Min-Cut queries are fundamental: Preprocess an undirected edge-weighted graph, to quickly report a minimum-weight cut that separates a query pair of nodes $s,t$. The best data structure known for this problem simply builds a cut-equivalent tree, discovered 60 years ago by Gomory and Hu, who also showed how to construct it using $n-1$ minimum $st$-cut computations. Using state-of-the-art algorithms for minimum $st$-cut (Lee and Sidford, FOCS 2014) arXiv:1312.6713, one can construct the tree in time $\tilde{O}(mn^{3/2})$, which is also the preprocessing time of the data structure. (Throughout, we focus on polynomially-bounded edge weights, noting that faster algorithms are known for small/unit edge weights.) Our main result shows the following equivalence: Cut-equivalent trees can be constructed in near-linear time if and only if there is a data structure for Min-Cut queries with near-linear preprocessing time and polylogarithmic (amortized) query time, and even if the queries are restricted to a fixed source. That is, equivalent trees are an essentially optimal solution for Min-Cut queries. This equivalence holds even for every minor-closed family of graphs, such as bounded-treewidth graphs, for which a two-decade old data structure (Arikati et al., J.~Algorithms 1998) implies the first near-linear time construction of cut-equivalent trees. Moreover, unlike all previous techniques for constructing cut-equivalent trees, ours is robust to relying on approximation algorithms. In particular, using the almost-linear time algorithm for $(1+\epsilon)$-approximate minimum $st$-cut (Kelner et al., SODA 2014), we can construct a $(1+\epsilon)$-approximate flow-equivalent tree (which is a slightly weaker notion) in time $n^{2+o(1)}$. This leads to the first $(1+\epsilon)$-approximation for All-Pairs Max-Flow that runs in time $n^{2+o(1)}$, and matches the output size almost-optimally.

中文翻译：

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切割等效树对于最小切割查询是最佳的

Min-Cut 查询是基本的：预处理一个无向边加权图，以快速报告一个最小权重切割，它将查询对节点 $s,t$ 分开。解决这个问题的最好的数据结构只是构建了一个割等效树，这是 60 年前由 Gomory 和 Hu 发现的，他们还展示了如何使用 $n-1$ 最小 $st$-cut 计算来构建它。使用最先进的最小 $st$-cut 算法（Lee 和 Sidford，FOCS 2014）arXiv:1312.6713，可以及时构建树 $\tilde{O}(mn^{3/2}) $，也是数据结构的预处理时间。（在整个过程中，我们专注于多项式有界边权重，并注意到更快的算法以小/单位边权重而闻名。）我们的主要结果显示了以下等价性：当且仅当存在用于具有近线性预处理时间和多对数（摊销）查询时间的 Min-Cut 查询的数据结构，并且即使查询仅限于固定来源。也就是说，等效树本质上是 Min-Cut 查询的最佳解决方案。这种等价性甚至适用于每一个小封闭图族，例如有界树宽图，对于这些图，两年前的数据结构（Arikati 等人，J.~Algorithms 1998）暗示了第一个近线性时间构造等价树。此外，与之前所有用于构造切割等效树的技术不同，我们的技术对依赖近似算法具有鲁棒性。特别是，对 $(1+\epsilon)$-approximate minimum $st$-cut (Kelner et al., SODA 2014），我们可以在时间 $n^{2+o(1)}$ 中构建一个 $(1+\epsilon)$-近似流等效树（这是一个稍微弱一点的概念）。这导致在时间 $n^{2+o(1)}$ 中运行的 All-Pairs Max-Flow 的第一个 $(1+\epsilon)$-近似值，并且几乎与输出大小匹配。