Given a weighted graph $G=(V,E,w)$, a partition of $V$ is $\Delta$-bounded if the diameter of each cluster is bounded by $\Delta$. A distribution over $\Delta$-bounded partitions is a $\beta$-padded decomposition if every ball of radius $\gamma\Delta$ is contained in a single cluster with probability at least $e^{-\beta\cdot\gamma}$. The weak diameter of a cluster $C$ is measured w.r.t. distances in $G$, while the strong diameter is measured w.r.t. distances in the induced graph $G[C]$. The decomposition is weak/strong according to the diameter guarantee. Formerly, it was proven that $K_r$ minor free graphs admit weak decompositions with padding parameter $O(r)$, while for strong decompositions only $O(r^2)$ padding parameter was known. Furthermore, for the case of a graph $G$, for which the induced shortest path metric $d_G$ has doubling dimension $d$, a weak $O(d)$-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong $O(r)$-padded decompositions for $K_r$ minor free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension $d$ we construct a strong $O(d)$-padded decomposition, which is also tight. We use this decomposition to construct strong $\left(O(d),\tilde{O}(d)\right)$ sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.

翻译：给定加权图 $G=(V,E,w)$，如果每个簇的直径以 $\Delta$ 为界，则 $V$ 的划分称为 $\Delta$-有界划分。若每个半径为 $\gamma\Delta$ 的球被包含在单个簇中的概率至少为 $e^{-\beta\cdot\gamma}$，则 $\Delta$-有界划分上的分布称为 $\beta$-填充分解。簇 $C$ 的弱直径是关于 $G$ 中距离度量的，而强直径是关于诱导子图 $G[C]$ 中距离度量的。根据直径保证，该分解分为弱分解和强分解。此前已证明，不含 $K_r$ 子式的图允许填充参数为 $O(r)$ 的弱分解，而对于强分解，已知的填充参数仅为 $O(r^2)$。此外，对于诱导最短路径度量 $d_G$ 具有倍增维数 $d$ 的图 $G$，可以构造弱 $O(d)$-填充分解，且该结果已知是紧的。而关于强直径的情形，尚未有类似结论。我们为不含 $K_r$ 子式的图构造了强 $O(r)$-填充分解，与弱分解的最新成果相匹配。类似地，对于具有倍增维数 $d$ 的图，我们构造了强 $O(d)$-填充分解，该结果也是紧的。利用这一分解，我们为此类图构造了强 $\left(O(d),\tilde{O}(d)\right)$ 稀疏覆盖方案。我们的新分解和覆盖方案对近似唯一游戏、轻量和稀疏支撑的构造以及路径报告距离预言机具有重要意义。