On the Average-Case Complexity of the Graph Reliability Problem on Gaussian Distributions

We introduce classes of narrow graphs (including grid strips of fixed width), for which the graph reliability problem admits a polynomial time algorithm. Using this algorithm, we show that graph reliability is computable in polynomial time for the average complexity with respect to a Gaussian distribution. The latter is defined as follows: the vertices are numbered by integers {1,2, ...n}, and the probability that an edge between i and j is present is e^−|i−j|2.