Temporally Flexible Transport Scheduling on Networks with Departure-Arrival Constriction and Nodal Capacity Limits

TL;DR

This paper extends optimal transport theory to networks with temporal and nodal constraints, proposing new formulations, analyzing solution properties, and developing scalable computational methods for complex transportation scenarios.

Contribution

It introduces a generalized OT framework with departure--arrival constraints, analyzes solution existence and uniqueness, and develops efficient algorithms using entropic regularization and Sinkhorn iterations.

Findings

Multi-marginal OT formulation for independent DA constraints

Alignment with unequal-dimensional OT for coupled DA constraints

Efficient scalable solutions with linear convergence demonstrated

Abstract

We investigate the optimal transport (OT) problem over networks, wherein supply and demand are conceptualized as temporal marginals governing departure rates of particles from source nodes and arrival rates at sink nodes. This setting extends the classical OT framework, where all mass is conventionally assumed to depart at $t = 0$ and arrive at $t = t_f$. Our generalization accommodates departures and arrivals at specified times, referred as departure--arrival(DA) constraints. In particular, we impose nodal-temporal flux constraints at source and sink nodes, characterizing two distinct scenarios: (i) Independent DA constraints, where departure and arrival rates are prescribed independently, and (ii) Coupled DA constraints, where each particle's transportation time span is explicitly specified. We establish that OT with independent DA constraints admits a multi-marginal optimal transport formulation, while the coupled DA case aligns with the unequal-dimensional OT framework. For line graphs, we analyze the existence and uniqueness of the solution path. For general graphs, we use a constructive path-based reduction and optimize over a prescribed set of paths. From a computational perspective, we consider entropic regularization of the original problem to efficiently provide solutions based on multi-marginal Sinkhorn method, making use of the graphical structure of the cost to further improve scalability. Our numerical simulation further illustrates the linear convergence rate in terms of marginal violation.