algorithm for guarding 1.5-dimensional terrains. In: Proceedings of the 7th Latin American Symposium on Theoretical Informatics, LATIN, vol. 3887, pp. 629–640 (2006)

  20. King, J., Krohn, E.: Terrain guarding is np-hard. SIAM J. Comput. 40(5), 1316–1339 (2011)

    Article  MathSciNet  Google Scholar 

  21. Lyu, Y., Üngör, A.: A fast 2-approximation algorithm for guarding orthogonal terrains. In: Proceedings of the 28th Canadian Conference on Computational Geometry, CCCG, pp. 161–167 (2016)

  22. Mehrabi, S.: Guarding the vertices of an orthogonal terrain using vertex guards (2015). arXiv:1512.08292

  23. O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford University Press Inc., Oxford (1987)

    MATH  Google Scholar 

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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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Author notes

  1. Akanksha Agrawal

    Present address: Indian Institute of Technology Madras, Chennai, India

  2. Sudeshna Kolay

    Present address: Indian Institute of Technology Kharagpur, Kharagpur, India

Authors and Affiliations

  1. Department of Computer Science, Ben-Gurion University, Beersheba, Israel

    Akanksha Agrawal, Sudeshna Kolay & Meirav Zehavi

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  1. Akanksha Agrawal
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  2. Sudeshna Kolay
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  3. Meirav Zehavi
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Correspondence to Akanksha Agrawal.

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A preliminary version of this article appeared in 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), 2020. During the work Agrawal was supported by PBC Fellowship Program for Outstanding Post-Doctoral Researchers from China and India. Zehavi is supported by Israel Science Foundation (ISF) individual research grant (No. 1176/18) and Binational Science Foundation (BSF) startup grant (No. 2018302).

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Agrawal, A., Kolay, S. & Zehavi, M. Parameter Analysis for Guarding Terrains. Algorithmica 84, 961–981 (2022). https://doi.org/10.1007/s00453-021-00913-9

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