A natural extension to the convex hull problem and a novel solution

TL;DR

This paper extends the convex hull problem to multiple polygons, introducing a new algorithm that efficiently computes minimal total perimeters of convex hulls after partitioning, with improved runtime in specific cases.

Contribution

It presents a novel algorithm for partitioning polygons to minimize total convex hull perimeters, extending the classic convex hull problem with practical optimization techniques.

Findings

Algorithm with cubic runtime in total vertices

Optimized runtime for disjoint polygons case

Effective partitioning strategy for minimal perimeter sum

Abstract

We study a natural extension to the well-known convex hull problem by introducing multiplicity: if we are given a set of convex polygons, and we are allowed to partition the set into multiple components and take the convex hull of each individual component, what is the minimum total sum of the perimeters of the convex hulls? We show why this problem is intriguing, and then introduce a novel algorithm with a run-time cubic in the total number of vertices. In the case that the input polygons are disjoint, we show an optimization that achieves a run-time that, in most cases, is cubic in the total number of polygons, within a logarithmic factor.