Approximating the Maximum Overlap of Polygons under Translation

TL;DR

This paper introduces an efficient approximation algorithm for finding the translation of one polygon that maximizes overlap with another, especially effective for polygons close to convex, running in near linear time.

Contribution

It presents a near linear time $(1- ext{epsilon})$-approximation algorithm for maximum polygon overlap under translation, applicable to polygons decomposable into few convex parts.

Findings

Algorithm runs in $O(c n)$ time, with $c$ depending on $k$ and $ ext{epsilon}$.

Effective for polygons close to convex, providing near linear time solutions.

Enables approximate maximum overlap computation efficiently for complex polygons.

Abstract

Let $P$ and $Q$ be two simple polygons in the plane of total complexity $n$, each of which can be decomposed into at most $k$ convex parts. We present an $(1-\varepsilon)$-approximation algorithm, for finding the translation of $Q$, which maximizes its area of overlap with $P$. Our algorithm runs in $O(c n)$ time, where $c$ is a constant that depends only on $k$ and $\varepsilon$. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time.