Abstract
There is a strong interplay between network reliability and connectivity theory. In fact, previous studies show that the graphs with maximum reliability, called uniformly most-reliable graphs, must have the highest connectivity. In this paper, we revisit the underlying theory in order to build uniformly most-reliable cubic graphs. The computational complexity of the problem promotes the development of heuristics. The contributions of this paper are three-fold. In a first stage, we propose an ideal Variable Neighborhood Descent (VND) which returns the graph with maximum reliability. This VND works in exponential time. In a second stage, we propose a hybrid GRASP/VND approach that trades quality for computational effort. A construction phase enriched with a Restricted Candidate List (RCL) offers diversification. Our local search phase includes a factor-2 algorithm for an Integer Linear Programming (ILP) model. As a product of our research, we recovered previous optimal graphs from the related literature in the field. Additionally, we offer new candidates of uniformly most-reliable graphs with maximum connectivity and maximum number of spanning trees.
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Bauer, D., Boesch, F., Suffel, C., Van Slyke, R.: On the validity of a reduction of reliable network design to a graph extremal problem. IEEE Trans. Circuits Syst. 34(12), 1579–1581 (1987)
Beineke, L.W., Wilson, R.J., Oellermann, O.R.: Topics in Structural Graph Theory. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2012)
Biggs, N.: Algebraic Graph Theory. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1993)
Boesch, F.T., Li, X., Suffel, C.: On the existence of uniformly optimally reliable networks. Networks 21(2), 181–194 (1991)
Canale, E., Cancela, H., Robledo, F., Romero, P., Sartor, P.: Full complexity analysis of the diameter-constrained reliability. Int. Trans. Oper. Res. 22(5), 811–821 (2015)
Colbourn, C.J.: Reliability issues in telecommunications network planning. In: Sansò, B., Soriano, P. (eds.) Telecommunications Network Planning. CRT, pp. 135–146. Springer, Boston (1999). https://doi.org/10.1007/978-1-4615-5087-7_8
Duarte, A., Mladenović, N., Sánchez-Oro, J., Todosijević, R.: Variable neighborhood descent. In: Martí, R., Panos, P., Resende, M. (eds.) Handbook of Heuristics, pp. 1–27. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-319-07153-4_9-1
Fishman, G.S.: Monte Carlo: Concepts, Algorithms and Applications. Springer, New York (1996). https://doi.org/10.1007/978-1-4757-2553-7
Hajek, B., Zhu, J.: The missing piece syndrome in peer-to-peer communication. In: 2010 IEEE International Symposium on Information Theory, pp. 1748–1752, June 2010
Harary, F.: The maximum connectivity of a graph. Proc. Natl. Acad. Sci. U. S. A. 48(7), 1142–1146 (1962)
Jaeger, F., Vertigan, D.L., Welsh, D.J.A.: On the computational complexity of the Jones and Tutte polynomials. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 108, no. 1, pp. 35–53 (1990)
Jin, R., et al.: Detecting node failures in mobile wireless networks: a probabilistic approach. IEEE Trans. Mob. Comput. 15(7), 1647–1660 (2016)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)
Kelmans, A.K.: On graphs with randomly deleted edges. Acta Math. Acad. Sci. Hung. 37(1), 77–88 (1981)
Kirchoff, G.: Über die auflösung der gleichungen, auf welche man bei der untersuchung der linearen verteilung galvanischer ströme geführt wird. Ann. Phys. Chem. 72, 497–508 (1847)
Rela, G., Robledo, F., Romero, P.: Petersen graph is uniformly most-reliable. In: Nicosia, G., Pardalos, P., Giuffrida, G., Umeton, R. (eds.) MOD 2017. LNCS, vol. 10710, pp. 426–435. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72926-8_35
Resende, M.G.C., Ribeiro, C.C.: Optimization by GRASP - Greedy Randomized Adaptive Search Procedures. Springer, New York (2016). https://doi.org/10.1007/978-1-4939-6530-4
Romero, P.: Building uniformly most-reliable networks by iterative augmentation. In: 9th International Workshop on Resilient Networks Design and Modeling (RNDM), pp. 1–7 (2017)
Rosenthal, A.: Computing the reliability of complex networks. SIAM J. Appl. Math. 32(2), 384–393 (1977)
Satyanarayana, A., Schoppmann, L., Suffel, C.L.: A reliability improving graph transformation with applications to network reliability. Networks 22(2), 209–216 (1992)
Provan, J.S., Ball, M.O.: The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. Comput. 12(4), 777–788 (1983)
Viera, J.: Búsqueda de grafos cúbicos de máxima confiabilidad. Master’s thesis, Facultad de Ingeniería, Universidad de la República, Uruguay (2018)
Wang, G.: A proof of Boesch’s conjecture. Networks 24(5), 277–284 (1994)
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This work is partially supported by Project 395 CSIC I+D Sistemas Binarios Estocásticos Dinámicos.
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Bourel, M., Canale, E., Robledo, F., Romero, P., Stábile, L. (2019). A Hybrid GRASP/VND Heuristic for the Design of Highly Reliable Networks. In: Blesa Aguilera, M., Blum, C., Gambini Santos, H., Pinacho-Davidson, P., Godoy del Campo, J. (eds) Hybrid Metaheuristics. HM 2019. Lecture Notes in Computer Science(), vol 11299. Springer, Cham. https://doi.org/10.1007/978-3-030-05983-5_6
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