Maximum-Weight Two Boxes Symmetric Difference Problem

2026-05-21

AI Summary

This study addresses the geometric optimization problem of selecting two axis-aligned rectangles in the plane such that the total weight of points in their symmetric difference is maximized, given a set of weighted points. We present the first exact algorithm for this problem, achieving a time complexity of $O(n^4 \log n)$ and space complexity of $O(n)$ by integrating efficient enumeration with pruning strategies, alongside techniques from computational geometry and combinatorial optimization. Beyond resolving the two-rectangle case, our approach establishes a general optimization framework that naturally extends to scenarios involving $k \geq 3$ rectangles or union-based objectives, thereby significantly broadening the scope of existing results in this domain.

Abstract

Let $P$ be a set of $n$ points in the plane, where each element of $P$ is assigned a weight $ω(p)$, 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.

Problem

• maximum-weight
• symmetric difference
• axis-aligned rectangles
• weighted points
• computational geometry

Innovation

• maximum-weight symmetric difference
• axis-aligned rectangles
• geometric optimization
• weighted point sets
• algorithmic framework