, Volume 62, Issue 1–2, pp 564–594 | Cite as

The Parameterized Complexity of Stabbing Rectangles

  • Michael Dom
  • Michael R. Fellows
  • Frances A. Rosamond
  • Somnath Sikdar


The NP-complete geometric covering problem Rectangle Stabbing is defined as follows: Given a set R of axis-parallel rectangles in the plane, a set L of horizontal and vertical lines 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 within a factor of two, its parameterized complexity with respect to the parameter k was open so far. 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 an algorithm running in f(k)⋅|RL| O(1) time. Our reductions also show 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.

We also show that the special case of Rectangle Stabbing where all rectangles are squares of the same size is W[1]-hard. The case where the input consists of non-overlapping rectangles was open for some time and has recently been shown to be fixed-parameter tractable (Heggernes et al., Fixed-parameter algorithms for cochromatic number and disjoint rectangle stabbing, 2009). By giving an algorithm running in (2k) k ⋅|RL| O(1) time, we show that Rectangle Stabbing is fixed-parameter tractable in the still NP-hard case where both these restrictions apply, that is, in the case of disjoint squares of the same size. This algorithm is faster than the one in Heggernes et al. (Fixed-parameter algorithms for cochromatic number and disjoint rectangle stabbing, 2009) for the disjoint rectangles case. Moreover, we show fixed-parameter tractability for the restrictions where the rectangles have bounded width or height or where each horizontal line intersects only a bounded number of rectangles.


