A Decomposition Theorem for Dynamic Flows

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.