Advertisement

Geometric Hitting Set for Segments of Few Orientations

  • Sándor P. Fekete
  • Kan Huang
  • Joseph S. B. Mitchell
  • Ojas Parekh
  • Cynthia A. PhillipsEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9499)

Abstract

We study several natural instances of the geometric hitting set problem for input consisting of sets of line segments (and rays, lines) having a small number of distinct slopes. These problems model path monitoring (e.g., on road networks) using the fewest sensors (the “hitting points”). We give approximation algorithms for cases including (i) lines of 3 slopes in the plane, (ii) vertical lines and horizontal segments, (iii) pairs of horizontal/vertical segments. We give hardness and hardness of approximation results for these problems. We prove that the hitting set problem for vertical lines and horizontal rays is polynomially solvable.

Notes

Acknowledgment

This work is supported by the Laboratory Directed Research and Development program at Sandia National Laboratories, a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. J. Mitchell acknowledges support from the US-Israel Binational Science Foundation (grant 2010074) and the National Science Foundation (CCF-1018388, CCF-1526406).

References

  1. 1.
    Alon, N.: A non-linear lower bound for planar epsilon-nets. Discrete Comput. Geom. 47, 235–244 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aronov, B., Ezra, E., Sharir, M.: Small-size \(\varepsilon \)-nets for axis-parallel rectangles and boxes. SIAM J. Computing 39, 3248–3282 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brimkov, V.E.: Approximability issues of guarding a set of segments. Int. J. Comput. Math. 90, 1653–1667 (2013)CrossRefzbMATHGoogle Scholar
  4. 4.
    Brimkov, V.E., Leach, A., Mastroianni, M., Wu, J.: Experimental study on approximation algorithms for guarding sets of line segments. In: Bebis, G., et al. (eds.) ISVC 2010, Part I. LNCS, vol. 6453, pp. 592–601. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Brimkov, V.E., Leach, A., Mastroianni, M., Wu, J.: Guarding a set of line segments in the plane. Theoret. Comput. Sci. 412, 1313–1324 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brimkov, V.E., Leach, A., Wu, J., Mastroianni, M.: Approximation algorithms for a geometric set cover problem. Discrete Appl. Math 160, 1039–1052 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brodén, B., Hammar, M., Nilsson, B.J.: Guarding lines and 2-link polygons is APX-hard. In: Proceedings of 13th Canadian Conference on Computational Geometry, pp. 45–48 (2001)Google Scholar
  8. 8.
    Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14, 263–279 (1995)CrossRefGoogle Scholar
  9. 9.
    Carr, R., Fujito, T., Konjevod, G., Parekh, O.: A 2 1/10-approximation algorithm for a generalization of the weighted edge-dominating set problem. In: Paterson, M. (ed.) ESA 2000. LNCS, vol. 1879, pp. 132–142. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  10. 10.
    Chvátal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4, 233–235 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Clarkson, K.L.: Algorithms for polytope covering and approximation. In: Dehne, F., Sack, J.-R., Santoro, N., Whitesdes, S. (eds.) Proceedings of 3rd Workshop Algorithms and Data Structures. LNCS, vol. 709, pp. 246–252. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  12. 12.
    Clarkson, K.L., Varadarajan, K.: Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37, 43–58 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dinur, I., Steurer, D.: Analytical approach to parallel repetition. In: Proceedings of the 46th ACM Symposium on Theory of Computing, pp. 624–633 (2014)Google Scholar
  14. 14.
    Dom, M., Fellows, M.R., Rosamond, F.A., Sikdar, S.: The parameterized complexity of stabbing rectangles. Algorithmica 62, 564–594 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Duh, R.-C., Fürer, M.: Approximation of k-set cover by semi-local optimization. In: Proceedings of the 29th ACM Symposium on Theory of Computing, pp. 256–264 (1997)Google Scholar
  16. 16.
    Dumitrescu, A., Jiang, M.: On the approximability of covering points by lines and related problems. Comput. Geom. 48, 703–717 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Even, G., Levi, R., Rawitz, D., Schieber, B., Shahar, S.M., Sviridenko, M.: Algorithms for capacitated rectangle stabbing and lot sizing with joint set-up costs. ACM Trans. Algorithms 4, 34:1–34:17 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Even, G., Rawitz, D., Shahar, S.M.: Hitting sets when the VC-dimension is small. Inf. Proc. Letters 95, 358–362 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gaur, D.R., Bhattacharya, B.: Covering points by axis parallel lines. In: Proceedings of 23rd European Workshop on Computational Geometry, pp. 42–45 (2007)Google Scholar
  20. 20.
    Gaur, D.R., Ibaraki, T., Krishnamurti, R.: Constant ratio approximation algorithms for the rectangle stabbing problem and the rectilinear partitioning problem. J. Algorithms 43, 138–152 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Giannopoulos, P., Knauer, C., Rote, G., Werner, D.: Fixed-parameter tractability and lower bounds for stabbing problems. Comput. Geom. 46, 839–860 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hassin, R., Megiddo, N.: Approximation algorithms for hitting objects with straight lines. Discrete Appl. Math. 30, 29–42 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Heednacram, A.: The NP-hardness of covering points with lines, paths and tours and their tractability with FPT-algorithms. Ph.D. Thesis, Griffith University (2010)Google Scholar
  24. 24.
    Hochbaum, D.S., Maas, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 130–136 (1985)CrossRefzbMATHGoogle Scholar
  25. 25.
    Joshi, A., Narayanaswamy, N.S.: Approximation algorithms for hitting triangle-free sets of line segments. In: Ravi, R., Gørtz, I.L. (eds.) SWAT 2014. LNCS, vol. 8503, pp. 357–367. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  26. 26.
    Kovaleva, S., Spieksma, F.C.: Approximation algorithms for rectangle stabbing and interval stabbing problems. SIAM J. Discrete Math. 20, 748–768 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kratsch, S., Philip, G., Ray, S.: Point line cover: the easy kernel is essentially tight. In: Proceedings of 25th ACM-SIAM Symposium on Discrete Algorithms, pp. 1596–1606 (2014)Google Scholar
  28. 28.
    Kumar, V.S.A., Arya, S., Ramesh, H.: Hardness of set cover with intersection 1. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 624–635. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  29. 29.
    Langerman, S., Morin, P.: Covering things with things. Discrete Comput. Geom. 33, 717–729 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Megiddo, N., Tamir, A.: On the complexity of locating linear facilities in the plane. Oper. Res. Lett. 1, 194–197 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mustafa, N.H., Ray, S.: Improved results on geometric hitting set problems. Discrete Comput. Geom. 44, 883–895 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    O’Rourke, J.: Art Gallery Theorems and Algorithms. The International Series of Monographs on Computer Science. Oxford University Press, New York (1987)zbMATHGoogle Scholar
  33. 33.
    Pach, J., Tardos, G.: Tight lower bounds for the size of epsilon-nets. J. Am. Math. Soc. 26, 645–658 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Urrutia, J.: Art Gallery and Illumination Problems. Handbook of Computational Geometry. Elsevier, Amsterdam (1999). Chap. 22Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Sándor P. Fekete
    • 1
  • Kan Huang
    • 2
  • Joseph S. B. Mitchell
    • 2
  • Ojas Parekh
    • 3
  • Cynthia A. Phillips
    • 3
    Email author
  1. 1.TU BraunschweigBraunschweigGermany
  2. 2.Stony Brook UniversityStony BrookUSA
  3. 3.Sandia National LabsAlbuquerqueUSA

Personalised recommendations