Exact Computation of Strongly Connected Reliability by Binary Decision Diagrams

  • Hirofumi SuzukiEmail author
  • Masakazu IshihataEmail author
  • Shin-ichi MinatoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)


Network reliability is the probability that a network system can perform a desired operation, such as communication between facilities, against stochastic equipment failures. On analyzing network systems that are represented by undirected graphs, the all-terminal reliability (ATR) is commonly used as one of the network reliability. As a natural extension of the ATR for the directed version, the strongly connected reliability (SCR) is known. The SCR should be computed on various network systems, such as ad-hoc network, that demand the property called strongly connected. Unfortunately, computing the SCR is known to be #P-complete, and little studies challenge the computation of the exact or an approximate SCR on limited graph classes. In this study, we propose the first practically efficient algorithm to compute the exact SCR in general. The algorithm constructs a binary decision diagram (BDD) representing all the strongly connected spanning subgraphs (SCSSs) in a given directed graph. Subsequently, the algorithm computes the exact SCR by a dynamic programming on the BDD. To efficiently construct BDDs, we designed a new variant of the frontier based search (FBS). We conducted computational experiments to evaluate the proposed algorithm. The results demonstrated that the proposed algorithm succeeded in computing the SCR in real-world networks with a few hundred edges within a reasonable time, which was previously impossible.


Network reliability All-terminal reliability Strongly connected reliability Binary decision diagram Dynamic programming 



This work was supported by JSPS KAKENHI Grant Number 15H05711.


  1. 1.
    Won, J.M., Karray, F.: Cumulative update of all-terminal reliability for faster feasibility decision. IEEE Trans. Reliab. 59(3), 551–562 (2010)CrossRefGoogle Scholar
  2. 2.
    Park, J.H.: All-terminal reliability analysis of wireless networks of redundant radio modules. IEEE Internet Things J. 3(2), 219–230 (2016)CrossRefGoogle Scholar
  3. 3.
    Brown, J., Li, X.: The strongly connected reliability of complete digraphs. Netw.: Int. J. 45, 165–168 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8, 410–421 (1979)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Karger, D.R.: A randomized fully polynomial time approximation scheme for the all terminal network reliability problem. In: Proceedings of the Twenty-seventh Annual ACM Symposium on Theory of Computing, STOC 1995, pp. 11–17. ACM, New York (1995)Google Scholar
  6. 6.
    Imai, H., Sekine, K., Imai, K.: Computational investigations of all-terminal network reliability via BDDs. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 82, 714–721 (1999)Google Scholar
  7. 7.
    Hwang, F.K., Wright, P.E., Hu, X.: Exact reliabilities of most reliable double-loop networks. Networks 30, 81–90 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Trans. Comput. 35(8), 677–691 (1986)CrossRefGoogle Scholar
  9. 9.
    Frederickson, G.N., JáJá, J.: Approximation algorithms for several graph augmentation problems. SIAM J. Comput. 10(2), 270–283 (1981)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Vincent, D., Cecile, B.: Transitive reduction for social network analysis and visualization. In: Proceedings of the 2005 IEEE/WIC/ACM International Conference on Web Intelligence, WI 2005, pp. 128–131. IEEE Computer Society, Washington, D.C. (2005)Google Scholar
  11. 11.
    Ardito, C.F., Paola, D.D., Gasparri, A.: Decentralized estimation of the minimum strongly connected subdigraph for robotic networks with limited field of view. In: 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), pp. 5304–5309, December 2012Google Scholar
  12. 12.
    Albert, R., DasGupta, B., Dondi, R., Sema Kachalo, E.S., Zelikovsky, A., Westbrooks, K.: A novel method for signal transduction network inference from indirect experimental evidence. J. Comput. Biol. 14(7), 927–949 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Aditya, S., DasGupta, B., Karpinski, M.: Algorithmic perspectives of network transitive reduction problems and their applications to synthesis and analysis of biological networks. Biology 3(1), 1–21 (2014)CrossRefGoogle Scholar
  14. 14.
    Kawahara, J., Inoue, T., Iwashita, H., Minato, S.: Frontier-based search for enumerating all constrained subgraphs with compressed representation. IEICE Trans. Fundam. E100-A(9), 1773–1784 (2017)CrossRefGoogle Scholar
  15. 15.
    Kinnersley, N.G.: The vertex separation number of a graph equals its path-width. Inf. Process. Lett. 42(6), 345–350 (1992)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Inoue, Y., Minato, S.: Acceleration of ZDD construction for subgraph enumeration via path-width optimization. TCS-TR-A-16-80. Hokkaido University (2016)Google Scholar
  17. 17.
    Yoshinaka, R., Kawahara, J., Denzumi, S., Arimura, H., Minato, S.: Counterexamples to the long-standing conjecture on the complexity of BDD binary operations. Inf. Process. Lett. 112, 636–640 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bergman, D., Ciré, A.A., van Hoeve, W., Hooker, J.N.: Decision Diagrams for Optimization. Artificial Intelligence: Foundations, Theory, and Algorithms. Springer, Cham (2016). Scholar
  19. 19.
    Maurer, P.: Generating strongly connected random graphs. In: Proceedings of the 2017 International Conference on Modeling, Simulation and Visualization Methods, MSV 2017, pp. 3–6. CSCE, Las Vegas (2017)Google Scholar
  20. 20.
    Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L.: Enumerating minimal dicuts and strongly connected subgraphs and related geometric problems. In: Bienstock, D., Nemhauser, G. (eds.) IPCO 2004. LNCS, vol. 3064, pp. 152–162. Springer, Heidelberg (2004). Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Graduate School of Information Science and TechnologyHokkaido UniversitySapporoJapan
  2. 2.NTT Communication Science LaboratoriesKyotoJapan
  3. 3.Graduate School of InformaticsKyoto UniversityKyotoJapan

Personalised recommendations