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

Author

Listed:
  • Isabel Haasler

    (Signal Processing Laboratory 4, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland)

  • Axel Ringh

    (Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden; Department of Mathematical Sciences, University of Gothenburg, SE-412 96 Gothenburg, Sweden)

  • Yongxin Chen

    (School of Aerospace Engineering, Georgia Institute of Technology, GA 30332 Atlanta, Georgia)

  • Johan Karlsson

    (Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden)

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.

Suggested Citation

  • Isabel Haasler & Axel Ringh & Yongxin Chen & Johan Karlsson, 2024. "Scalable Computation of Dynamic Flow Problems via Multimarginal Graph-Structured Optimal Transport," Mathematics of Operations Research, INFORMS, vol. 49(2), pages 986-1011, May.
    • Handle: RePEc:inm:ormoor:v:49:y:2024:i:2:p:986-1011
    DOI: 10.1287/moor.2021.0148
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