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

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

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 m1+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 Õ(n2) 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.

Original languageEnglish
Title of host publicationProceedings - 2023 IEEE 64th Annual Symposium on Foundations of Computer Science, FOCS 2023
PublisherIEEE Computer Society
Pages2204-2212
Number of pages9
ISBN (Electronic)9798350318944
DOIs
StatePublished - 2023
Event64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023 - Santa Cruz, United States
Duration: 6 Nov 20239 Nov 2023

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023
Country/TerritoryUnited States
CitySanta Cruz
Period6/11/239/11/23

All Science Journal Classification (ASJC) codes

  • General Computer Science

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