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

2025-10-31

AI Summary

This paper addresses the efficient construction of Gomory–Hu trees. We propose the first randomized reduction framework that is optimal up to polylogarithmic factors, cleanly reducing Gomory–Hu tree construction to a polylogarithmic number of maximum flow computations. For unweighted graphs, the total size of all reduced instances and the additional preprocessing time are both $ ilde{O}(m)$; for weighted graphs and unweighted hypergraphs, both bounds improve to $ ilde{O}(n^2)$—matching the current state-of-the-art. Our approach integrates modern maximum flow algorithms with extended subtree decomposition techniques, achieving simplicity, generality, and scalability. Notably, this is the first framework to yield tight reductions for the all-pairs minimum cut problem across three fundamental graph classes: unweighted graphs, weighted graphs, and hypergraphs. The result significantly simplifies algorithm design and improves computational efficiency.

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 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 $ 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 maxflow computations
• Achieving tight polylog factors in unweighted graphs
• Extending reduction to hypergraphs while maintaining efficiency

Innovation

• Reduces Gomory-Hu tree to polylog maxflow computations
• Uses randomized reduction for unweighted and weighted graphs
• Extends reduction to hypergraphs maintaining tight polylog factors