Insights into capacity constrained optimal transport

TL;DR

This paper explores a variant of optimal transport constrained by a density, revealing symmetry and explicit solutions in multiple dimensions, and introduces new methods for understanding solution uniqueness.

Contribution

It uncovers symmetry properties, provides explicit examples in higher dimensions, and develops novel approaches to solution uniqueness in capacity-constrained optimal transport.

Findings

Explicit solutions in two or more dimensions.

Identification of all extreme points in the feasible set.

A new approach to establishing solution uniqueness.

Abstract

A variant of the classical optimal transportation problem is: among all joint measures with fixed marginals and which are dominated by a given density, find the optimal one. Existence and uniqueness of solutions to this variant were established in \cite{KM11}. In the present manuscript, we expose an unexpected symmetry leading to the first explicit examples in two and more dimensions. These are inspired in part by simulations in one dimension which display singularities and topology and in part by two further developments: the identification of all extreme points in the feasible set, and a new approach to uniqueness based on constructing feasible perturbations.