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Branched Optimal Transport (BOT) is a generalization of optimal transport inwhich transportation costs along an edge are subadditive. This subadditivitymodels an increase in transport efficiency when shipping mass along the sameroute, favoring branched transportation networks. We here study the NP-hardoptimization 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 aBOT network for many sources and sinks, given a topology. Second, we argue thata topology with more than three edges meeting at a branching point is neveroptimal. Third, we show that the results obtained for the Euclidean planegeneralize directly to optimal transportation networks on two-dimensionalRiemannian manifolds. Finally, we present a simple but effective approximateBOT solver combining geometric optimization with a combinatorial optimizationof the network topology.

Theory and Approximate Solvers for Branched Optimal Transport with Multiple Sources