A Decomposition Theorem for Dynamic Flows

TL;DR

This work develops a decomposition theorem for dynamic flows, showing that every integrable dynamic edge $s$,$d$-flow can be represented as a nonnegative combination of $s$,$d$-walk inflows and zero-transit-time cycles. The authors introduce autonomous network loadings to rigorously handle flow subtraction steps and prove algorithmic convergence to a dynamic circulation, with finite termination under travel-time lower bounds and finite-support conditions. They provide a constructive Flow Decomposition Algorithm, prove existence of optimal solutions for optimization problems involving autonomous loadings, and give a precise characterization for when a pure $s$,$d$-walk decomposition exists. These results bridge edge- and path-based dynamic equilibria, enabling inverse problems, toll design insights, and potential extensions to broad dynamic-flow models beyond classical FIFO settings. The framework also yields structural results that could inform future studies on dynamic traffic assignment and infinite-dimensional optimization in network contexts.

Abstract

The famous flow decomposition theorem of Gallai (1985) states that any static edge $s$,$d$-flow in a directed graph can be decomposed into a nonnegative linear combination of incidence vectors of paths and cycles. In this paper, we study the decomposition problem for the setting of dynamic edge $s$,$d$-flows assuming a quite general dynamic flow propagation model. We prove the following decomposition theorem: For any integrable dynamic edge $s$,$d$-flow, there exists a decomposition into a nonnegative linear combination of $s$,$d$-walk inflows and cycles of zero transit time. We show that a variant of the classical algorithmic approach of iteratively subtracting walk inflows from the current dynamic edge flow converges to a dynamic circulation and that every such circulation can be induced by inflows into cycles of zero transit time. The algorithm terminates in finite time, if there is a lower bound on the minimum edge travel times and the flow is finitely supported. We further characterize those dynamic edge flows which can be decomposed purely into nonnegative linear combinations of $s$,$d$-walk inflows. The proofs rely on the new concept of autonomous network loadings which allows us to describe how particles of a different walk flow would hypothetically propagate throughout the network under the fixed travel times induced by the given edge flow. We show several technical properties of this type of network loading and, as a byproduct, we also derive some general results on dynamic flows which could be of interest outside the context of this paper as well.

Figures (2)

• Figure 1: A schematic overview over the structure of this paper: The top part contains our main results (for auton. flows in the middle and for non-auton. flows on the right) as well as the main building blocks for their proofs (on the left). Implication arrows indicate that one result is the main ingredient for proving another. Single arrows indicate that a result is used within the corresponding proof. The bottom half contains more technical results on auton. network loadings, which are used in many proofs. Not all relations between these results and other results are explicitly drawn. The following abbreviations and notations are used: auton. for auton. [,] flow cons. for flow conservation, flow decomp. for flow decomposition and $\ell_{w}(h_w)$ and $\ell_\mathcal{W}(h)$ for the auton. network loading of a single walk inflow $h_w$ and a whole walk inflow vector $h$, respectively.
• Figure 2: An edge $s$,${d}$-flow that has no pure flow decomposition even though every simple zero-cycle has a flow-carrying outgoing edge. All edges in this network have a constant traversal time of zero. The edge-labels denote the edge inflow rates of the given flow.

Theorems & Definitions (16)

• Example 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1
• proof
• Theorem 3.3
• Lemma 3.4
• Theorem 3.5