Faster (and Still Pretty Simple) Unbiased Estimators for Network (Un)reliability

Consider the problem of estimating the (un)reliability of an n-vertex graph when edges fail with probability p. We show that the Recursive Contraction Algorithms for minimum cuts, essentially unchanged and running in n
^2+o(1)
time, yields an unbiased estimator of constant relative variance (and thus an FPRAS with the same time bound) whenever p
^c
<; n
^-2
. For larger p, we show that reliable graphs-where failures are rare so seemingly harder to find-effectively act like small graphs and can thus be analyzed quickly. Combining these ideas gives us an unbiased estimator for unreliability running in Õ(n
^2.78
) time, an improvement on the previous Õ(n
³
) time bound.