The Intractibility of the 2-critical Flow Problem in K-flow Graphs

A k-ow-graph is a single-source/single-sink acyclic graph which is the union of k arc-disjoint source-sink paths. Arcs have unit capacity and are subject to i.i.d. random failures. It is shown that the problem of computing the probability that the operating arcs support a ow of value at least k ? 2 is #P-complete. This also shows that the \dual" problem of counting the number of subsets of arcs in a k-ow-graph that are contained in the union of at most 2 arc-disjoint source-sink paths is also #P-complete. This result has important negative consequences for computing network throughput reliability, and is in contrast to the fact that both problems are polynomial when G is in addition planar, or in any k-ow graph when the target ow value is k or k ? 1, or dually, in any acyclic graph when it is required to count those arc-sets that are subsets of single source-sink paths.