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

TL;DR

This paper introduces a simple, efficient randomized reduction from Gomory-Hu trees to maxflow computations, achieving near-optimal size and runtime for unweighted graphs and extending to weighted graphs and hypergraphs.

Contribution

It presents the first tight reduction from Gomory-Hu trees to maxflow, applicable to unweighted, weighted, and hypergraph cases, with improved efficiency.

Findings

Reduces Gomory-Hu trees to maxflow with near-linear size and time for unweighted graphs.

Extension of reduction to weighted graphs with increased size and runtime.

First tight reduction for hypergraph Gomory-Hu trees to hypergraph maxflow.

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.