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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


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics