A Geometry-Based Approach for Solving the Transportation Problem with Euclidean Cost

TL;DR

This paper introduces a geometric method for solving the semi-discrete optimal transport problem with Euclidean cost, establishing existence, uniqueness, and providing an algorithm based on Voronoi partitions.

Contribution

It presents a novel geometric approach linking optimal transport maps to weighted Voronoi diagrams, with an explicit construction and an algorithm for computation.

Findings

Proves existence and uniqueness of the transport map.

Establishes a one-to-one correspondence with weighted Voronoi partitions.

Provides an algorithm to compute the optimal partition.

Abstract

In the semi-discrete version of Monge's problem one tries to find a transport map $T$ with minimum cost from an absolutely continuous measure $\mu$ on $\mathbb{R}^d$ to a discrete measure $\nu$ that is supported on a finite set in $\mathbb{R}^d$. The problem is considered for the case of the Euclidean cost function. Existence and uniqueness is shown by an explicit construction which yields a one-to-one mapping between the optimal $T$ and an additively weighted Voronoi partition of $\mathbb{R}^d$. From the proof an algorithm is derived to compute this partition.