Maximizing the area of intersection of rectangles

Abstract

This paper attacks the following problem. We are given a large number $N$ of rectangles in the plane, each with horizontal and vertical sides, and also a number $r<N$. The given list of $N$ rectangles may contain duplicates. The problem is to find $r$ of these rectangles, such that, if they are discarded, then the intersection of the remaining $(N-r)$ rectangles has an intersection with as large an area as possible. We will find an upper bound, depending only on $N$ and $r$, and not on the particular data presented, for the number of steps needed to run the algorithm on (a mathematical model of) a computer. In fact our algorithm is able to determine, for each $s\le r$, $s$ rectangles from the given list of $N$ rectangles, such that the remaining $(N-s)$ rectangles have as large an area as possible, and this takes hardly any more time than taking care only of the case $s=r$. Our algorithm extends to $d$-dimensional rectangles. Our method is to exhaustively examine all possible intersections---this is much faster than it sounds, because we do not need to examine all $\binom Ns$ subsets in order to find all possible intersection rectangles. For an extreme example, suppose the rectangles are nested, for example concentric squares of distinct sizes, then the only intersections examined are the smallest $s+1$ rectangles.