APMF < APSP? Gomory-Hu Tree for Unweighted Graphs in Almost-Quadratic Time*

We design an n 2+o(1)-time algorithm that constructs a cut-equivalent (Gomory-Hu) tree of a simple graph on n nodes. This bound is almost-optimal in terms of n, and it improves on the recent tildeO(n 2.5}) bound by the authors (STOC 2021), which was the first to break the cubic barrier. Consequently, the All-Pairs Maximum-Flow (APMF) problem has time complexity n 2+o(1), and for the first time in history, this problem can be solved faster than All-Pairs Shortest Paths (APSP). We further observe that an almost-linear time algorithm (in terms of the number of edges m) is not possible without first obtaining a subcubic algorithm for multigraphs. Finally, we derandomize our algorithm, obtaining the first subcubic deterministic algorithm for Gomory-Hu Tree in simple graphs, showing that randomness is not necessary for beating the n-1 times max-flow bound from 1961. The upper bound is tildeO(n 223}}) and it would improve to n 2+o(1)\mathbf{imathbf{f there is a deterministic single-pair maximum-flow algorithm that is almost-linear. The key novelty is in using a 'dynamic pivot' technique instead of the randomized pivot selection that was central in recent works.