The authors present a number of techniques based on tie-sets and cut-sets for bounding and approximating the 2-terminal reliability of a communications network model, with the knowledge that the 2-terminal reliability problem is NP-complete. Three classes of approximations are treated theoretically and in terms of examples. One method uses all the network cut-sets, but truncates the reliability expansion, leading to upper and lower bounds. Another method chooses subsets of the tie-sets and cut-sets, including the shorter (fewer element) tie-sets and cut-sets, and carries out the entire expansion. The third approximation technique incorporates features of both of the first two methods. The reduced tie-set and cut-set method, although exponential in the number of tie-sets and cut-sets of the network in theory, is polynomial in practice and provides lower and upper bounds of the same order of magnitude in most instances. The combined method has been shown to be polynomial in the number of 1, 2, and 3-edge cut-sets used and to run faster than the reduced tie-set and cut-set method.
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