On the Rectangle Escape Problem

Motivated by a bus routing application, we study the following rectangle escape problem: Given a set S of n rectangles inside a rectangular region R, extend each rectangle in S toward one of the four borders of R so that the maximum density over the region R is minimized, where the density of each point p ∈ R is defined as the number of extended rectangles containing p. We show that the problem is hard to approximate to within a factor better than 3/2 in general. When the optimal density is sufficiently large, we provide a randomized algorithm that achieves an approximation factor of 1 + ε with high probability improving upon the current best 4-approximation algorithm available for the problem. When the optimal density is one, we provide an exact algorithm that finds an optimal solution in O(n) time, improving upon the current best O(n)-time algorithm.