Irrelevant Components and Exact Computation of the Diameter Constrained Reliability

Let G = (V;E) be a simple graph with jVj = n nodes and jEj = m links, a subset K V of terminals, a vector p = (p1;:::;pm)2 [0; 1] m and a positive integer d, called diameter. We assume nodes are perfect but links fail stochastically and independently, with probabilities qi = 1 pi. The diameter-constrained reliability (DCR for short), is the probability that the terminals of the resulting subgraph remain connected by paths composed by d links, or less. This number is denoted by R d(p). The general computation of the parameter R d K;G(p) belongs to the class of NP-Hard problems, since is subsumes the complexity that a random graph is connected. A discussion of the computational complexity for DCR-subproblems is provided in terms of the number of terminal nodes k =jKj and diameter d. Either when d = 1 or when d = 2 and k is xed, the DCR is inside

[1]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[2]  Pablo Enrique Sartor Del Giudice.  Diameter-constrained network reliability : properties and computation. (Propriétés et méthodes de calcul de la fiabilité diamètre-bornée des réseaux) , 2013 .

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  S. H. Ahmad.  A Simple Technique for Computing Network Reliability , 1982, IEEE Transactions on Reliability.

[5]  A. Rosenthal.  Computing the Reliability of Complex Networks , 1977 .

[6]  R. K. Wood,et al.  Note on "A Linear-Time Algorithm for Computing K-Terminal Reliability in a Series-Parallel Network" , 1996, SIAM J. Comput..

[7]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[8]  LOUIS PETINGI.  Diameter-related properties of graphs and applications to network reliability theory , 2013 .

[9]  Gerardo Rubino,et al.  Rare Event Simulation using Monte Carlo Methods , 2009 .

[10]  Michael O. Ball,et al.  Computational Complexity of Network Reliability Analysis: An Overview , 1986, IEEE Transactions on Reliability.

[11]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[12]  Héctor Cancela,et al.  Reliability of communication networks with delay constraints: computational complexity and complete topologies , 2004, Int. J. Math. Math. Sci..

[13]  Eduardo Alberto Canale,et al.  Diameter Constrained Reliability: Computational Complexity in terms of the diameter and number of terminals , 2014, ArXiv.

[14]  Charles J. Colbourn.  Reliability Issues In Telecommunications Network Planning , 1999 .

[15]  Héctor Cancela,et al.  Polynomial-Time Topological Reductions That Preserve the Diameter Constrained Reliability of a Communication Network , 2011, IEEE Transactions on Reliability.

[16]  J. Scott Provan,et al.  The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected , 1983, SIAM J. Comput..

[17]  Gerardo Rubino,et al.  On computing the 2-diameter -constrained K -reliability of networks , 2013, Int. Trans. Oper. Res..

[18]  Kishor S. Trivedi,et al.  A survey of efficient reliability computation using disjoint products approach , 1995, Networks.

[19]  Richard M. Karp,et al.  Reducibility among combinatorial problems" in complexity of computer computations , 1972 .

[20]  Fred Moskowitz,et al.  The analysis of redundancy networks , 1958, Transactions of the American Institute of Electrical Engineers, Part I: Communication and Electronics.

[21]  Appajosyula Satyanarayana,et al.  A Linear-Time Algorithm for Computing K-Terminal Reliability in Series-Parallel Networks , 1985, SIAM J. Comput..

[22]  Eduardo Alberto Canale,et al.  Monte Carlo methods in diameter-constrained reliability , 2014, Opt. Switch. Netw..