Finding Largest Rectangles in Convex Polygons

TL;DR

This paper presents exact and approximation algorithms for finding the largest-area and perimeter rectangles within convex polygons, achieving polynomial and near-linear time complexities.

Contribution

It introduces the first exact $O(n^3)$ algorithms and efficient $(1- ext{epsilon})$-approximation algorithms for the problem.

Findings

Exact algorithms run in $O(n^3)$ time.

Approximation algorithms run in near-linear time.

Algorithms effectively find maximum rectangles in convex polygons.

Abstract

We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with $n$ vertices. We give exact algorithms that solve these problems in time $O(n^3)$. We also give $(1-\varepsilon)$-approximation algorithms that take time $O(\varepsilon^{-3/2}+ \varepsilon^{-1/2} \log n)$.