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 wellstudied 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[1]-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.