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A polynomial time algorithm for the minimum flow problem in time-varying networks

Author

Listed:
  • S. Khodayifar

    (Institute for Advanced Studies in Basic Sciences (IASBS))

  • M. A. Raayatpanah

    (Kharazmi University)

  • P. M. Pardalos

    (University of Florida
    National Research University Higher School of Economics)

Abstract

Flow variations over time generalize standard network flows by introducing an element of time. In contrast to the classical case of static flows, a flow over time in such a network specifies a flow rate entering an arc for each point in time. In this setting, the capacity of an arc limits the rate of flow into the arc at each point in time. Traditionally, flows over time are computed in time-expanded networks that contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static network flows, its drawback is the enormous size of the time-expanded network. In this paper, we extend the results about the minimum flow problem to network flows (with n nodes and m arcs) in which the time-varying lower bounds can involve both the source and the sink nodes (as in Fathabadi et al.) and also one additional node other than the source and the sink nodes. It is shown that this problem for the set $$\{0,1,\ldots ,T\}$$ { 0 , 1 , … , T } of time points can be solved by at most n minimum flow computations, by suitably extending the dynamic minimum flow algorithm and reoptimization techniques. The running time of the presented algorithm is $$O(n^2m)$$ O ( n 2 m ) .

Suggested Citation

  • S. Khodayifar & M. A. Raayatpanah & P. M. Pardalos, 2019. "A polynomial time algorithm for the minimum flow problem in time-varying networks," Annals of Operations Research, Springer, vol. 272(1), pages 29-39, January.
  • Handle: RePEc:spr:annopr:v:272:y:2019:i:1:d:10.1007_s10479-017-2450-2
    DOI: 10.1007/s10479-017-2450-2
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    References listed on IDEAS

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    1. Opasanon, Sathaporn & Miller-Hooks, Elise, 2006. "Multicriteria adaptive paths in stochastic, time-varying networks," European Journal of Operational Research, Elsevier, vol. 173(1), pages 72-91, August.
    2. Ronald Koch & Ebrahim Nasrabadi & Martin Skutella, 2011. "Continuous and discrete flows over time," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(3), pages 301-337, June.
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    5. Koch, Ronald & Nasrabadi, Ebrahim, 2014. "Flows over time in time-varying networks: Optimality conditions and strong duality," European Journal of Operational Research, Elsevier, vol. 237(2), pages 580-589.
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    Cited by:

    1. Laura Ciupala & Adrian Deaconu, 2021. "Incremental Minimum Flow Algorithms," Mathematics, MDPI, vol. 9(9), pages 1-13, May.

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