The Parameterized Complexity of the Rectangle Stabbing Problem and Its Variants

  • Michael Dom
  • Somnath Sikdar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5059)

Abstract

We study an NP-complete geometric covering problem called d-Dimensional Rectangle Stabbing, where, given a set of axis-parallel d-dimensional hyperrectangles, a set of axis-parallel (d − 1)-dimensional hyperplanes and a positive integer k, the question is whether one can select at most k of the hyperplanes such that every hyperrectangle is intersected by at least one of these hyperplanes. This problem is well-studied from the approximation point of view, while its parameterized complexity remained unexplored so far. Here we show, by giving a nontrivial reduction from a problem called Multicolored Clique, that for d ≥ 3 the problem is W[1]-hard with respect to the parameter k. For the case d = 2, whose parameterized complexity is still open, we consider several natural restrictions and show them to be fixed-parameter tractable.

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References

  1. 1.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  2. 2.
    Even, G., Rawitz, D., Shahar, S.: Approximation algorithms for capacitated rectangle stabbing. In: Calamoneri, T., Finocchi, I., Italiano, G.F. (eds.) CIAC 2006. LNCS, vol. 3998, pp. 18–29. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Fellows, M.R.: Personal communication (September 2007)Google Scholar
  4. 4.
    Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems (manuscript, 2007)Google Scholar
  5. 5.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)Google Scholar
  6. 6.
    Gaur, D.R., Ibaraki, T., Krishnamurti, R.: Constant ratio approximation algorithms for the rectangle stabbing problem and the rectilinear partitioning problem. J. Algorithms 43(1), 138–152 (2002)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Giannopoulos, P., Knauer, C., Whitesides, S.: Parameterized complexity of geometric problems. The Computer Journal (2007), doi:10.1093/comjnl/bxm053Google Scholar
  8. 8.
    Hassin, R., Megiddo, N.: Approximation algorithms for hitting objects with straight lines. Discrete Appl. Math. 30, 29–42 (1991)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kovaleva, S., Spieksma, F.C.R.: Approximation of a geometric set covering problem. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 493–501. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Kovaleva, S., Spieksma, F.C.R.: Approximation algorithms for rectangle stabbing and interval stabbing problems. SIAM J. Discrete Math. 20(3), 748–768 (2006)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Langerman, S., Morin, P.: Covering things with things. Discrete Comput. Geom. 33(4), 717–729 (2005)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Mecke, S., Schöbel, A., Wagner, D.: Station location – complexity and approximation. In: Proc. 5th ATMOS, IBFI Dagstuhl, Germany (2005)Google Scholar
  13. 13.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Michael Dom
    • 1
  • Somnath Sikdar
    • 2
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany
  2. 2.The Institute of Mathematical SciencesC.I.T CampusChennaiIndia

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