A fully polynomial randomized approximation scheme for hypergraph unreliability

We give a fully polynomial randomized approximation scheme for the probability that an explicitly represented undirected hypergraph becomes disconnected when its hyperedges fail independently with arbitrary rational probabilities. The rank is unrestricted. The estimator uses the paired representative construction of Cen, Li, and Panigrahi: after duplicating each edge, distinct heads provide a lower bound of the minimum cut on the rate at which any surviving representative pair is destroyed. We turn this construction into an executable importance sampler by suppressing transitions that destroy a uniformly chosen pair, preserving the original rates of all other transitions, and correcting by the reciprocal number of final surviving pairs. The resulting ideal estimator has relative second moment at most n(n-1)/2. For common failure probability p, we normalize by the exact rational p^lambda, replace the residual likelihood by an absorbing cemetery transition, and simulate a rational Euler discretization with a fixed random-bit budget. For nonidentical probabilities, certified dyadic rate rounding, an exact rational weighted minimum cut, and relative rational normalization give the extension. Explicit error and confidence bounds establish polynomial worst-case running time in the incidence input size, the probability encoding length, epsilon^(-1), and log(1/delta). The result concerns all-terminal unreliability, not its complement or a prescribed terminal-pair event.