Temporal Flows in Temporal Networks

  • Eleni C. Akrida
  • Jurek Czyzowicz
  • Leszek Gąsieniec
  • Łukasz Kuszner
  • Paul G. Spirakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10236)


We introduce temporal flows on temporal networks [17, 19], i.e., networks the links of which exist only at certain moments of time. Such networks are ephemeral in the sense that no link exists after some time. Our flow model is new and differs from the “flows over time” model, also called “dynamic flows” in the literature. We show that the problem of finding the maximum amount of flow that can pass from a source vertex \({s}\) to a sink vertex \({t}\) up to a given time is solvable in Polynomial time, even when node buffers are bounded. We then examine mainly the case of unbounded node buffers. We provide a simplified static Time-Extended network (\(\mathrm {STEG}\)), which is of polynomial size to the input and whose static flow rates are equivalent to the respective temporal flow of the temporal network; using \(\mathrm {STEG}\), we prove that the maximum temporal flow is equal to the minimum temporal\({s}\text {-}{t}\)cut. We further show that temporal flows can always be decomposed into flows, each of which moves only through a journey, i.e., a directed path whose successive edges have strictly increasing moments of existence. We partially characterise networks with random edge availabilities that tend to eliminate the \({s}\rightarrow {t}\) temporal flow. We then consider mixed temporal networks, which have some edges with specified availabilities and some edges with random availabilities; we show that it is #P-hard to compute the tails and expectations of the maximum temporal flow (which is now a random variable) in a mixed temporal network.


  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall Inc., Upper Saddle River (1993)MATHGoogle Scholar
  2. 2.
    Akrida, E.C., Czyzowicz, J., Gasieniec, L., Kuszner, L., Spirakis, P.G.: Flows in temporal networks. CoRR abs/1606.01091 (2016)Google Scholar
  3. 3.
    Akrida, E.C., Gasieniec, L., Mertzios, G.B., Spirakis, P.G.: Ephemeral networks with random availability of links: the case of fast networks. J. Parallel Distrib. Comput. 87, 109–120 (2016)CrossRefGoogle Scholar
  4. 4.
    Akrida, E.C., Gąsieniec, L., Mertzios, G.B., Spirakis, P.G.: On temporally connected graphs of small cost. In: Sanità, L., Skutella, M. (eds.) WAOA 2015. LNCS, vol. 9499, pp. 84–96. Springer, Cham (2015). doi:10.1007/978-3-319-28684-6_8 CrossRefGoogle Scholar
  5. 5.
    Akrida, E.C., Spirakis, P.G.: On verifying and maintaining connectivity of interval temporal networks. In: Bose, P., Gąsieniec, L.A., Römer, K., Wattenhofer, R. (eds.) ALGOSENSORS 2015. LNCS, vol. 9536, pp. 142–154. Springer, Cham (2015). doi:10.1007/978-3-319-28472-9_11 CrossRefGoogle Scholar
  6. 6.
    Aronson, J.E.: A survey of dynamic network flows. Ann. Oper. Res. 20(1–4), 1–66 (1989)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Avin, C., Koucký, M., Lotker, Z.: How to explore a fast-changing world (cover time of a simple random walk on evolving graphs). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5125, pp. 121–132. Springer, Heidelberg (2008). doi:10.1007/978-3-540-70575-8_11 CrossRefGoogle Scholar
  8. 8.
    Batra, J., Garg, N., Kumar, A., Mömke, T., Wiese, A.: New approximation schemes for unsplittable flow on a path. In: Indyk, P. (ed.) Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, 4–6 January 2015, pp. 47–58. SIAM (2015)Google Scholar
  9. 9.
    Casteigts, A., Flocchini, P., Quattrociocchi, W., Santoro, N.: Time-varying graphs, dynamic networks. Int. J. Parallel Emerg. Distrib. Syst. (IJPEDS) 27(5), 387–408 (2012)CrossRefGoogle Scholar
  10. 10.
    Chaintreau, A., Mtibaa, A., Massoulié, L., Diot, C.: The diameter of opportunistic mobile networks. In: Proceedings of the 2007 ACM Conference on Emerging Network Experiment and Technology, CoNEXT 2007, New York, NY, USA, 10–13 December 2007, p. 12 (2007)Google Scholar
  11. 11.
    Clementi, A.E.F., Macci, C., Monti, A., Pasquale, F., Silvestri, R.: Flooding time of edge-Markovian evolving graphs. SIAM J. Discret. Math. (SIDMA) 24(4), 1694–1712 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Erlebach, T., Hoffmann, M., Kammer, F.: On temporal graph exploration. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 444–455. Springer, Heidelberg (2015). doi:10.1007/978-3-662-47672-7_36 Google Scholar
  13. 13.
    Ford, D.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (2010)MATHGoogle Scholar
  14. 14.
    Hoppe, B., Tardos, E.: The quickest transshipment problem. In: Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1995, pp. 512–521. Society for Industrial and Applied Mathematics, Philadelphia (1995)Google Scholar
  15. 15.
    Hoppe, B.E.: Efficient dynamic network flow algorithms. Ph.D. thesis (1995)Google Scholar
  16. 16.
    Kamiyama, N., Katoh, N.: The universally quickest transshipment problem in a certain class of dynamic networks with uniform path-lengths. Discret. Appl. Math. 178, 89–100 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kempe, D., Kleinberg, J.M., Kumar, A.: Connectivity and inference problems for temporal networks. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC), pp. 504–513 (2000)Google Scholar
  18. 18.
    Madry, A.: Fast approximation algorithms for cut-based problems in undirected graphs. In: 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, Las Vegas, Nevada, USA, 23–26 October 2010, pp. 245–254 (2010)Google Scholar
  19. 19.
    Mertzios, G.B., Michail, O., Chatzigiannakis, I., Spirakis, P.G.: Temporal network optimization subject to connectivity constraints. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7966, pp. 657–668. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39212-2_57 Google Scholar
  20. 20.
    Orlin, J.B.: Max flows in O(nm) time, or better. In: Symposium on Theory of Computing Conference, STOC 2013, Palo Alto, CA, USA, 1–4 June 2013, pp. 765–774 (2013)Google Scholar
  21. 21.
    Papadimitriou, C.M.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  22. 22.
    Powell, W.B., Jaillet, P., Odoni, A.: Stochastic and dynamic networks and routing. Handb. Oper. Res. Manag. Sci. 8, 141–295 (1995)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Radzik, T.: Faster algorithms for the generalized network flow problem. Math. Oper. Res. 23(1), 69–100 (1998)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Serna, M.J.: Randomized parallel approximations to max flow. In: Kao, M.-Y. (ed.) Encyclopedia of Algorithms, pp. 1750–1753. Springer, New York (2016)CrossRefGoogle Scholar
  25. 25.
    Skutella, M.: An introduction to network flows over time. In: Cook, W., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, pp. 451–482. Springer, Heidelberg (2008)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Eleni C. Akrida
    • 1
  • Jurek Czyzowicz
    • 2
  • Leszek Gąsieniec
    • 1
  • Łukasz Kuszner
    • 3
  • Paul G. Spirakis
    • 1
    • 4
  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  2. 2.Dépt. d’informatiqueUniversité du Québec en OutaouaisGatineauCanada
  3. 3.Faculty of Electronics, Telecommunications and InformaticsGdańsk University of TechnologyGdańskPoland
  4. 4.Computer Technology Institute and Press “Diophantus” (CTI)PatrasGreece

Personalised recommendations