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

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## 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 \(\{0,1,\ldots ,T\}\) 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 \(O(n^2m)\).

## Keywords

Network flows Combinatorial optimization (Dynamic) Minimum flow (Dynamic) Preflow-pull algorithm## Notes

### Acknowledgements

The authors thank the respected editors and the anonymous referees for their valuable and constructive suggestions on this work, which improved the content substantially.

## References

- Ahuja, R., Magnati, T., & Orlin, J. (1993).
*Network flows. Theory, algorithms and applications*. Englewood Cliffs, NJ: Prentice Hall Inc.Google Scholar - Aronson, J. (1989). A survey of dynamic network flows.
*Annals of Operations Research*,*20*, 1–66.CrossRefGoogle Scholar - Cai, X., Sha, D., & Wong, C. K. (2001). Time-varying minimum cost flow problems.
*European Journal of Operational Research*,*131*, 352–374.CrossRefGoogle Scholar - Ciurea, E., & Ciupala, L. (2004). Sequential and parallel algorithms for minimum flow.
*Journal of Applied Mathematics and Computing*,*15*, 53–75.CrossRefGoogle Scholar - Fathabadi, H. S., Khezri, S., & Khodayifar, S. (2015). A simple algorithm for reliability evaluation in dynamic networks with stochastic transit times.
*Journal of Industrial and Production Engineering*,*32*(2), 115–120.CrossRefGoogle Scholar - Fathabadi, H. S., Khodayifar, S., & Raayatpanah, M. A. (2012). Minimum flow problem on network flows with time-varying bounds.
*Applied Mathematical Modelling*,*36*, 4414–4421.CrossRefGoogle Scholar - Fleischer, L. K., & Skutella, M. (2007). Quickest flows over time.
*SIAM Journal on Computing*,*36*, 1600–1630.CrossRefGoogle Scholar - Fleischer, L. K., & Tardos, E. (1998). Efficient continuous-time dynamic network flow algorithms.
*Operations Research Letters*,*23*, 71–80.CrossRefGoogle Scholar - Fonoberova, M., & Lozovanu, D. (2007).
*Optimal dynamic flows in networks and applications*. In*The international symposium the issues of calculation optimization: Communications, Crimea, Ukraine*(pp. 292–293).Google Scholar - Ford, L. R., & Fulkerson, D. R. (1958). Constructing maximal dynamic flows from static flows.
*Operations Research*,*6*, 419–433.CrossRefGoogle Scholar - Gallo, G., Grigoriadis, M. D., & Tarjan, R. E. (1989). A fast parametric maximum flow algorithm and applications.
*SIAM Journal on Computing*,*18*, 30–55.CrossRefGoogle Scholar - Goldberg, A. V., & Tarjan, R. E. (1986). A new approach to the maximum flow problem. In
*Proceedings on 18th annual ACM symposium on theory of computing*(pp. 136–146).Google Scholar - Grazia, M. (2007). A note on the parametric maximum flow problem and some related reoptimization issues.
*Annals of Operations Research*,*150*, 231–244.CrossRefGoogle Scholar - Hall, A., Hippler, S., & Skutella, M. (2003). Multicommodity flows over time: Efficient algorithms and complexity. In J.C.M. Baeten, J.K. Lenstra, J. Parrow, & G.J. Woeginger (Eds.),
*Automata, languages and programming. Lecture Notes in Computer Science*(Vol. 2719, pp. 397–409). Berlin: Springer.Google Scholar - Hofman, A. J. (1960). Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In R. Bellman & M. Hall Jr. (Eds.),
*Proceedings of symposia in applied mathematics, combinatorial analysis*(Vol. X, pp. 113–127). Providence, RI: American Mathematical Society.Google Scholar - Hoppe, B. (1995). Efficient dynamic network flow algorithms.
*Ph.D. Thesis*, Cornell University.Google Scholar - Klinz, B., & Woeginger, G. J. (2004). Minimum-cost dynamic flows: The series-parallel case.
*Networks*,*43*, 153–162.CrossRefGoogle Scholar - Koch, R., & Nasrabadi, E. (2014). Flows over time in time-varying networks: Optimality conditions and strong duality.
*European Journal of Operational Research*,*237*(2), 580–589.CrossRefGoogle Scholar - Koch, R., Nasrabadi, E., & Skutella, M. (2011). Continuous and discrete flows over time.
*Mathematical Methods of Operations Research*,*73*, 301–337.CrossRefGoogle Scholar - Opasanon, S., & Miller-Hooks, E. (2006). Multicriteria adaptive paths in stochastic, time-varying networks.
*European Journal of Operational Research*,*173*(1), 72–91.CrossRefGoogle Scholar - Orlin, J. B. (2013). Max flows in O (nm) time, or better. In
*Proceedings of the forty-fifth annual ACM symposium on theory of computing*(pp. 765–774).Google Scholar - Powell, W., Jaillet, P., & Odoni, A. (1995). Stochastic and dynamic networks and routing. In M.O. Ball, T.L. Magnanti, C.L. Monma, & G.L. Nemhauser (Eds.),
*Network routing, handbooks in operations research and management science, North-Holland, Amsterdam, The Netherlands*(Vol. 8, pp. 141–295). (**Chapter 3**).Google Scholar - Skutella, M. (2009). An introduction to network flows over time. In W. Cook, L. Lovasz, & J. Vygen (Eds.),
*Research trends in combinatorial optimization*(pp. 451–482). Berlin: Springer.CrossRefGoogle Scholar