Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport

Abstract

In this work, we develop a new framework for dynamic network flow problems based on optimal transport theory. We show that the dynamic multicommodity minimum-cost network flow problem can be formulated as a multimarginal optimal transport problem, where the cost function and the constraints on the marginals are associated with a graph structure. By exploiting these structures and building on recent advances in optimal transport theory, we develop an efficient method for such entropy-regularized optimal transport problems. In particular, the graph structure is utilized to efficiently compute the projections needed in the corresponding Sinkhorn iterations, and we arrive at a scheme that is both highly computationally efficient and easy to implement. To illustrate the performance of our algorithm, we compare it with a state-of-the-art linear programming (LP) solver. We achieve good approximations to the solution at least one order of magnitude faster than the LP solver. Finally, we showcase the methodology on a traffic routing problem with a large number of commodities. Funding: This work was supported by KTH Digital Futures, Knut och Alice Wallenbergs Stiftelse [Grants KAW 2018.0349, KAW 2021.0274, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation], Vetenskapsrådet [Grant 2020-03454], and the National Science Foundation [Grants 1942523 and 2206576].