A Representation and Query Structure to Approximate Mincuts of a Network

this paper, applications requiring a structured knowledge of approximate mincuts can take advantage of the fast running time of the Karger--Stein algorithm as well

Approximating s-t Minimum Cuts in Õ(n²) Time

We improve on random sampling techniques for approximately solving problems that involve cuts in graphs. We give a linear-time construction that transforms any graph on n vertices into an O(n log n)-edge graphon the same vertices whose cuts have approximately the same value as the original graph's. In this new graph, for example, we can run the Õ(mn)-time maximum flow algorithm of Goldberg and Tarjan to find an s-t minimum cut in Õ(n²) time. This corresponds to a (1 + ɛ)-times minimum s-t cut in the original graph. In a similar way, we can approximate a sparsest cut in Õ(n²) time