Approximation Algorithms for Capacitated Rectangle Stabbing

In the rectangle stabbing problem we are given a set of axis parallel rectangles and a set of horizontal and vertical lines, and our goal is to find a minimum size subset of lines that intersect all the rectangles. In this paper we study the capacitated version of this problem in which the input includes an integral capacity for each line. The capacity of a line bounds the number of rectangles that the line can cover. We consider two versions of this problem. In the first, one is allowed to use only a single copy of each line (hard capacities), and in the second, one is allowed to use multiple copies of every line provided that multiplicities are counted in the size of the solution (soft capacities). For the case of d-dimensional rectangle stabbing with soft capacities, we present a 6dapproximation algorithm and a 2-approximation algorithm when d = 1. For d-dimensional rectangle stabbing problem with hard capacities, we present a bi-criteria algorithm that computes 16d-approximate solutions that use at most two copies of every line. For the one dimensional case, an 8-approximation algorithm for hard capacities is presented. Finally, we present hardness results for rectangle stabbing when the dimension is part of the input and for a two-dimensional weighted version with hard capacities.