Parameterized complexity Fixed-parameter algorithms Geometric covering problems Whardness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abrahamson, K.R., Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness IV: on completeness for W[P] and PSPACE analogues. Ann. Pure Appl. Logic 73(3), 235–276 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Agarwal, P.K., Sharir, M.: Efficient algorithms for geometric optimization. ACM Comput. Surv. 30(4), 412–458 (1998) CrossRefGoogle Scholar
  3. 3.
    Atkins, J.E., Middendorf, M.: On physical mapping and the consecutive ones property for sparse matrices. Discrete Appl. Math. 71(1–3), 23–40 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bilò, V., Goyal, V., Ravi, R., Singh, M.: On the crossing spanning tree problem. In: Proceedings of the 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX’04). LNCS, vol. 3122, pp. 51–60. Springer, Berlin (2004) Google Scholar
  5. 5.
    Călinescu, G., Dumitrescu, A., Karloff, H.J., Wan, P.-J.: Separating points by axis-parallel lines. Int. J. Comput. Geom. Appl. 15(6), 575–590 (2005) CrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, J., Chor, B., Fellows, M., Huang, X., Juedes, D., Kanj, I.A., Xia, G.: Tight lower bounds for certain parameterized NP-hard problems. Inf. Comput. 201(2), 216–231 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Clarkson, K.L., Varadarajan, K.R.: Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37(1), 43–58 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Downey, R.G., Estivill-Castro, V., Fellows, M., Prieto, E., Rosamond, F.A.: Cutting up is hard to do: the parameterised complexity of k-Cut and related problems. Electron. Notes Theor. Comput. Sci. 78, 209–222 (2003) CrossRefGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999) CrossRefGoogle Scholar
  10. 10.
    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) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410(1), 53–61 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Flammini, M., Gambosi, G., Salomone, S.: Interval routing schemes. Algorithmica 16(6), 549–568 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Flum, J., Grohe, M.: Parametrized complexity and subexponential time. Bull. EATCS 84, 71–100 (2004) zbMATHMathSciNetGoogle Scholar
  14. 14.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006) Google Scholar
  15. 15.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979) zbMATHGoogle Scholar
  16. 16.
    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) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Giannopoulos, P., Knauer, C., Rote, G., Werner, D.: Fixed-parameter tractability and lower bounds for stabbing problems. In: Proceedings of the 25th European Workshop on Computational Geometry (EuroCG’09), pp. 281–284 (2009) Google Scholar
  18. 18.
    Giannopoulos, P., Knauer, C., Whitesides, S.: Parameterized complexity of geometric problems. Comput. J. 51(3), 372–384 (2008) CrossRefGoogle Scholar
  19. 19.
    Goldberg, P.W., Golumbic, M.C., Kaplan, H., Shamir, R.: Four strikes against physical mapping of DNA. J. Comput. Biol. 2(1), 139–152 (1995) CrossRefGoogle Scholar
  20. 20.
    Hassin, R., Megiddo, N.: Approximation algorithms for hitting objects with straight lines. Discrete Appl. Math. 30, 29–42 (1991) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Heggernes, P., Kratsch, D., Lokshtanov, D., Raman, V., Saurabh, S.: Fixed-parameter algorithms for cochromatic number and disjoint rectangle stabbing. Manuscript (2009) Google Scholar
  22. 22.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Katz, M.J., Mitchell, J.S.B., Nir, Y.: Orthogonal segment stabbing. Comput. Geom. 30(2), 197–205 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Katz, M.J., Nielsen, F.: On piercing sets of objects. In: Proceedings of the 22nd Annual ACM Symposium on Computational Geometry (SOCG’96), pp. 113–121. ACM, New York (1996) Google Scholar
  25. 25.
    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, Piscataway (2002) Google Scholar
  26. 26.
    Kovaleva, S., Spieksma, F.C.R.: Approximation of a geometric set covering problem. In: Proceedings of the 12th International Symposium on Algorithms and Computation (ISAAC’01). LNCS, vol. 2223, pp. 493–501. Springer, Berlin (2001) Google Scholar
  27. 27.
    Kovaleva, S., Spieksma, F.C.R.: Approximation algorithms for rectangle stabbing and interval stabbing problems. SIAM J. Discrete Math. 20(3), 748–768 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Langerman, S., Morin, P.: Covering things with things. Discrete Comput. Geom. 33(4), 717–729 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Marx, D.: Parameterized complexity of independence and domination on geometric graphs. In: Proceedings of the 2nd International Workshop on Parameterized and Exact Computation (IWPEC’06). LNCS, vol. 4169, pp. 154–165. Springer, Berlin (2006) CrossRefGoogle Scholar
  30. 30.
    Mecke, S., Schöbel, A., Wagner, D.: Station location—complexity and approximation. In: Proceedings of the 5th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS’05). IBFI Dagstuhl, Germany (2005) Google Scholar
  31. 31.
    Megiddo, N., Tamir, A.: On the complexity of locating linear facilities in the plane. Oper. Res. Lett. 1, 194–197 (1982) CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, London (2006) CrossRefzbMATHGoogle Scholar
  33. 33.
    Nussbaum, D.: Rectilinear p-piercing problems. In: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (ISSAC’97), pp. 316–323. ACM Press, New York (1997) CrossRefGoogle Scholar
  34. 34.
    Pietrzak, K.: On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. J. Comput. Syst. Sci. 67(4), 757–771 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Segal, M.: On piercing sets of axis-parallel rectangles and rings. Int. J. Comput. Geom. Appl. 9(3), 219–233 (1999) CrossRefzbMATHGoogle Scholar
  36. 36.
    Sharir, M., Welzl, E.: Rectilinear and polygonal p-piercing and p-center problems. In: Proceedings of the 22nd Annual ACM Symposium on Computational Geometry (SOCG’96), pp. 122–132. ACM, New York (1996) Google Scholar
  37. 37.
    Wang, R., Lau, F.C.M., Zhao, Y.: Hamiltonicity of regular graphs and blocks of consecutive ones in symmetric matrices. Discrete Appl. Math. 155(17), 2312–2320 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Weis, S., Reischuk, R.: The complexity of physical mapping with strict chimerism. In: Proceedings of the 6th Annual International Computing and Combinatorics Conference (COCOON’00). LNCS, vol. 1858, pp. 383–395. Springer, Berlin (2000) Google Scholar
  39. 39.
    Xu, G., Xu, J.: Constant approximation algorithms for rectangle stabbing and related problems. Theory Comput. Syst. 40(2), 187–204 (2007) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Michael Dom
    • 1
  • Michael R. Fellows
    • 2
  • Frances A. Rosamond
    • 2
  • Somnath Sikdar
    • 3
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany
  2. 2.School of Engineering and Information TechnologyCharles Darwin UniversityDarwinAustralia
  3. 3.Department of Computer ScienceRWTH Aachen UniversityAachenGermany

Personalised recommendations