On Temporally Connected Graphs of Small Cost

  • Eleni C. AkridaEmail author
  • Leszek Gąsieniec
  • George B. Mertzios
  • Paul G. Spirakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9499)


We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of n vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex u to vertex v is a path from u to v where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a (uv)-journey for any pair of vertices \(u,v,~u\not = v\). We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can freely choose availability instances for all edges and aims for temporal connectivity with very small cost; the cost is the total number of availability instances used. We achieve this via a simple polynomial-time procedure which derives designs of cost linear in n, and at most the optimal cost plus 2. To show this, we prove a lower bound on the cost for any undirected graph. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead given a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless \(P=NP\). On the positive side, we show that in dense graphs with random edge availabilities, all but \(\varTheta (n)\) labels are redundant whp. A temporal design may, however, be minimal, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least \(n \log {n}\) labels.



We wish to thank Thomas Gorry for co-implementing the code used in the proof of Theorem 4.


  1. 1.
    Akrida, E.C., Gąsieniec, L., Mertzios, G.B., Spirakis, P.G.: Ephemeral networks with random availability of links: Diameter and connectivity. In: Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA) (2014)Google Scholar
  2. 2.
    Alimonti, P., Kann, V.: Hardness of approximating problems on cubic graphs. In: Bongiovanni, G., Bovet, D.P., Di Battista, G. (eds.) CIAC 1997. LNCS, vol. 1203, pp. 288–298. Springer, Heidelberg (1997) CrossRefGoogle Scholar
  3. 3.
    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, Part I. LNCS, vol. 5125, pp. 121–132. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  4. 4.
    Bui-Xuan, B.-M., Ferreira, A., Jarry, A.: Computing shortest, fastest, and foremost journeys in dynamic networks. Int. J. Found. Comput. Sci. 14(2), 267–285 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Casteigts, A., Flocchini, P.: Deterministic Algorithms in Dynamic Networks: Formal Models and Metrics. Defence R&D Canada, Technical report, April 2013Google Scholar
  6. 6.
    Casteigts, A., Flocchini, P.: Deterministic Algorithms in Dynamic Networks: Problems, Analysis, and Algorithmic Tools. Defence R&D Canada, Technical report, April 2013Google Scholar
  7. 7.
    Casteigts, A., Flocchini, P., Quattrociocchi, W., Santoro, N.: Time-varying graphs and dynamic networks. Int. J. Parallel Emergent Distrib. Syst. (IJPEDS) 27(5), 387–408 (2012)CrossRefGoogle Scholar
  8. 8.
    Clementi, A.E.F., Macci, C., Monti, A., Pasquale, F., Silvestri, R.: Flooding time of edge-markovian evolving graphs. SIAM J. Discrete Math. (SIDMA) 24(4), 1694–1712 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Dutta, C., Pandurangan, G., Rajaraman, R., Sun, Z., Viola, E.: On the complexity of information spreading in dynamic networks. In: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 717–736 (2013)Google Scholar
  10. 10.
    Fleischer, L., Skutella, M.: Quickest flows over time. SIAM J. Comput. 36(6), 1600–1630 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Fleischer, L., Tardos, É.: Efficient continuous-time dynamic network flow algorithms. Oper. Res. Lett. 23(3–5), 71–80 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Gupta, A., Krishnaswamy, R., Ravi, R.: Online and stochastic survivable network design. SIAM J. Comput. 41(6), 1649–1672 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    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
  14. 14.
    Klinz, B., Woeginger, G.J.: One, two, three, many, or: complexity aspects of dynamic network flows with dedicated arcs. Oper. Res. Lett. 22(4–5), 119–127 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Koch, R., Nasrabadi, E., Skutella, M.: Continuous and discrete flows over time - A general model based on measure theory. Math. Meth. of OR 73(3), 301–337 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Kontogiannis, S., Zaroliagis, C.: Distance oracles for time-dependent networks. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 713–725. Springer, Heidelberg (2014) Google Scholar
  17. 17.
    Kuhn, F., Lynch, N.A., Oshman, R.: Distributed computation in dynamic networks. In: Proceedings of the 42nd Annual ACM Symposium on Theory of Computing (STOC), pp. 513–522 (2010)Google Scholar
  18. 18.
    Lau, L.C., Naor, J., Salavatipour, M.R., Singh, M.: Survivable network design with degree or order constraints. SIAM J. Comput. 39(3), 1062–1087 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Lau, L.C., Singh, M.: Additive approximation for bounded degree survivable network design. SIAM J. Comput. 42(6), 2217–2242 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Mertzios, G.B., Michail, O., Chatzigiannakis, I., Spirakis, P.G.: Temporal network optimization subject to connectivity constraints. In: Fomin, F.V., Freivalds, R.U., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 657–668. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  21. 21.
    Michail, O., Chatzigiannakis, I., Spirakis, P.G.: Causality, influence, and computation in possibly disconnected synchronous dynamic networks. In: Baldoni, R., Flocchini, P., Binoy, R. (eds.) OPODIS 2012. LNCS, vol. 7702, pp. 269–283. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  22. 22.
    O’Dell, R., Wattenhofer, R.: Information dissemination in highly dynamic graphs. In: Proceedings of the 2005 Joint Workshop on Foundations of Mobile Computing (DIALM-POMC), pp. 104–110 (2005)Google Scholar
  23. 23.
    Scheideler, C.: Models and techniques for communication in dynamic networks. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 27–49. Springer, Heidelberg (2002) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Eleni C. Akrida
    • 1
    Email author
  • Leszek Gąsieniec
    • 1
  • George B. Mertzios
    • 2
  • Paul G. Spirakis
    • 1
  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  2. 2.School of Engineering and Computing SciencesDurham UniversityDurhamUK

Personalised recommendations