Deterministic Min-cut in Poly-logarithmic Max-flows
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We give a deterministic algorithm for finding the minimum (weight) cut of an undirected graph on n vertices and m edges using polylog(n) calls to any maximum flow subroutine. Using the current best deterministic maximum flow algorithms, this yields an overall running time of Õ(m ·min(√(m), n^2/3)) for weighted graphs, and m^4/3+o(1) for unweighted (multi)-graphs. This marks the first improvement for this problem since a running time bound of Õ(mn) was established by several papers in the early 1990s. To obtain this result, we introduce a new tool for finding minimum cuts of an undirected graph: *isolating cuts*. Given a set of vertices R, this entails finding cuts of minimum weight that separate (or isolate) each individual vertex v∈ R from the rest of the vertices R∖{v}. Naïvely, this can be done using |R| maxflow calls, but we show that just O(log |R|) suffice for finding isolating cuts for any set of vertices R. We call this the *isolating cut lemma*.
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