Advertisement

Algorithmica

pp 1–29 | Cite as

Guarding Orthogonal Art Galleries with Sliding k-Transmitters: Hardness and Approximation

  • Therese Biedl
  • Timothy M. Chan
  • Stephanie Lee
  • Saeed Mehrabi
  • Fabrizio Montecchiani
  • Hamideh Vosoughpour
  • Ziting Yu
Article
  • 70 Downloads

Abstract

A sliding k-transmitter inside an orthogonal polygon P, for a fixed \(k\ge 0\), is a point guard that travels along an axis-parallel line segment s in P. The sliding k-transmitter can see a point \(p\in P\) if the perpendicular from p onto s intersects the boundary of P in at most k points. In the Minimum Sliding k-Transmitters (\({\hbox {ST}}_k\)) problem, the objective is to guard P with the minimum number of sliding k-transmitters. In this paper, we give a constant-factor approximation algorithm for the \({\hbox {ST}}_k\) problem on P for any fixed \(k\ge 0\). Moreover, we show that the \({\hbox {ST}}_0\) problem is NP-hard on orthogonal polygons with holes even if only horizontal sliding 0-transmitters are allowed. For \(k>0\), the problem is NP-hard even in the extremely restricted case where P is simple and monotone. Finally, we study art gallery theorems; i.e., we give upper and lower bounds on the number of sliding transmitters required to guard P relative to the number of vertices of P.

