The Parameterized Complexity of the Rectangle Stabbing Problem and Its Variants
We study an NP-complete geometric covering problem called d-Dimensional Rectangle Stabbing, where, given a set of axis-parallel d-dimensional hyperrectangles, a set of axis-parallel (d − 1)-dimensional hyperplanes and a positive integer k, the question is whether one can select at most k of the hyperplanes such that every hyperrectangle is intersected by at least one of these hyperplanes. This problem is well-studied from the approximation point of view, while its parameterized complexity remained unexplored so far. Here we show, by giving a nontrivial reduction from a problem called Multicolored Clique, that for d ≥ 3 the problem is W-hard with respect to the parameter k. For the case d = 2, whose parameterized complexity is still open, we consider several natural restrictions and show them to be fixed-parameter tractable.
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