Fair Cuts, Approximate Isolating Cuts, and Approximate Gomory-Hu Trees in Near-Linear Time
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In this paper, we introduce a robust notion of (1+ϵ)-approximate (s, t)-mincuts in undirected graphs where every cut edge can be simultaneously saturated (in the same direction) to a 1/1+ϵ fraction by an (s, t)-flow. We call these (1+ϵ)-fair cuts. Unlike arbitrary approximate (s, t)-mincuts, fair cuts can be uncrossed, which is a key property of (s, t)-mincuts used in many algorithms. We also give a near-linear Õ(m/ϵ^3)-time algorithm for computing an (s, t)-fair cut. This offers a general tool for trading off a (1+ϵ)-approximation for near-linear running time in mincut based algorithms. As an application of this new concept, we obtain a near-linear time algorithm for constructing a (1+ϵ)-approximate Gomory-Hu tree, thereby giving a nearly optimal algorithm for the (1+ϵ)-approximate all-pairs max-flows (APMF) problem in undirected graphs. Our result is obtained via another intermediate tool of independent interest. We obtain a near-linear time algorithm for finding (1+ϵ)-approximate isolating cuts in undirected graphs, a concept that has gained wide traction over the past year.
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