# Minimum Perimeter-Sum Partitions in the Plane

## TL;DR

This paper introduces an efficient $O(n ext{log}^2 n)$ exact algorithm and a faster approximation algorithm for partitioning points in the plane to minimize the sum of convex hull perimeters, improving over previous quadratic solutions.

## Contribution

The paper presents the first subquadratic exact algorithm and a near-linear approximation algorithm for the minimum perimeter-sum partition problem.

## Key findings

- Exact algorithm runs in $O(n ext{log}^2 n)$ time.
- Approximation algorithm achieves $(1+ ext{epsilon})$-factor in near-linear time.
- Significantly improves computational efficiency over previous methods.

## Abstract

Let $P$ be a set of $n$ points in the plane. We consider the problem of partitioning $P$ into two subsets $P_1$ and $P_2$ such that the sum of the perimeters of $\text{CH}(P_1)$ and $\text{CH}(P_2)$ is minimized, where $\text{CH}(P_i)$ denotes the convex hull of $P_i$. The problem was first studied by Mitchell and Wynters in 1991 who gave an $O(n^2)$ time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in $O(n \log^2 n)$ time and a $(1+\varepsilon)$-approximation algorithm running in $O(n + 1/\varepsilon^2\cdot\log^2(1/\varepsilon))$ time.