# A near-linear time approximation scheme for geometric transportation with arbitrary supplies and spread

## TL;DR

This paper introduces the first near-linear time approximation scheme for the geometric transportation problem with arbitrary supplies, achieving high-probability near-optimal solutions efficiently regardless of data spread or supply magnitude.

## Contribution

It presents the first algorithm with near-linear time complexity for geometric transportation that is independent of data spread and supply magnitude, providing high-probability $(1 + ext{epsilon})$-approximations.

## Key findings

- Achieves near-linear time complexity $n ext{epsilon}^{-O(d)} ext{log}^{O(d)} n$.
- Independent of data spread and supply magnitude.
- Provides high-probability $(1 + ext{epsilon})$-approximate solutions.

## Abstract

The geometric transportation problem takes as input a set of points $P$ in $d$-dimensional Euclidean space and a supply function $\mu : P \to \mathbb{R}$. The goal is to find a transportation map, a non-negative assignment $\tau : P \times P \to \mathbb{R}_{\geq 0}$ to pairs of points, so the total assignment leaving each point is equal to its supply, i.e., $\sum_{r \in P} \tau(q, r) - \sum_{p \in P} \tau(p, q) = \mu(q)$ for all points $q \in P$. The goal is to minimize the weighted sum of Euclidean distances for the pairs, $\sum_{(p, q) \in P \times P} \tau(p, q) \cdot ||q - p||_2$. We describe the first algorithm for this problem that returns, with high probability, a $(1 + \varepsilon)$-approximation to the optimal transportation map in $n\varepsilon^{-O(d)}\log^{O(d)}{n}$ time. In contrast to the previous best algorithms for this problem, our near-linear running time bound is independent of the spread of $P$ and the magnitude of its real-valued supplies.