Advertisement

Parameterized Complexity of Stabbing Rectangles and Squares in the Plane

  • Michael Dom
  • Michael R. Fellows
  • Frances A. Rosamond
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5431)

Abstract

The NP-complete geometric covering problem Rectangle Stabbing is defined as follows: Given a set of horizontal and vertical lines in the plane, a set of rectangles in the plane, and a positive integer k, select at most k of the lines such that every rectangle is intersected by at least one of the selected lines.

While it is known that the problem can be approximated in polynomial time with a factor of two, its parameterized complexity with respect to the parameter k was open so far—only its generalization to three or more dimensions was known to be W[1]-hard. Giving two fixed-parameter reductions, one from the W[1]-complete problem Multicolored Clique and one to the W[1]-complete problem Short Turing Machine Acceptance, we prove that Rectangle Stabbing is W[1]-complete with respect to the parameter k, which in particular means that there is no hope for fixed-parameter tractability with respect to the parameter k. Our reductions show also the W[1]-completeness of the more general problem Set Cover on instances that “almost have the consecutive-ones property”, that is, on instances whose matrix representation has at most two blocks of 1s per row.

For the special case of Rectangle Stabbing where all rectangles are squares of the same size we can also show W[1]-hardness, while the parameterized complexity of the special case where the input consists of rectangles that do not overlap is open. By giving an algorithm running in (4k + 1) k ·n O(1) time, we show that Rectangle Stabbing is fixed-parameter tractable in the still NP-hard case where both these restrictions apply.

Keywords

Parameterized Complexity Vertex Cover Edge Color Vertex Color Blue Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Călinescu, G., Dumitrescu, A., Karloff, H.J., Wan, P.-J.: Separating points by axis-parallel lines. Internat. J. Comput. Geom. Appl. 15(6), 575–590 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Dom, M., Sikdar, S.: The parameterized complexity of the rectangle stabbing problem and its variants. In: Preparata, F.P., Wu, X., Yin, J. (eds.) FAW 2008. LNCS, vol. 5059, pp. 288–299. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefMATHGoogle Scholar
  4. 4.
    Even, G., Levi, R., Rawitz, D., Schieber, B., Shahar, S., Sviridenko, M.: Algorithms for capacitated rectangle stabbing and lot sizing with joint set-up costs. ACM Trans. Algorithms 4(3), Article 34 (2008)Google Scholar
  5. 5.
    Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems (manuscript, 2008)Google Scholar
  6. 6.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)MATHGoogle Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)MATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Giannopoulos, P., Knauer, C., Whitesides, S.: Parameterized complexity of geometric problems. The Computer Journal 51(3), 372–384 (2008)CrossRefGoogle Scholar
  10. 10.
    Hassin, R., Megiddo, N.: Approximation algorithms for hitting objects with straight lines. Discrete Appl. Math. 30, 29–42 (1991)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Koushanfar, F., Slijepcevic, S., Potkonjak, M., Sangiovanni-Vincentelli, A.: Error-tolerant multimodal sensor fusion. In: Proceedings of the IEEE CAS Workshop on Wireless Communications and Networking. IEEE CAS, Los Alamitos (2002)Google Scholar
  12. 12.
    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
  13. 13.
    Kovaleva, S., Spieksma, F.C.R.: Approximation algorithms for rectangle stabbing and interval stabbing problems. SIAM J. Discrete Math. 20(3), 748–768 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Langerman, S., Morin, P.: Covering things with things. Discrete Comput. Geom. 33(4), 717–729 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Marx, D.: Parameterized complexity of independence and domination on geometric graphs. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 154–165. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Mecke, S., Schöbel, A., Wagner, D.: Station location – complexity and approximation. In: Kroon, L.G., Möhring, R.H. (eds.) Proc. 5th ATMOS, IBFI Dagstuhl, Germany (2005)Google Scholar
  17. 17.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefMATHGoogle Scholar
  18. 18.
    Xu, G., Xu, J.: Constant approximation algorithms for rectangle stabbing and related problems. Theory Comput. Syst. 40(2), 187–204 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Michael Dom
    • 1
  • Michael R. Fellows
    • 2
  • Frances A. Rosamond
    • 2
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany
  2. 2.PC Research Unit, Office of DVC (Research)University of NewcastleCallaghanAustralia

Personalised recommendations