Given a metric space $(X,d_X)$, a $(\beta,s,\Delta)$-sparse cover is a collection of clusters $\mathcal{C}\subseteq P(X)$ with diameter at most $\Delta$, such that for every point $x\in X$, the ball $B_X(x,\frac\Delta\beta)$ is fully contained in some cluster $C\in \mathcal{C}$, and $x$ belongs to at most $s$ clusters in $\mathcal{C}$. Our main contribution is to show that the shortest path metric of every $K_r$-minor free graphs admits $(O(r),O(r^2),\Delta)$-sparse cover, and for every $\epsilon>0$, $(4+\epsilon,O(\frac1\epsilon)^r,\Delta)$-sparse cover (for arbitrary $\Delta>0$). We then use this sparse cover to show that every $K_r$-minor free graph embeds into $\ell_\infty^{\tilde{O}(\frac1\epsilon)^{r+1}\cdot\log n}$ with distortion $3+\eps$ (resp. into $\ell_\infty^{\tilde{O}(r^2)\cdot\log n}$ with distortion $O(r)$). Further, we provide applications of these sparse covers into padded decompositions, sparse partitions, universal TSP / Steiner tree, oblivious buy at bulk, name independent routing, and path reporting distance oracles.

翻译：给定度量空间$(X,d_X)$，一个$(\beta,s,\Delta)$-稀疏覆盖是由直径不超过$\Delta$的簇构成的集合$\mathcal{C}\subseteq P(X)$，使得对每个点$x\in X$，球$B_X(x,\frac\Delta\beta)$完全包含于某个簇$C\in \mathcal{C}$，且$x$至多属于$\mathcal{C}$中的$s$个簇。我们的主要贡献是证明：每个$K_r$-极小自由图的最短路径度量允许$(O(r),O(r^2),\Delta)$-稀疏覆盖，并且对任意$\epsilon>0$，允许$(4+\epsilon,O(\frac1\epsilon)^r,\Delta)$-稀疏覆盖（对任意$\Delta>0$）。我们进而利用该稀疏覆盖证明：每个$K_r$-极小自由图可嵌入$\ell_\infty^{\tilde{O}(\frac1\epsilon)^{r+1}\cdot\log n}$，畸变为$3+\epsilon$（分别嵌入$\ell_\infty^{\tilde{O}(r^2)\cdot\log n}$，畸变为$O(r)$）。此外，我们提供了这些稀疏覆盖在填充分解、稀疏分割、通用旅行商/斯坦纳树、松散批量购买、无名称路由以及路径报告距离预言机中的应用。