Robust flows over time: models and complexity results

Abstract

We study dynamic network flows with uncertain input data under a robust optimization perspective. In the dynamic maximum flow problem, the goal is to maximize the flow reaching the sink within a given time horizon T, while flow requires a certain travel time to traverse an edge. In our setting, we account for uncertain travel times of flow. We investigate maximum flows over time under the assumption that at most \(\varGamma \) travel times may be prolonged simultaneously due to delay. We develop and study a mathematical model for this problem. As the dynamic robust flow problem generalizes the static version, it is NP-hard to compute an optimal flow. However, our dynamic version is considerably more complex than the static version. We show that it is NP-hard to verify feasibility of a given candidate solution. Furthermore, we investigate temporally repeated flows and show that in contrast to the non-robust case (that is, without uncertainties) they no longer provide optimal solutions for the robust problem, but rather yield a worst case optimality gap of at least T. We finally show that the optimality gap is at most \(O(\eta k \log T)\), where \(\eta \) and k are newly introduced instance characteristics and provide a matching lower bound instance with optimality gap \(\varOmega (\log T)\) and \(\eta = k = 1\). The results obtained in this paper yield a first step towards understanding robust dynamic flow problems with uncertain travel times.

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Acknowledgements

We thank the reviewers for their very careful reading of the manuscript and their valuable comments. We thank the DFG for their support within Project B06 in CRC TRR 154.

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Correspondence to Andreas Wierz.

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Gottschalk, C., Koster, A.M.C.A., Liers, F. et al. Robust flows over time: models and complexity results. Math. Program. 171, 55–85 (2018). https://doi.org/10.1007/s10107-017-1170-3

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Mathematics Subject Classification

  • 05C21 Flows in graphs
  • 90C05 Linear programming
  • 90C59 Approximation methods and heuristics
  • 90C46 Optimality conditions, duality