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The Parameterized Complexity of Stabbing Rectangles

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Abstract

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)⋅|R∪L|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⋅|R∪L|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.

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References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  2. Agarwal, P.K., Sharir, M.: Efficient algorithms for geometric optimization. ACM Comput. Surv. 30(4), 412–458 (1998)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Clarkson, K.L., Varadarajan, K.R.: Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37(1), 43–58 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  9. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)

    Book  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  12. Flammini, M., Gambosi, G., Salomone, S.: Interval routing schemes. Algorithmica 16(6), 549–568 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Flum, J., Grohe, M.: Parametrized complexity and subexponential time. Bull. EATCS 84, 71–100 (2004)

    MATH  MathSciNet  Google Scholar 

  14. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)

    Google Scholar 

  15. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Giannopoulos, P., Knauer, C., Whitesides, S.: Parameterized complexity of geometric problems. Comput. J. 51(3), 372–384 (2008)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Heggernes, P., Kratsch, D., Lokshtanov, D., Raman, V., Saurabh, S.: Fixed-parameter algorithms for cochromatic number and disjoint rectangle stabbing. Manuscript (2009)

  22. Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Katz, M.J., Mitchell, J.S.B., Nir, Y.: Orthogonal segment stabbing. Comput. Geom. 30(2), 197–205 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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. Kovaleva, S., Spieksma, F.C.R.: Approximation algorithms for rectangle stabbing and interval stabbing problems. SIAM J. Discrete Math. 20(3), 748–768 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Langerman, S., Morin, P.: Covering things with things. Discrete Comput. Geom. 33(4), 717–729 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  32. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, London (2006)

    Book  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  35. Segal, M.: On piercing sets of axis-parallel rectangles and rings. Int. J. Comput. Geom. Appl. 9(3), 219–233 (1999)

    Article  MATH  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Xu, G., Xu, J.: Constant approximation algorithms for rectangle stabbing and related problems. Theory Comput. Syst. 40(2), 187–204 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07743, Jena, Germany

    Michael Dom

  2. School of Engineering and Information Technology, Charles Darwin University, Darwin, NT, 0909, Australia

    Michael R. Fellows & Frances A. Rosamond

  3. Department of Computer Science, RWTH Aachen University, 52074, Aachen, Germany

    Somnath Sikdar

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  1. Michael Dom
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  3. Frances A. Rosamond
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  4. Somnath Sikdar
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Correspondence to Somnath Sikdar.

Additional information

Parts of this article have been published in the Proceedings of the 2nd International Frontiers of Algorithmics Workshop (FAW’08) (June 19–21, 2008, Changsha, China), vol. 5059 of LNCS, pp. 288–299, Springer, 2008, and in the Proceedings of the 3rd International Workshop on Algorithms and Computation (WALCOM’09) (February 18–20, 2009, Kolkata, India), vol. 5431 of LNCS, pp. 298–309, Springer, 2009.

Research of M. Dom and S. Sikdar was done at the Friedrich-Schiller-Universität Jena, Germany, and The Institute of Mathematical Sciences, Chennai, India and supported by the DAAD-DST exchange program D/05/57666 of the Deutscher Akademischer Austauschdienst and the Department of Science and Technology, Government of India.

The research of M.R. Fellows and F.A. Rosamond is supported by the Australian Research Council. Research done at the Friedrich-Schiller-Universität Jena, Germany, where Michael R. Fellows was supported by the Alexander von Humboldt-Foundation, Bonn, Germany, as a recipient of the Humboldt Research Award.

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Dom, M., Fellows, M.R., Rosamond, F.A. et al. The Parameterized Complexity of Stabbing Rectangles. Algorithmica 62, 564–594 (2012). https://doi.org/10.1007/s00453-010-9471-4

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