All-Pairs Max-Flow is no Harder than Single-Pair Max-Flow: Gomory-Hu Trees in Almost-Linear Time
Amir Abboud, Jason Li, Debmalya Panigrahi, Thatchaphol Saranurak · 2023
A Gomory-Hu tree (also called a cut tree) succinctly represents $(s, t)$ min-cuts (and therefore, $(s, t)$ max-flow values) of all pairs of vertices $s, t$ in an undirected graph. In this paper, we give an $m^{1+o(1)}$-time algorithm for constructing a Gomory-Hu tree for a graph with m edges. This shows that the all-pairs max-flows problem has the same running time as the single-pair max-flow problem, up to a subpolynomial factor. Prior to our work, the best known Gomory-Hu tree algorithm was obtained in recent work by Abboud et al. (FOCS 2022) and requires $\tilde{O}\left(n^{2}\right)$ time for a graph with n vertices. Our result marks a natural culmination of over 60 years of research into the all-pairs maxflows problem that started with Gomory and Hu’s pathbreaking result introducing the Gomory-Hu tree in 1961.