Networked systems make the reliability assessment of critical infrastructure computationally challenging given the combinatorial nature of system-level states. Several methods from numerical schemes to analytical approaches, such as Monte Carlo Simulation (MCS) and recursive decomposition algorithms (RDA), respectively, have been applied to this stochastic network problem. Despite progress over several decades, the problem remains open because of its intrinsic computational complexity. As the structural facilities of infrastructure systems continue to in terconnect in network forms, their study steers analysts to develop system reliability assessment methods based on graph theory and network science. A fully polynomial randomized approximation scheme (FPRAS) based on Karger’s graph contraction algorithm is an approximating method for reliability evaluation, which has a unique property rarely exploited in engineering reliability: that by performing a number of experiments in polynomial time (as a function of system size), it provides an a priori theoretical guarantee that the reliability estimate falls into the ϵ-neighborhood of its true value with (1−δ) confidence. We build upon the FPRAS ideas to develop an s-t reliability version that has practical appeal. Focusing on the relevant-cut enumeration stage of the FPRAS, we find correlations between the recurrence frequencies of links in minimum cuts within the randomization phase of the contraction algorithm, and typical network topological properties. We employ LASSO regression analysis to approximate the relationship between link recurrence frequencies and such topological metrics. With the topology-informed link recurrence frequencies, obtained at a much lower computational cost, we use a new biased contraction probability yielding 16.9% more distinct minimum cuts (MinCuts) than the original random contraction scheme. The biased contraction scheme proposed here can significantly improve the efficiency of reliability evaluation of networked infrastructure systems, while supporting infrastructure systems design, maintenance and restoration given its ability to offer error guarantees, which are ideal for future prescriptive guidelines in practice.