Skip to main content
SpringerLink
Log in

Geometric Hitting Set for Segments of Few Orientations

  • Conference paper
  • First Online:
Approximation and Online Algorithms (WAOA 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9499))

Included in the following conference series:

  • 496 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alon, N.: A non-linear lower bound for planar epsilon-nets. Discrete Comput. Geom. 47, 235–244 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aronov, B., Ezra, E., Sharir, M.: Small-size \(\varepsilon \)-nets for axis-parallel rectangles and boxes. SIAM J. Computing 39, 3248–3282 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brimkov, V.E.: Approximability issues of guarding a set of segments. Int. J. Comput. Math. 90, 1653–1667 (2013)

    Article  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14, 263–279 (1995)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  10. Chvátal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4, 233–235 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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. Dom, M., Fellows, M.R., Rosamond, F.A., Sikdar, S.: The parameterized complexity of stabbing rectangles. Algorithmica 62, 564–594 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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. Dumitrescu, A., Jiang, M.: On the approximability of covering points by lines and related problems. Comput. Geom. 48, 703–717 (2015)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  18. Even, G., Rawitz, D., Shahar, S.M.: Hitting sets when the VC-dimension is small. Inf. Proc. Letters 95, 358–362 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Giannopoulos, P., Knauer, C., Rote, G., Werner, D.: Fixed-parameter tractability and lower bounds for stabbing problems. Comput. Geom. 46, 839–860 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hassin, R., Megiddo, N.: Approximation algorithms for hitting objects with straight lines. Discrete Appl. Math. 30, 29–42 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  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. Hochbaum, D.S., Maas, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 130–136 (1985)

    Article  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  26. Kovaleva, S., Spieksma, F.C.: Approximation algorithms for rectangle stabbing and interval stabbing problems. SIAM J. Discrete Math. 20, 748–768 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  29. Langerman, S., Morin, P.: Covering things with things. Discrete Comput. Geom. 33, 717–729 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Megiddo, N., Tamir, A.: On the complexity of locating linear facilities in the plane. Oper. Res. Lett. 1, 194–197 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mustafa, N.H., Ray, S.: Improved results on geometric hitting set problems. Discrete Comput. Geom. 44, 883–895 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. O’Rourke, J.: Art Gallery Theorems and Algorithms. The International Series of Monographs on Computer Science. Oxford University Press, New York (1987)

    MATH  Google Scholar 

  33. Pach, J., Tardos, G.: Tight lower bounds for the size of epsilon-nets. J. Am. Math. Soc. 26, 645–658 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Urrutia, J.: Art Gallery and Illumination Problems. Handbook of Computational Geometry. Elsevier, Amsterdam (1999). Chap. 22

    Google Scholar 

Download references

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).

Author information

Authors and Affiliations

  1. TU Braunschweig, Braunschweig, Germany

    Sándor P. Fekete

  2. Stony Brook University, Stony Brook, NY, USA

    Kan Huang & Joseph S. B. Mitchell

  3. Sandia National Labs, Albuquerque, NM, USA

    Ojas Parekh & Cynthia A. Phillips

Authors
  1. Sándor P. Fekete
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kan Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Joseph S. B. Mitchell
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ojas Parekh
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Cynthia A. Phillips
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cynthia A. Phillips .

Editor information

Editors and Affiliations

  1. University of Waterloo, Waterloo, Canada

    Laura Sanità

  2. Technische Universität Berlin, Berlin, Berlin, Germany

    Martin Skutella

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Fekete, S.P., Huang, K., Mitchell, J.S.B., Parekh, O., Phillips, C.A. (2015). Geometric Hitting Set for Segments of Few Orientations. In: Sanità, L., Skutella, M. (eds) Approximation and Online Algorithms. WAOA 2015. Lecture Notes in Computer Science(), vol 9499. Springer, Cham. https://doi.org/10.1007/978-3-319-28684-6_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-28684-6_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-28683-9

  • Online ISBN: 978-3-319-28684-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation