We consider the problem to transport resources/mass while abiding by constraints on the flow through constrictions along their path between specified terminal distributions. Constrictions, conceptualized as toll stations at specified points, limit the flow rate across. We quantify flow-rate constraints via a bound on a sought probability density of the times that mass-elements cross toll stations and cast the transportation scheduling in a Kantorovich-type of formalism. Recent work by our team focused on the existence of Monge maps for similarly constrained transport minimizing average kinetic energy. The present formulation in this paper, besides being substantially more general, is cast as a (generalized) multi-marginal transport problem - a problem of considerable interest in modern-day machine learning literature and motivated extensive computational analyses. An enabling feature of our formalism is the representation of an average quadratic cost on the speed of transport as a convex constraint that involves crossing times.

翻译：本文研究在指定终端分布之间运输资源/质量时，需遵守流经路径中瓶颈处流量约束的问题。将瓶颈概念化为指定点处的收费站点，限制其流通速率。我们通过限定质量单元穿越收费站点时间的概率密度函数来量化流量约束，并将运输调度问题纳入Kantorovich形式体系。团队近期工作聚焦于存在Monge映射以满足类似约束下最小化平均动能的最优运输问题。本文提出的形式化方法不仅更具普适性，更被构建为(广义)多边缘运输问题——这类问题在现代机器学习文献中备受关注，并引发了大量计算分析。本形式体系的关键特征在于：将运输速度的二次平均代价表示为涉及穿越时间的凸约束。