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

06/05/2021
by   Amir Abboud, et al.
0

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 Õ(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 Õ(n^22/3) and it would improve to n^2+o(1) if 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.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset