A $(\beta,\delta,\Delta)$-padded decomposition of an edge-weighted graph $G = (V,E,w)$ is a stochastic decomposition into clusters of diameter at most $\Delta$ such that for every vertex $v\in V$, the probability that $\rm{ball}_G(v,\gamma\Delta)$ is entirely contained in the cluster containing $v$ is at least $e^{-\beta\gamma}$ for every $\gamma \in [0,\delta]$. Padded decompositions have been studied for decades and have found numerous applications, including metric embedding, multicommodity flow-cut gap, multicut, and zero extension problems, to name a few. In these applications, parameter $\beta$, called the padding parameter, is the most important parameter since it decides either the distortion or the approximation ratios. For general graphs with $n$ vertices, $\beta = \Theta(\log n)$. Klein, Plotkin, and Rao showed that $K_r$-minor-free graphs have padding parameter $\beta = O(r^3)$, which is a significant improvement over general graphs when $r$ is a constant. A long-standing conjecture is to construct a padded decomposition for $K_r$-minor-free graphs with padding parameter $\beta = O(\log r)$. Despite decades of research, the best-known result is $\beta = O(r)$, even for graphs with treewidth at most $r$. In this work, we make significant progress toward the aforementioned conjecture by showing that graphs with treewidth $\rm{tw}$ admit a padded decomposition with padding parameter $O(\log \rm{tw})$, which is tight. As corollaries, we obtain an exponential improvement in dependency on treewidth in a host of algorithmic applications: $O(\sqrt{ \log n \cdot \log(\rm{tw})})$ flow-cut gap, max flow-min multicut ratio of $O(\log(\rm{tw}))$, an $O(\log(\rm{tw}))$ approximation for the 0-extension problem, an $\ell^{O(\log n)}_\infty$ embedding with distortion $O(\log \rm{tw})$, and an $O(\log \rm{tw})$ bound for integrality gap for the uniform sparsest cut.

翻译：对于边赋权图 $G = (V,E,w)$，一个 $(\beta,\delta,\Delta)$-填充分解是一种随机分解，它将图分解为直径至多为 $\Delta$ 的簇，并且对于每个顶点 $v\in V$，以及每个 $\gamma \in [0,\delta]$，满足 $\rm{ball}_G(v,\gamma\Delta)$ 完全包含在 $v$ 所属簇内的概率至少为 $e^{-\beta\gamma}$。填充分解已被研究数十年，并在诸多领域得到应用，例如度量嵌入、多商品流割间隙、多割以及零延拓问题等。在这些应用中，参数 $\beta$（称为填充参数）是最重要的参数，因为它决定了失真度或近似比率。对于具有 $n$ 个顶点的一般图，$\beta = \Theta(\log n)$。Klein、Plotkin 和 Rao 证明了 $K_r$-minor-free 图具有填充参数 $\beta = O(r^3)$，当 $r$ 为常数时，这相对于一般图是一个显著的改进。一个长期存在的猜想是为 $K_r$-minor-free 图构造填充参数 $\beta = O(\log r)$ 的填充分解。尽管经过数十年的研究，即使对于树宽至多为 $r$ 的图，最著名的结果仍是 $\beta = O(r)$。在本工作中，我们朝着上述猜想迈出了重要一步，证明了树宽为 $\rm{tw}$ 的图允许填充参数为 $O(\log \rm{tw})$ 的填充分解，并且该结果是紧的。作为推论，我们在众多算法应用中获得了对树宽依赖性的指数级改进：$O(\sqrt{ \log n \cdot \log(\rm{tw})})$ 的流割间隙，$O(\log(\rm{tw}))$ 的最大流-最小多割比率，0-延拓问题的 $O(\log(\rm{tw}))$ 近似，失真度为 $O(\log \rm{tw})$ 的 $\ell^{O(\log n)}_\infty$ 嵌入，以及均匀稀疏割整数性间隙的 $O(\log \rm{tw})$ 上界。