We study optimal transport between probability measures supported on the same finite metric space, where the ground cost is a distance induced by a weighted connected graph. Building on recent work showing that the resulting Kantorovich distance can be expressed as a minimization problem over the set of spanning trees of this underlying graph, we investigate the implications of this reformulation on the construction of an optimal transport plan and a dual potential based on the solution of such an optimization problem. In this setting, we derive an explicit formula for the Kantorovich potential in terms of the imbalanced cumulative mass (a generalization of the cumulative distribution in R) along an optimal spanning tree solving such a minimization problem, under a weak non-degeneracy condition on the pair of measures that guarantees the uniqueness of a dual potential. Our second contribution establishes the existence of an optimal transport plan that can be computed efficiently by a dynamic programming procedure once an optimal spanning tree is known. Finally, we propose a stochastic algorithm based on simulated annealing on the space of spanning trees to compute such an optimal spanning tree. Numerical experiments illustrate the theoretical results and demonstrate the practical relevance of the proposed approach for optimal transport on finite metric spaces.

翻译：我们研究同一有限度量空间上概率测度间的最优输运问题，其中基础成本为由加权连通图导出的距离。基于近期研究表明，由此产生的Kantorovich距离可表示为该底层图生成树集合上的最小化问题，我们探讨此重构对构建最优输运方案及基于此类优化问题解的对偶势的影响。在此框架下，我们导出了Kantorovich势的显式公式，该公式通过沿求解此类最小化问题的最优生成树上的非平衡累积质量（ℝ中累积分布函数的推广）表达，其成立条件是对测度对满足保证对偶势唯一性的弱非退化条件。我们的第二个贡献证明了最优输运方案的存在性，该方案在已知最优生成树时可通过动态规划程序高效计算。最后，我们提出一种基于生成树空间模拟退火的随机算法来计算此类最优生成树。数值实验验证了理论结果，并证明了所提方法在有限度量空间最优输运问题中的实际应用价值。