Theory and Approximate Solvers for Branched Optimal Transport with Multiple Sources
This work addresses the computational challenge of designing efficient branched transportation networks, which is incremental as it builds on existing BOT theory with new theoretical insights and algorithms.
The paper tackles the NP-hard optimization of Branched Optimal Transport (BOT) networks with multiple sources and sinks in Euclidean space, showing that topologies with more than three edges at branching points are never optimal and generalizing results to Riemannian manifolds, while presenting an approximate solver combining geometric and combinatorial optimization.
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.