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.

翻译：给定一个无向图$G=(V,E,w)$，Gomory-Hu树$T$（Gomory与Hu，1961）是定义在顶点集$V$上的一棵树，它精确地保留了$G$中所有顶点对之间的最小割。本文提出了一种简单高效的随机化归约方法，将Gomory-Hu树的构建问题归约为多对数次最大流计算。在无权图上，该归约将问题转化为总实例规模为$\tilde{O}(m)$的图上的最大流计算，且算法仅需$\tilde{O}(m)$的额外时间。这是首个在多项式对数因子内达到紧界的归约方法。该归约可自然推广到加权图，但此时实例规模和运行时间将增至$\tilde{O}(n^2)$。最后，我们展示了如何将归约方法扩展至无权超图，将超图的Gomory-Hu树构建问题归约为超图上的最大流计算。此归约同样是首个在多项式对数因子内达到紧界的结果。