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 $3+\epsilon$ (resp. into $\ell_\infty^{\tilde{O}(r^2)\cdot\log n}$ with distortion $O(r)$). Further, among other applications, this sparse cover immediately implies an algorithm for the oblivious buy-at-bulk problem in fixed minor free graphs with the tight approximation factor $O(\log n)$ (previously nothing beyond general graphs was known).

翻译：给定度量空间$(X,d_X)$，$(\beta,s,\Delta)$-稀疏覆盖是指一族直径至多为$\Delta$的簇$\mathcal{C}\subseteq P(X)$，使得对于任意点$x\in X$，球$B_X(x,\frac\Delta\beta)$完全包含于某个簇$C\in \mathcal{C}$中，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$K_r$-无小图的**最短路径度量**均存在$(O(r),O(r^2),\Delta)$-稀疏覆盖，且对于任意$\epsilon>0$，存在$(4+\epsilon,O(\frac1\epsilon)^r,\Delta)$-稀疏覆盖（其中$\Delta>0$可任意选取）。随后，我们利用该稀疏覆盖证明了：每个$K_r$-无小图可嵌入到$\ell_\infty^{\tilde{O}(\frac1\epsilon)^{r+1}\cdot\log n}$中且失真度为$3+\epsilon$（相应地，可嵌入到$\ell_\infty^{\tilde{O}(r^2)\cdot\log n}$中且失真度为$O(r)$）。此外，该稀疏覆盖在其他应用中直接导出了**固定无小图**中** oblivious buy-at-bulk 问题**的算法，其近似比达到紧界$O(\log n)$（此前仅知一般图上的结果，而无小图类未有突破）。