On the d-Runaway Rectangle Escape Problem

In the Rectangle Escape problem, we are given a set of rectangles S in a rectangular region R, and we would like to extend these rectangles to one of the four sides of R while ensuring that the maximum number of overlaps is minimized. More formally, define the density of a point p in R as the number of extended rectangles that contain p. The question is then to find an extension with the smallest maximum density. We consider the problem of maximizing the number of rectangles that can be extended when the maximum density allowed is at most d. It is known that this problem is polynomially solvable for d = 1, and NP-hard for any d ≥ 2. We consider approximation and exact algorithms for fixed values of d. We also show that a very special case of this problem, when all the rectangles are unit squares from a grid, continues to be NP-hard for d = 2.