A \emph{tree cut-sparsifier} $T$ of quality $α$ of a graph $G$ is a single tree that preserves the capacities of all cuts in the graph up to a factor of $α$. A \emph{tree flow-sparsifier} $T$ of quality $α$ guarantees that every demand that can be routed in $T$ can also be routed in $G$ with congestion at most $α$. We present a near-linear time algorithm that, for any undirected capacitated graph $G=(V,E,c)$, constructs a tree cut-sparsifier $T$ of quality $O(\log^{2} n \log\log n)$, where $n=|V|$. This nearly matches the quality of the best known polynomial construction of a tree cut-sparsifier, of quality $O(\log^{1.5} n \log\log n)$ [Räcke and Shah, ESA~2014]. By the flow-cut gap, our result yields a tree flow-sparsifier (and congestion-approximator) of quality $O(\log^{3} n \log\log n)$. This improves on the celebrated result of [Räcke, Shah, and Täubig, SODA~2014] (RST) that gave a near-linear time construction of a tree flow-sparsifier of quality $O(\log^{4} n)$. Our algorithm builds on a recent \emph{expander decomposition} algorithm by [Agassy, Dorfman, and Kaplan, ICALP~2023], which we use as a black box to obtain a clean and modular foundation for tree cut-sparsifiers. This yields an improved and simplified version of the RST construction for cut-sparsifiers with quality $O(\log^{3} n)$. We then introduce a near-linear time \emph{refinement phase} that controls the load accumulated on boundary edges of the sub-clusters across the levels of the tree. Combining the improved framework with this refinement phase leads to our final $O(\log^{2} n \log\log n)$ tree cut-sparsifier.

翻译：图$G$的$\alpha$质量\emph{树割稀疏化器}$T$是一棵能够将图中所有割的容量保持至多$\alpha$倍的单一树。$\alpha$质量\emph{树流稀疏化器}$T$保证任何在$T$中可路由的需求，在$G$中至多以$\alpha$的拥塞进行路由。本文提出一种近线性时间算法，对于任意无向赋权图$G=(V,E,c)$，构造出质量$O(\log^{2} n \log\log n)$的树割稀疏化器$T$，其中$n=|V|$。该结果几乎匹配了已知最佳多项式时间构造的树割稀疏化器质量$O(\log^{1.5} n \log\log n)$[Räcke and Shah, ESA~2014]。通过流割间隙，我们的结果可得到一个质量$O(\log^{3} n \log\log n)$的树流稀疏化器（及拥塞近似器）。这改进了[Räcke, Shah, and Täubig, SODA~2014]（RST）的经典工作，该工作给出了质量$O(\log^{4} n)$的树流稀疏化器的近线性时间构造。我们的算法基于[Agassy, Dorfman, and Kaplan, ICALP~2023]近期提出的\emph{扩展图分解}算法，将其作为黑盒使用，为树割稀疏化器提供了一个清晰且模块化的基础框架。由此得到了RST构造的改进简化版本，用于构建质量$O(\log^{3} n)$的割稀疏化器。我们随后引入了一个近线性时间的\emph{精化阶段}，用于控制在树的各个层级上子簇边界边积累的负载。将改进的框架与此精化阶段相结合，最终得到了我们质量$O(\log^{2} n \log\log n)$的树割稀疏化器。