Roughly, a metric space has padding parameter $\beta$ if for every $\Delta>0$, there is a stochastic decomposition of the metric points into clusters of diameter at most $\Delta$ such that every ball of radius $\gamma\Delta$ is contained in a single cluster with probability at least $e^{-\gamma\beta}$. The padding parameter is an important characteristic of a metric space with vast algorithmic implications. In this paper we prove that the shortest path metric of every $K_r$-minor-free graph has padding parameter $O(\log r)$, which is also tight. This resolves a long standing open question, and exponentially improves the previous bound. En route to our main result, we construct sparse covers for $K_r$-minor-free graphs with improved parameters, and we prove a general reduction from sparse covers to padded decompositions.

翻译：粗略而言，若度量空间具有填充参数$\beta$，则对任意$\Delta>0$，存在将度量点随机分解为直径不超过$\Delta$的聚类的方式，使得每个半径为$\gamma\Delta$的球以至少$e^{-\gamma\beta}$的概率被包含在单个聚类内。填充参数是度量空间的重要特征，具有广泛的算法意义。本文证明每个$K_r$-无小图的短路径度量具有$O(\log r)$的填充参数，且该界是紧的。这解决了一个长期悬而未决的问题，并将先前界指数级改进。在研究过程中，我们为$K_r$-无小图构造了具有改进参数的稀疏覆盖，并证明了从稀疏覆盖到填充分解的一般性归约。