Optimal time-varying flows on congested networks
Operations Research
A survey of dynamic network flows
Annals of Operations Research
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
The Quickest Transshipment Problem
Mathematics of Operations Research
Minimum Cost Dynamic Flows: The Series-Parallel Case
Proceedings of the 4th International IPCO Conference on Integer Programming and Combinatorial Optimization
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Time-Expanded Graphs for Flow-Dependent Transit Times
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Approximating earliest arrival flows with flow-dependent transit times
Discrete Applied Mathematics
Flows in dynamic networks with aggregate arc capacities
Information Processing Letters
Traffic Networks and Flows over Time
Algorithmics of Large and Complex Networks
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About forty years ago, Ford and Fulkerson studied maximum 'dynamic' s-t-flows in networks with fixed transit times on the edges and a fixed time horizon. They showed that there always exists an optimal solution which sends flow on certain s-t-paths at a constant rate as long as there is enough time left for the flow along a path to arrive at the sink.Although this result does not hold for the more general setting of flows over time with load-dependent transit times on the edges, we prove that there always exists a provably good solution of this structure. Moreover, such a solution can be determined very efficiently by only one minimum convex cost flow computation on the underlying 'static' network. Finally, we show that the time-dependent flow problem under consideration is NP-hard and even cannot be approximated with arbitrary precision in polynomial time, unless P=NP.