Karger (SIAM Journal on Computing, 1999) developed the first fully-polynomial
approximation scheme to estimate the probability that a graph G becomes
disconnected, given that its edges are removed independently with probability
p. This algorithm runs in n5+o(1)ϵ−3 time to obtain an
estimate within relative error ϵ.
We improve this run-time through algorithmic and graph-theoretic advances.
First, there is a certain key sub-problem encountered by Karger, for which a
generic estimation procedure is employed, we show that this has a special
structure for which a much more efficient algorithm can be used. Second, we
show better bounds on the number of edge cuts which are likely to fail. Here,
Karger's analysis uses a variety of bounds for various graph parameters, we
show that these bounds cannot be simultaneously tight. We describe a new graph
parameter, which simultaneously influences all the bounds used by Karger, and
obtain much tighter estimates of the cut structure of G. These techniques
allow us to improve the runtime to n3+o(1)ϵ−2, our results also
rigorously prove certain experimental observations of Karger & Tai (Proc.
ACM-SIAM Symposium on Discrete Algorithms, 1997). Our rigorous proofs are
motivated by certain non-rigorous differential-equation approximations which,
however, provably track the worst-case trajectories of the relevant parameters.
A key driver of Karger's approach (and other cut-related results) is a bound
on the number of small cuts: we improve these estimates when the min-cut size
is "small" and odd, augmenting, in part, a result of Bixby (Bulletin of the
AMS, 1974)