Maximum-Weight Two Boxes Symmetric Difference Problem

TL;DR

This paper introduces an algorithm to find two overlapping axis-aligned rectangles that maximize the total weight of points in their symmetric difference, with potential extensions to multiple boxes.

Contribution

It presents a novel $O(n^4 ext{log} n)$ time algorithm for the maximum-weight symmetric difference problem for two rectangles, adaptable to multiple boxes.

Findings

Algorithm efficiently finds optimal rectangles for weighted points.

Framework can be extended to multiple boxes and unions.

Provides theoretical bounds and practical adaptability.

Abstract

Let $P$ be a set of $n$ points in the plane, where each element of $P$ is assigned a weight $\omega(p)$, positive or negative. In this paper, we present an algorithm that runs in $O(n^4\log n)$ time and $O(n)$ space to find two possibly overlapping axis-aligned rectangles $A$ and $B$ so as to maximize the total weight of the points contained in the symmetric difference of $A$ and $B$. The same optimization framework can easily be adapted to solve related problems such as to maximize the total weight in the symmetric difference of $k \geq 3$ boxes and/or in the union of $k \geq 2$ boxes.