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.

翻译：Gallai (1985) 著名的流分解定理指出，有向图中任何静态边 $s$,$d$-流都可以分解为路径和环的关联向量的非负线性组合。在本文中，我们研究动态边 $s$,$d$-流的分解问题，假设一个相当一般的动态流传播模型。我们证明了以下分解定理：对于任何可积的动态边 $s$,$d$-流，都存在一个分解，将其表示为 $s$,$d$-行走流入量和零传输时间环的非负线性组合。我们证明了经典迭代算法的一种变体——从当前动态边流中迭代减去行走流入量——会收敛到一个动态环流，并且每一个这样的环流都可以由零传输时间环的流入量所诱导。如果存在最小边旅行时间的下界且流是有限支撑的，则该算法会在有限时间内终止。我们进一步刻画了那些可以纯粹分解为 $s$,$d$-行走流入量的非负线性组合的动态边流。证明依赖于自主网络负载这一新概念，它使我们能够描述不同行走流的粒子在给定边流所诱导的固定旅行时间下，将如何在网络中假设性地传播。我们展示了这类网络负载的几个技术性质，并且作为副产品，我们还推导出一些关于动态流的一般性结果，这些结果在本文的语境之外也可能具有研究价值。