Theory and Approximate Solvers for Branched Optimal Transport with Multiple Sources

Peter Lippmann, Enrique Fita Sanmartín, Fred A Hamprecht

Keywords: Combinatorial Optimization, Optimal Transport, Irrigation Networks, Structured Prediction, Steiner Tree Problem, Branched Optimal Transport, Transportation Networks

TL;DR: We lay out the theory and practice of devising optimal transportation routes with subadditive edge costs as a generalization of optimal transport, encouraging solutions with branched structure.

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.