We study multi-marginal optimal transport (MOT) problems where the underlying cost has a graphical structure. These graphical multi-marginal optimal transport problems have found applications in several domains including traffic flow control and regression problems in the Wasserstein space. MOT problem can be approached through two aspects: a single big MOT problem, or coupled minor OT problems. In this paper, we focus on the latter approach and demonstrate it has efficiency gain from the parallelization. For tree-structured MOT problems, we introduce a novel parallelizable algorithm that significantly reduces computational complexity. Additionally, we adapt this algorithm for general graphs, employing the modified junction trees to enable parallel updates. Our contributions, validated through numerical experiments, offer new avenues for MOT applications and establish benchmarks in computational efficiency.

翻译：本文研究具有图结构成本的多边际最优传输问题。这类图形多边际最优传输问题已在交通流控制、Wasserstein空间回归等多个领域得到应用。处理MOT问题可从两个角度切入：作为单一大型MOT问题求解，或分解为耦合的次要OT问题处理。本文聚焦于后一种方法，并证明其通过并行化可获得效率提升。针对树状结构的MOT问题，我们提出了一种创新的并行算法，能显著降低计算复杂度。此外，我们将该算法推广至一般图结构，采用改进的连接树实现并行更新。通过数值实验验证的贡献为MOT应用开辟了新途径，并在计算效率方面建立了新基准。