Maximum Overlap Area of Several Convex Polygons Under Translations

TL;DR

This paper introduces an efficient algorithm for translating multiple convex polygons in the plane to maximize their intersection area, providing both a single optimal placement and all such optimal translations.

Contribution

It presents the first algorithm with near-linear time complexity for finding maximum overlap of multiple convex polygons under translation.

Findings

Achieves $O(n ext{log}^{2k-3}n)$ time complexity for the optimization problem.

Provides an $O(n)$ algorithm to compute all optimal translations once a solution is found.

Extends previous work by handling multiple polygons simultaneously.

Abstract

Let $k \geq 2$ be a constant. Given any $k$ convex polygons in the plane with a total of $n$ vertices, we present an $O(n\log^{2k-3}n)$ time algorithm that finds a translation of each of the polygons such that the area of intersection of the $k$ polygons is maximized. Given one such placement, we also give an $O(n)$ time algorithm which computes the set of all translations of the polygons which achieve this maximum.