Annals of Operations Research

, Volume 272, Issue 1–2, pp 29–39

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

Advances in Theoretical and Applied Combinatorial Optimization

## 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

## References

1. Ahuja, R., Magnati, T., & Orlin, J. (1993). Network flows. Theory, algorithms and applications. Englewood Cliffs, NJ: Prentice Hall Inc.Google Scholar
2. Aronson, J. (1989). A survey of dynamic network flows. Annals of Operations Research, 20, 1–66.
3. Cai, X., Sha, D., & Wong, C. K. (2001). Time-varying minimum cost flow problems. European Journal of Operational Research, 131, 352–374.
4. Ciurea, E., & Ciupala, L. (2004). Sequential and parallel algorithms for minimum flow. Journal of Applied Mathematics and Computing, 15, 53–75.
5. 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.
6. 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.
7. Fleischer, L. K., & Skutella, M. (2007). Quickest flows over time. SIAM Journal on Computing, 36, 1600–1630.
8. Fleischer, L. K., & Tardos, E. (1998). Efficient continuous-time dynamic network flow algorithms. Operations Research Letters, 23, 71–80.
9. 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
10. Ford, L. R., & Fulkerson, D. R. (1958). Constructing maximal dynamic flows from static flows. Operations Research, 6, 419–433.
11. Gallo, G., Grigoriadis, M. D., & Tarjan, R. E. (1989). A fast parametric maximum flow algorithm and applications. SIAM Journal on Computing, 18, 30–55.
12. 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
13. Grazia, M. (2007). A note on the parametric maximum flow problem and some related reoptimization issues. Annals of Operations Research, 150, 231–244.
14. 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
15. 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
16. Hoppe, B. (1995). Efficient dynamic network flow algorithms. Ph.D. Thesis, Cornell University.Google Scholar
17. Klinz, B., & Woeginger, G. J. (2004). Minimum-cost dynamic flows: The series-parallel case. Networks, 43, 153–162.
18. 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.
19. Koch, R., Nasrabadi, E., & Skutella, M. (2011). Continuous and discrete flows over time. Mathematical Methods of Operations Research, 73, 301–337.
20. Opasanon, S., & Miller-Hooks, E. (2006). Multicriteria adaptive paths in stochastic, time-varying networks. European Journal of Operational Research, 173(1), 72–91.
21. 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
22. 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
23. 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.

© Springer Science+Business Media New York 2017

## Authors and Affiliations

• S. Khodayifar
• 1
• M. A. Raayatpanah
• 2
• P. M. Pardalos
• 3
• 4
1. 1.Department of MathematicsInstitute for Advanced Studies in Basic Sciences (IASBS)ZanjanIran
2. 2.Faculty of Mathematical Sciences and ComputerKharazmi UniversityTehranIran
3. 3.Department of Industrial and Systems Engineering, Center for Applied OptimizationUniversity of FloridaGainesvilleUSA
4. 4.Laboratory of Algorithms and Technologies for Network AnalysisNational Research University Higher School of EconomicsMoscowRussia