Sample average
Simulate M random subgraphs. Count fraction where s connected to t. Standard Monte Carlo. Slow but easy.
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Importance sampling
Bias toward configurations near reliability boundary. Karger's algorithm variants for rare-event simulation.
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Bounds
Upper bound: min cut probability (probability every edge in min cut fails). Lower bound: max flow rounded.
Cactus of cuts
All min cuts encoded as cactus graph. Reliability computable in polynomial time on cactus.