Keywords

Art gallery problem Sliding k-transmitters \(\epsilon \)-nets 

References

  1. 1.
    Aggarwal, A.: The art gallery theorem: its variations, applications and algorithmic aspects. PhD thesis, Johns Hopkins University, (1984)Google Scholar
  2. 2.
    Aichholzer, O., Monroy, R.F., Flores-Peñaloza, D., Hackl, T., Urrutia, J., Vogtenhuber, B.: Modem illumination of monotone polygons. Comput. Geom. 68, 101–118 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Avis, D., Toussaint, G.T.: An optimal algorithm for determining the visibility of a polygon from an edge. IEEE Trans. Comput. 30(12), 910–914 (1981)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ballinger, B., Benbernou, N., Bose, P., Damian, M., Demaine, E.D., Dujmovic, V., Flatland, R.Y., Hurtado, F., Iacono, J., Lubiw, A., Morin, P., Adinolfi, V.S., Souvaine, D.L., Uehara, R.: Coverage with k-transmitters in the presence of obstacles. J. Comb. Optim. 25(2), 208–233 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Biedl, T., Chan, T.M., Lee, S., Mehrabi, S., Montecchiani, F., Vosoughpour, H.: On guarding orthogonal polygons with sliding cameras. In: Poon, S.-H., et al. (eds.) Proceedings of the 11th International Workshop on Algorithms and Computation (WALCOM 2017), volume 10167 of Lecture Notes in Computer Science, pp. 1–12. Springer, (2017)Google Scholar
  6. 6.
    Biedl, T., Kant, G., Kaufmann, M.: On triangulating planar graphs under the four-connectivity constraint. Algorithmica 19(4), 427–446 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Biedl, T., Mehrabi, S.: On r-guarding thin orthogonal polygons. In: 27th International Symposium on Algorithms and Computation (ISAAC 2016) vol. 64 of LIPIcs, pp. 17:1–17:13 (2016)Google Scholar
  8. 8.
    Biedl, T., Mehrabi, S., Yu, Z.: Sliding \(k\)-transmitters: Hardness and approximation. In: Proceedings of the 28th Canadian Conference on Computational Geometry (CCCG 2016), Vancouver, Canada (2016)Google Scholar
  9. 9.
    Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discret. Comput. Geom. 14(4), 463–479 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cannon, S., Fai, T.G., Iwerks, J., Leopold, U., Schmidt, C.: Combinatorics and complexity of guarding polygons with edge and point 2-transmitters. Comput. Geom. 68, 89–100 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chvátal, V.: A combinatorial theorem in plane geometry. J. Comb. Theory Ser. B 18, 39–41 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Clarkson, K.L., Varadarajan, K.R.: Improved approximation algorithms for geometric set cover. Discret. Comput. Geom. 37(1), 43–58 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    de Berg, M., Durocher, S., Mehrabi, S.: Guarding monotone art galleries with sliding cameras in linear time. J. Discret. Algorithms 44, 39–47 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Durocher, S., Filtser, O., Fraser, R., Mehrabi, A.D., Mehrabi, S.: A (7/2)-approximation algorithm for guarding orthogonal art galleries with sliding cameras. In: Proceedings of Latin-American Symposium (LATIN 2014), vol. 8392 of LNCS, pp. 294–305 (2014)Google Scholar
  15. 15.
    Durocher, S., Filtser, O., Fraser, R., Mehrabi, A.D., Mehrabi, S.: Guarding orthogonal art galleries with sliding cameras. Comput. Geom. 65, 12–26 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Durocher, S., Mehrabi, S.: Guarding orthogonal art galleries using sliding cameras: algorithmic and hardness results. In: Proceedings of Mathematical Foundations of Computer Science (MFCS 2013), vol. 8087 of LNCS, pp. 314–324 (2013)Google Scholar
  17. 17.
    Eidenbenz, S., Stamm, C., Widmayer, P.: Inapproximability results for guarding polygons and terrains. Algorithmica 31(1), 79–113 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Garey, M.R., Johnson, D.S.: The rectilinear Steiner tree problem in NP-complete. SIAM J. Appl. Math. 32, 826–834 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hoffmann, F., Kaufmann, M., Kriegel, K.: The art gallery theorem for polygons with holes. In: Proceedings of Foundations of Computer Science (FOCS 1991), pp. 39–48 (1991)Google Scholar
  20. 20.
    Kahn, J., Klawe, M.M., Kleitman, D.J.: Traditional galleries require fewer watchmen. SIAM J. Algebr. Discret. Methods 4(2), 194–206 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Katz, M.J., Mitchell, J.S.B., Nir, Y.: Orthogonal segment stabbing. Comput. Geom. 30(2), 197–205 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Katz, M.J., Morgenstern, G.: Guarding orthogonal art galleries with sliding cameras. Int. J. Comp. Geom. Appl. 21(2), 241–250 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kirkpatrick, D.G.: An \({O}(\lg \lg {OPT})\)-approximation algorithm for multi-guarding galleries. Discret. Comput. Geom. 53(2), 327–343 (2015)CrossRefzbMATHGoogle Scholar
  24. 24.
    Krohn, E., Nilsson, B.J.: Approximate guarding of monotone and rectilinear polygons. Algorithmica 66(3), 564–594 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lee, D.T., Lin, A.K.: Computational complexity of art gallery problems. IEEE Trans. Inf. Theory 32(2), 276–282 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lubiw, A.: Decomposing polygonal regions into convex quadrilaterals. In: Proceedings of the ACM Symposium on Computational Geometry (SoCG 1985), pp. 97–106 (1985)Google Scholar
  27. 27.
    Mehrabi, S.: Geometric optimization problems on orthogonal polygons: hardness results and approximation algorithms. PhD thesis, University of Manitoba, Winnipeg, Canada (2015)Google Scholar
  28. 28.
    O’Rourke, J.: The complexity of computing minimum convex covers for polygons. In: 20th Allerton Conference on Communication Control Computing, pp. 75–84 (1982)Google Scholar
  29. 29.
    O’Rourke, J.: An alternate proof of the rectilinear art gallery theorem. J. Geom. 21, 118–130 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    O’Rourke, J.: Galleries need fewer mobile guards: a variation to Chvátal’s theorem. Geometriae Dedicata 14, 273–283 (1983)MathSciNetzbMATHGoogle Scholar
  31. 31.
    O’Rourke, J.: Art Gallery Theorems and Algorithms. The International Series of Monographs on Computer Science. Oxford University Press, New York (1987)Google Scholar
  32. 32.
    Schuchardt, D., Hecker, H.: Two NP-hard art-gallery problems for ortho-polygons. Math. Logic Q. 41(2), 261–267 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Tamassia, R., Tollis, I.G.: A unified approach to visibility representation of planar graphs. Discret. Comput. Geom. 1, 321–341 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Tomás, A.P.: Guarding thin orthogonal polygons is hard. In: Proceedings of Fundamentals of Computation Theory (FCT 2013), vol. 8070 of LNCS, pp. 305–316 (2013)Google Scholar
  35. 35.
    Urrutia, J.: Art gallery and illumination problems. In: Handbook of Computational Geometry, pp. 973–1027. North-Holland, (2000)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Therese Biedl
    • 1
  • Timothy M. Chan
    • 1
  • Stephanie Lee
    • 1
  • Saeed Mehrabi
    • 1
  • Fabrizio Montecchiani
    • 2
  • Hamideh Vosoughpour
    • 1
  • Ziting Yu
    • 1
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Department of EngineeringUniversity of PerugiaPerugiaItaly

Personalised recommendations