An Introduction to Network Flows over Time

We give an introduction into the fascinating area of flows over time—also called “dynamic flows” in the literature. Starting from the early work of Ford and Fulkerson on maximum flows over time, we cover many exciting results that have been obtained over the last fifty years. One purpose of this chapter is to serve as a possible basis for teaching network flows over time in an advanced course on combinatorial optimization. Flow variation over time is an important feature in network flow problems arising in various applications such as road or air traffic control, production systems, communication networks (e. g., the Internet), and financial flows. In such applications, flow values on arcs are not constant but may change over time. Moreover, there is a second temporal dimension in these applications. Usually, flow does not travel instantaneously through a network but requires a certain amount of time to travel through each arc. In particular, when routing decisions are being made in one part of a network, the effects can be seen in other parts of the network only after a certain time delay. Not only the amount of flow to be transmitted but also the time needed for the transmission plays an essential role. The above mentioned aspects of network flows are not captured by the classic static network flow models. This is where network flows over time come into play. They include a temporal dimension and therefore provide a more realistic modeling tool for numerous real-world applications. Only few textbooks on combinatorial optimization and network flows, however, mention this topic at all; see, e.g., Ford and Fulkerson [15, Chapter III.9], Ahuja, Magnanti, and Orlin [1, Chapter 19.6], Korte and Vygen [25, Chapter 9.7], and Schrijver [34, Chapter 12.5c]. The following treatment of the topic has been developed for the purpose of teaching in an advanced course on combinatorial optimization. We concentrate on flows over time (also called “dynamic flows” in the literature) with finite time horizon and constant capacities and constant transit times in a continuous time model . For a broader overview of flows over time we refer to the survey papers by Aronson [5], Powell, Jaillet, and Odoni [33], Kotnyek [26], and Lovetskii and Melamed [27]. We also refer to the PhD thesis of Hoppe [20] for an easily accessible and detailed treatment of the topic based on the discrete time model . The paper is organized as follows. In Section 1 we introduce notation and shortly repeat some basic definitions and results from static network flow theory that are of particular importance for our purposes. For a more detailed treatment of those issues we refer to standard textbooks such as, e.g., [1] and [25]. In Section 2 we present a classical result of Ford and Fulkerson on the Maximum Flow Over Time Problem. This problem can be reduced to a static min-cost flow computation. Section 3 is devoted to a special class of flows over time called earliest arrival flows which can be used to model evacuation scenarios. We present a classical result that shows how to compute an earliest arrival flow with the Successive Shortest Path Algorithm. In Section 4 ∗This work was supported by DFG Research Center Matheon in Berlin.