Parameterized Complexity of Stabbing Rectangles and Squares in the Plane
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-hard. Giving two fixed-parameter reductions, one from the W-complete problem Multicolored Clique and one to the W-complete problem Short Turing Machine Acceptance, we prove that Rectangle Stabbing is W-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-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-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.
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- 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.Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems (manuscript, 2008)Google Scholar
- 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
- 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