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A Simple Deterministic Reduction From Gomory-Hu Tree to Maxflow and Expander Decomposition

Maximilian Probst Gutenberg, Weixuan Yuan

TL;DR

The paper addresses computing Gomory-Hu trees by reducing the problem to two core graph primitives: maxflow and expander decomposition. It delivers a deterministic reduction for unweighted graphs with near-linear total input size $\tilde{O}(m)$ and overhead $\tilde{O}(m)$, and extends to weighted graphs and unweighted hypergraphs; an approximate GH-tree variant is provided with $ (1+\epsilon) $ accuracy and near-linear costs. The approach avoids the intricate SSMC subroutine, favoring a simple, recursive decomposition that composes subinstances via balanced mincuts. The results unify and simplify the practical computation of GH trees, enabling near-linear-time reductions to well-studied primitives and offering a clear pathway to deterministic and randomized implementations.

Abstract

Given an undirected graph $G=(V,E,w)$, a Gomory-Hu tree $T$ (Gomory and Hu, 1961) is a tree on $V$ that preserves all-pairs mincuts of $G$ exactly. We present a simple and efficient randomized reduction from Gomory-Hu trees to polylog maxflow computations. On unweighted graphs, our reduction reduces to maxflow computations on graphs of total instance size $\tilde{O}(m)$ and the algorithm requires only $\tilde{O}(m)$ additional time. Our reduction is the first that is tight up to polylog factors. The reduction also seamlessly extends to weighted graphs, however, instance sizes and runtime increase to $\tilde{O}(n^2)$. Finally, we show how to extend our reduction to reduce Gomory-Hu trees for unweighted hypergraphs to maxflow in hypergraphs. Again, our reduction is the first that is tight up to polylog factors.

A Simple Deterministic Reduction From Gomory-Hu Tree to Maxflow and Expander Decomposition

TL;DR

The paper addresses computing Gomory-Hu trees by reducing the problem to two core graph primitives: maxflow and expander decomposition. It delivers a deterministic reduction for unweighted graphs with near-linear total input size and overhead , and extends to weighted graphs and unweighted hypergraphs; an approximate GH-tree variant is provided with accuracy and near-linear costs. The approach avoids the intricate SSMC subroutine, favoring a simple, recursive decomposition that composes subinstances via balanced mincuts. The results unify and simplify the practical computation of GH trees, enabling near-linear-time reductions to well-studied primitives and offering a clear pathway to deterministic and randomized implementations.

Abstract

Given an undirected graph , a Gomory-Hu tree (Gomory and Hu, 1961) is a tree on that preserves all-pairs mincuts of exactly. We present a simple and efficient randomized reduction from Gomory-Hu trees to polylog maxflow computations. On unweighted graphs, our reduction reduces to maxflow computations on graphs of total instance size and the algorithm requires only additional time. Our reduction is the first that is tight up to polylog factors. The reduction also seamlessly extends to weighted graphs, however, instance sizes and runtime increase to . Finally, we show how to extend our reduction to reduce Gomory-Hu trees for unweighted hypergraphs to maxflow in hypergraphs. Again, our reduction is the first that is tight up to polylog factors.

Paper Structure

This paper contains 17 sections, 10 theorems, 8 equations, 8 algorithms.

Table of Contents

  1. Introduction
  2. Fast Algorithms for Gomory-Hu trees.
  3. Towards Simple and Practical Algorithms.
  4. Maxflow.
  5. Expander Decomposition.
  6. Remark.
  7. Our Contribution
  8. Preliminaries
  9. Isolating Cuts Lemma.
  10. Hit-And-Miss Theorems.
  11. Expander Decomposition for General Demands: A Blackbox Reduction
  12. Gomory-Hu tree
  13. $\tau$-Connectivity and Large Component
  14. Gomory-Hu Tree Step
  15. Exact Gomory-Hu Tree
  16. ...and 2 more sections

Key Result

Theorem 1.2

Given an $m$-edge, undirected, unweighted graph $G=(V,E)$, there is a deterministic algorithm that reduces Gomory-Hu tree to maxflow and expander decomposition calls on instances with total size $\tilde{O}(m)$ edges, with additional time $\tilde{O}(m)$.All maxflow and expander decomposition calls ar

Theorems & Definitions (33)

  • Definition 1.1: Expander Decomposition
  • Theorem 1.2: Deterministic GH tree
  • Theorem 1.3: Approximate GH Tree
  • Lemma 2.1: Isolating Cuts Lemma
  • Theorem 2.2: see Definition 4 and Theorem 3.1 in cs2024deterministic
  • Definition 3.1: General Expander Decomposition
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • Definition 4.1
  • ...and 23 more