# Algorithms for capacitated rectangle stabbing and lot sizing with joint set-up costs

## Abstract

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 article, 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, but the multiplicities are counted in the size (or weight) of the solution (soft capacities). We present an exact polynomial-time algorithm for the weighted one dimensional case with hard capacities that can be extended to the one dimensional weighted case with soft capacities. This algorithm is also extended to solve a certain capacitated multi-item lot-sizing inventory problem with joint set-up costs. For the case of d-dimensional rectangle stabbing with soft capacities, we present a 3d-approximation algorithm for the unweighted case. For d-dimensional rectangle stabbing problem with hard capacities, we present a bi-criteria algorithm that computes 4d-approximate solutions that use at most two copies of every line. 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. © 2008 ACM.