Robust flows over time: models and complexity results


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. 1.

    Aneja, Y.P., Chandrasekaran, R., Nair, K.: Maximizing residual flow under an arc destruction. Networks 38(4), 194–198 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Aronson, J.E.: A survey of dynamic network flows. Ann. Oper. Res. 20(1), 1–66 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)

    Google Scholar 

  4. 4.

    Bertsimas, D., Nasrabadi, E., Stiller, S.: Robust and adaptive network flows. Oper. Res. 61(5), 1218–1242 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bertsimas, D., Sim, M.: Robust discrete optimization and network flows. Math. Program. Ser. B 98, 49–71 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51, 161–166 (1950)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Disser, Y., Matuschke, J.: The complexity of computing a robust flow (2017). arXiv:1704.08241

  8. 8.

    Du, D., Chandrasekaran, R.: The maximum residual flow problem: NP-hardness with two-arc destruction. Networks 50(3), 181–182 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Ford Jr., L.R., Fulkerson, D.R.: Constructing maximal dynamic flows from static flows. Oper. Res. 6(3), 419–433 (1958)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theoret. Comput. Sci. 10(2), 111–121 (1980)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Gupta, U.I., Lee, D.T., Leung, J.T.: Efficient algorithms for interval graphs and circular-arc graphs. Networks 12(4), 459–467 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Springer, Berlin (1972)

  14. 14.

    Koch, R., Nasrabadi, E., Skutella, M.: Continuous and discrete flows over time. A general model based on measure theory. Math. Methods Oper. Res. 73(3), 301–337 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Koch, T., Hiller, B., Pfetsch, M.E., Schewe, L.: Evaluating Gas Network Capacities. SIAM, Philadelphia (2015)

    Google Scholar 

  16. 16.

    Köhler, E., Skutella, M.: Flows over time with load-dependent transit times. SIAM J. Optim. 15(4), 1185–1202 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Matuschke, J., McCormick, T.S., Oriolo, G., Peis, B., Skutella, M.: (2015)

  18. 18.

    Skutella, M.: An introduction to network flows over time. In: Cook, W.J., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, pp. 451–482. Springer, Berlin (2009)

    Google Scholar 

  19. 19.

    Wood, R.K.: Deterministic network interdiction. Math. Comput. Model. 17(2), 1–18 (1993)

    MathSciNet  Article  MATH  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to Andreas Wierz.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation

Mathematics Subject Classification

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