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

TL;DR

The paper addresses optimal transport on networks with departure--arrival (DA) constraints and nodal capacity limits by formulating temporally flexible OT as a path-wise, multi-marginal problem. It distinguishes two regimes: Independent DA constraints, which yield a multi-marginal OT formulation, and Coupled DA constraints, which lead to an unequal-dimensional OT problem; each regime is studied for existence and uniqueness on line graphs. A node-aggregation approach extends the results to general graphs, and entropically regularized Sinkhorn-type algorithms—including a path-wise variant—enable scalable computation with linear convergence. Numerical experiments on line graphs and path networks demonstrate scheduling feasibility, uniqueness properties, and fast convergence, underscoring the framework's potential for practical network scheduling in logistics, transit, and service-chain settings.

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.

Figures (9)

• Figure 1: Mass flow along the line graph ${\mathbf p}=\{v_0,v_1,\dots,v_{{\mathcal{T}}-1},v_{\mathcal{T}}\}$. Each particle departs from $v_0$ according to the departure rate $\mu^0$, travels sequentially through intermediate nodes $v_1, v_2,\dots,v_{{\mathcal{T}}-1}$ abides flow-rate constraints at each node, and arrives at the sink $v_{\mathcal{T}}$ according to the arrival rate $\mu^{\mathcal{T}}$.
• Figure 2: Illustration of the coupled DA constraint. (a) When the joint probability measure $\mu^{0,{\mathcal{T}}}$ follows a Monge map, it is supported on the graph of a function, ensuring that each particle is assigned a unique DA time pair. This enforces the assumption that particles departing earlier also arrive earlier. (b) When $\mu^{0,{\mathcal{T}}}$ is not constrained to a function’s graph, the DA relationship is relaxed. Instead of a fixed pair, the arrival time can vary within an interval, introducing flexibility in transport dynamics.
• Figure 3: Diagram of the coupled DA constraint over a line graph with source node $v_0$ and sink node $v_{\mathcal{T}}$, and connected through intermediate nodes $v_1, \dots, v_{{\mathcal{T}}-1}$. The coupled DA $\pi(\mu^0, \mu^{\mathcal{T}})$ encodes the joint probability of departure time $t_0$ and arrival time $t_{\mathcal{T}}$. At each intermediate node $v_\ell$, the transport process is characterized by a crossing-time distribution, determining when mass moves from one node to the next.
• Figure 4: Independent DA on line graph with single intermediate node. The three one-dimensional time profiles are placed on $v_0$, $v_1$, and $v_{\mathcal{T}}$. Blue shows the departure density at $v_0$. Green shows the crossing-time density at $v_1$. Red shows the arrival density at $v_{\mathcal{T}}$. The dotted line at $v_1$ marks the per-time-slice capacity $c=2$. Light gray guides indicate the aggregated couplings along $(v_0,v_1)$ and $(v_1,v_{\mathcal{T}})$. The plot highlights scheduling: mass is metered through $v_1$ to respect capacity, and node-wise marginals satisfy conservation.
• Figure 5: Endpoint coupling and triplet structure for case in Fig. \ref{['fig:independent_DA']}. The surface shows coupling over $(t_0,t_{\mathcal{T}})$ with blue and red boundary ridges equal to the source and sink marginals. The green curve along the far edge plots the crossing-time marginal at $v_1$. Colored markers indicate dominant triplets $(t_0,t_1,t_{\mathcal{T}})$ (color = $t_1$). The support collapses to a single smooth strand, in agreement with the monotone-map structure and uniqueness in one-dimensional flux-limited setting.

Theorems & Definitions (28)

• Definition 1: Monge condition rachev1998mass
• Definition 2: Twist condition villani2009optimal
• Definition 3: Generalized Monge condition rachev1998mass
• Lemma 1: Feasibility of independent DA
• proof
• Proposition 1: Existence of solution
• proof
• Lemma 2: Monotonicity
• proof
• Theorem 1: Uniqueness