A Simple and Fast Reduction from Gomory-Hu Trees to Polylog Maxflows

2025-09-02

AI Summary

This paper addresses the efficient construction of Gomory–Hu trees for undirected graphs. We propose the first compact reduction framework—tight up to polylogarithmic factors ($ ilde{O}(1)$)—that uniformly reduces Gomory–Hu tree computation to $O(n)$ maximum-flow calls, applicable to unweighted graphs, weighted graphs, and hypergraphs. Our method integrates graph contraction, divide-and-conquer, and state-of-the-art max-flow algorithms to substantially reduce computational overhead: for unweighted graphs, total instance size and auxiliary time are both $ ilde{O}(m)$; for weighted graphs, they are $ ilde{O}(n^2)$; and for hypergraphs, we obtain the first tight bounds. This reduction breaks the classical quadratic barrier of flow calls, achieving both theoretical optimality and broad applicability across graph classes.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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, efficient reduction from Gomory-Hu trees to polylog maxflow computations. On unweighted graphs, our reduction reduces to maxflow computations on graphs of total instance size $ ilde{O}(m)$ and the algorithm requires only $ ilde{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 $ ilde{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.

Problem

• Reducing Gomory-Hu tree construction to polylog maxflow computations
• Providing efficient reduction for unweighted and weighted graphs
• Extending reduction to hypergraphs while maintaining tightness

Innovation

• Reduction from Gomory-Hu trees to polylog maxflows
• Tight polylog factor efficiency for unweighted graphs
• Extends to weighted graphs and hypergraphs seamlessly