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

2026-02-16

AI Summary

This study addresses the network optimal transport problem with temporal flexibility, where flows at source and sink nodes are subject to time-specific departure–arrival constraints and nodal capacity limits. By introducing two classes of temporal constraints—decoupled and coupled—the problem is respectively formulated as multi-marginal optimal transport and unbalanced-dimensional optimal transport, thereby establishing, for the first time, a unified framework incorporating both temporal constraints and capacity restrictions. Methodologically, the authors propose a path-based reduced multi-marginal Sinkhorn algorithm that leverages entropy regularization and exploits the graph structure of the cost function to accelerate computation, enabling efficient solutions on both line graphs and general graphs. Numerical experiments demonstrate that the algorithm achieves linear convergence in terms of marginal constraint violation, significantly enhancing scalability for large-scale network transport scheduling.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• optimal transport
• temporal scheduling
• departure-arrival constraints
• nodal capacity
• network flows

Innovation

• Optimal Transport
• Temporal Scheduling
• Departure-Arrival Constraints
• Multi-marginal OT
• Entropic Regularization