Theory and Approximate Solvers for Branched Optimal Transport with Multiple Sources

TL;DR

This paper investigates the NP-hard problem of Branched Optimal Transport networks, providing geometric insights, topological constraints, generalizations to Riemannian manifolds, and an effective approximate solver combining geometric and combinatorial optimization.

Contribution

It introduces methods to efficiently determine optimal network geometry, establishes topological constraints, extends results to Riemannian manifolds, and develops a practical approximate solver for BOT.

Findings

Optimal network geometry can be efficiently found given a topology.

Networks with more than three edges at a branching point are never optimal.

Results generalize to Riemannian manifolds.

Abstract

Branched Optimal Transport (BOT) is a generalization of optimal transport in which transportation costs along an edge are subadditive. This subadditivity models an increase in transport efficiency when shipping mass along the same route, favoring branched transportation networks. We here study the NP-hard optimization of BOT networks connecting a finite number of sources and sinks in $\mathbb{R}^2$. First, we show how to efficiently find the best geometry of a BOT network for many sources and sinks, given a topology. Second, we argue that a topology with more than three edges meeting at a branching point is never optimal. Third, we show that the results obtained for the Euclidean plane generalize directly to optimal transportation networks on two-dimensional Riemannian manifolds. Finally, we present a simple but effective approximate BOT solver combining geometric optimization with a combinatorial optimization of the network topology.