Deterministic Almost-Linear-Time Gomory-Hu Trees

2025-07-27

AI Summary

This paper addresses the construction of Gomory–Hu (GH) trees for undirected weighted graphs (G = (V, E, w)), which exactly encode all pairwise minimum cuts. Prior to this work, the best deterministic algorithm ran in (O(nm^{1+o(1)})) time—a significant bottleneck. We present the first deterministic (m^{1+o(1)})-time algorithm, achieving an almost-linear breakthrough. Our method comprises two key innovations: (1) a deterministic reduction from the all-pairs minimum cut problem to the single-source minimum cut problem, incurring only subpolynomial overhead; and (2) the first almost-linear-time deterministic algorithm for single-source minimum cut. These advances combine deterministic graph sparsification, compression, and divide-and-conquer techniques, leveraging recent almost-linear-time maximum flow algorithms. Our result resolves a long-standing open problem in GH tree construction and, as a corollary, yields the first deterministic almost-linear-time algorithm for computing (k)-edge-connected components—fundamental in graph connectivity analysis.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Given an $m$-edge, undirected, weighted graph $G=(V,E,w)$, a Gomory-Hu tree $T$ (Gomory and Hu, 1961) is a tree over the vertex set $V$ such that all-pairs mincuts in $G$ are preserved exactly in $T$. In this article, we give the first almost-optimal $m^{1+o(1)}$-time deterministic algorithm for constructing a Gomory-Hu tree. Prior to our work, the best deterministic algorithm for this problem dated back to the original algorithm of Gomory and Hu that runs in $nm^{1+o(1)}$ time (using current maxflow algorithms). In fact, this is the first almost-linear time deterministic algorithm for even simpler problems, such as finding the $k$-edge-connected components of a graph. Our new result hinges on two separate and novel components that each introduce a distinct set of de-randomization tools of independent interest: - a deterministic reduction from the all-pairs mincuts problem to the single-souce mincuts problem incurring only subpolynomial overhead, and - a deterministic almost-linear time algorithm for the single-source mincuts problem.

Problem

• Construct Gomory-Hu trees in near-linear deterministic time
• Improve upon decades-old deterministic mincut algorithms
• Develop novel deterministic tools for graph connectivity problems

Innovation

• Deterministic almost-linear-time Gomory-Hu tree construction
• Subpolynomial reduction to single-source mincuts
• Novel deterministic single-source mincuts algorithm