Discrete & Computational Geometry

, Volume 44, Issue 4, pp 883–895 | Cite as

Improved Results on Geometric Hitting Set Problems



We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P = NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are half-spaces in ℝ3 and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local-search algorithm which iterates over local improvements only.


Hitting sets Local search Epsilon nets Greedy algorithms Transversals Approximation algorithms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agarwal, P.K., Mustafa, N.H.: Independent set of intersection graphs of convex objects in 2d. Comput. Geom. 34(2), 83–95 (2006) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ambühl, C., Erlebach, T., Mihalák, M., Nunkesser, M.: Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs. In: APPROX-RANDOM, pp. 3–14 (2006) Google Scholar
  3. 3.
    Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for k-median and facility location problems. In: STOC, pp. 21–29 (2001) Google Scholar
  4. 4.
    Bronnimann, H., Goodrich, M.: Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14(4), 463–479 (1995) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Carmi, P., Katz, M., Lev-Tov, N.: Covering points by unit disks of fixed location. In: ISAAC, pp. 644–655 (2007) Google Scholar
  6. 6.
    Călinescu, G., Mandoiu, I.I., Wan, P.-J., Zelikovsky, A.Z.: Selecting forwarding neighbors in wireless ad hoc networks. Mob. Netw. Appl. 9(2), 101–111 (2004) CrossRefGoogle Scholar
  7. 7.
    Chan, T., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. In: Proceedings of Symposium on Computational Geometry (2009) Google Scholar
  8. 8.
    Chan, T.M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. In: Symposium on Computational Geometry, pp. 333–340 (2009) Google Scholar
  9. 9.
    Clarkson, K., Varadarajan, K.: Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37, 43–58 (2007) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Even, G., Rawitz, D., Shahar, S.: Hitting sets when the VC-dimension is small. Inf. Process. Lett. 95, 358–362 (2005) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16(6), 1004–1022 (1987) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979) MATHGoogle Scholar
  13. 13.
    Haussler, D., Welzl, E.: Epsilon-nets and simplex range queries. Discrete Comput. Geom. 2, 127–151 (1987) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hochbaum, D.S., Maass, W.: Fast approximation algorithms for a nonconvex covering problem. J. Algorithms 8(3), 305–323 (1987) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kanungo, T., Mount, D.M., Netanyahu, N.S., Piatko, C.D., Silverman, R., Wu, A.Y.: A local search approximation algorithm for k-means clustering. In: Symposium on Computational Geometry, pp. 10–18 (2002) Google Scholar
  16. 16.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972) Google Scholar
  17. 17.
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. Technical Report, Stanford University, Stanford, CA, USA (1977) Google Scholar
  18. 18.
    Matousek, J.: Lectures in Discrete Geometry. Springer, New York (2002) Google Scholar
  19. 19.
    Matousek, J., Seidel, R., Welzl, E.: How to net a lot with little: Small epsilon-nets for disks and halfspaces. In: Proceedings of Symposium on Computational Geometry, pp. 16–22 (1990) Google Scholar
  20. 20.
    Mustafa, N., Ray, S.: Improved results on geometric hitting set problems. In: Proceedings of Symposium on Computational Geometry (2009) Google Scholar
  21. 21.
    Narayanappa, S., Vojtechovský, P.: An improved approximation factor for the unit disk covering problem. In: CCCG (2006) Google Scholar
  22. 22.
    Pach, J., Agarwal, P.K.: Combinatorial Geometry. Wiley, New York (1995) MATHGoogle Scholar
  23. 23.
    Pach, J., Woeginger, G.: Some new bounds for epsilon-nets. In: Symposium on Computational Geometry, pp. 10–15 (1990) Google Scholar
  24. 24.
    Pyrga, E., Ray, S.: New existence proofs for epsilon-nets. In: Proceedings of Symposium on Computational Geometry, pp. 199–207 (2008) Google Scholar
  25. 25.
    Raz, R., Safra, M.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of STOC, pp. 475–484 (1997) Google Scholar
  26. 26.
    Varadarajan, K.: Weighted geometric set cover via quasi uniform sampling. In: STOC’ 10 Proceedings of the 42nd ACM Symposium on Theory of Computing, Cambridge, pp. 641–648. ACM, New York (2010) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dept. of Computer ScienceLUMSLahorePakistan
  2. 2.Max-Plank-Institut für InformatikSaarbrückenGermany

Personalised recommendations