A note on network reliability Institute for Advanced Study, Princeton, NJ 08540 Summary: A note on network reliability Noga Alon Institute for Advanced Study, Princeton, NJ 08540 and Department of Mathematics Tel Aviv University, Tel Aviv, Israel Let G = (V, E) be a loopless undirected multigraph, with a probability pe, 0 pe 1 assigned to every edge e E. Let Gp be the random subgraph of G obtained by deleting each edge e of G, randomly and independently, with probability qe = 1 - pe. For any nontrivial subset S V let (S, S) denote, as usual, the cut determined by S, i.e., the set of all edges of G with an end in S and an end in its complement S. Define P(S) = e(S,S) pe, and observe that P(S) is simply the expected number of edges of Gp that lie in the cut (S, S). In this note we prove the following. Theorem 1 For every positive constant b there exists a constant c = c(b) > 0 so that if P(S) c log n for every nontrivial S V , then the probability that Gp is disconnected is at most 1/nb . The assertion of this theorem (in an equivalent form) was conjectured by Dimitris Bertsimas, who was motivated by the study of a class of approximation graph algorithms based on a randomized rounding technique of solutions of appropriately formulated linear programming relaxations. Observe that the theorem is sharp, up to the multiplicative factor c, by the Collections: Mathematics