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Flows over time generalize classical ``static'' network flows by introducing a temporal dimension. They can thus be used to model non-instantaneous travel times for flow and variation of flow values over time, both of which are crucial characteristics in many real-world routing problems. There exist two different models of flows over time with respect to flow conservation: one where flow might be stored temporarily at intermediate nodes and a stricter model where flow entering an intermediate node must instantaneously progress to the next arc. While the first model is in general easier to handle, the second model is often more realistic since in applications like, e.\,g., road traffic, storage of flow at intermediate nodes is undesired or even prohibited. The main contribution of this paper is a fully polynomial time approximation scheme (FPTAS) for (min-cost) multi-commodity flows over time without intermediate storage. This improves upon the best previously known $(2+\varepsilon)$-approximation algorithm presented 10 years ago by Fleischer and Skutella (IPCO~2002).
Flows over time and generalized flows are two advanced network flow models of utmost importance, as they incorporate two crucial features occurring in numerous real-life networks. Flows over time feature time as a problem dimension and allow to realistically model the fact that commodities (goods, information, etc.) are routed through a network over time. Generalized flows allow for gain/loss factors on the arcs that model physical transformations of a commodity due to leakage, evaporation, breeding, theft, or interest rates. Although the latter effects are usually time-bound, generalized flow models featuring a temporal dimension have never been studied in the literature.
In this paper we introduce the problem of computing a generalized maximum flow over time in networks with both gain factors and transit times on the arcs. While generalized maximum flows and maximum flows over time can be computed efficiently, our combined problem turns out to be NP-hard and even completely non-approximable. A natural special case is given by lossy networks where the loss rate per time unit is identical on all arcs. For this case we present a (practically efficient) FPTAS that also reveals a surprising connection to so-called earliest arrival flows.