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Preprints 2009

Continuous and Discrete Flows Over Time: A General Model Based on Measure Theory
Citation key Report-016-2009
Author Ronald Koch and Ebrahim Nasrabadi and Martin Skutella
Year 2009
Number 016
Month jun
Institution Technische Universität Berlin, Institut für Mathematik
Abstract Network flows over time form a fascinating area of research. They model the temporal dynamics of network flow problems occurring in a wide variety of applications. Research in this area has been pursued in two different and mainly independent directions with respect to time modeling: discrete and continuous time models. In this paper we deploy measure theory in order to introduce a general model of network flows over time combining both discrete and continuous aspects into a single model. Here, the flow on each arc is modeled as a Borel measure on the real line (time axis) which assigns to each suitable subset a real value, interpreted as the amount of flow entering the arc over the subset. We focus on the maximum flow problem formulated in a network where capacities on arcs are also given as Borel measures and storage might be allowed at the nodes of the network. We generalize the concept of cuts to the case of these Borel Flows and extend the famous MaxFlow-MinCut Theorem.
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