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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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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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