A partition $\mathcal{P}$ of a weighted graph $G$ is $(\sigma,\tau,\Delta)$-sparse if every cluster has diameter at most $\Delta$, and every ball of radius $\Delta/\sigma$ intersects at most $\tau$ clusters. Similarly, $\mathcal{P}$ is $(\sigma,\tau,\Delta)$-scattering if instead for balls we require that every shortest path of length at most $\Delta/\sigma$ intersects at most $\tau$ clusters. Given a graph $G$ that admits a $(\sigma,\tau,\Delta)$-sparse partition for all $\Delta>0$, Jia et al. [STOC05] constructed a solution for the Universal Steiner Tree problem (and also Universal TSP) with stretch $O(\tau\sigma^2\log_\tau n)$. Given a graph $G$ that admits a $(\sigma,\tau,\Delta)$-scattering partition for all $\Delta>0$, we construct a solution for the Steiner Point Removal problem with stretch $O(\tau^3\sigma^3)$. We then construct sparse and scattering partitions for various different graph families, receiving many new results for the Universal Steiner Tree and Steiner Point Removal problems.

翻译：加权图$G$的一个划分$\mathcal{P}$称为$(\sigma,\tau,\Delta)$-稀疏的，若每个簇的直径至多为$\Delta$，且半径为$\Delta/\sigma$的每个球与至多$\tau$个簇相交。类似地，$\mathcal{P}$称为$(\sigma,\tau,\Delta)$-散射的，若对于球的要求替换为：长度至多为$\Delta/\sigma$的每条最短路径与至多$\tau$个簇相交。给定一个对所有$\Delta>0$均存在$(\sigma,\tau,\Delta)$-稀疏划分的图$G$，Jia等人[STOC05]构造了通用斯坦纳树问题（以及通用旅行商问题）的一个解，其拉伸比为$O(\tau\sigma^2\log_\tau n)$。给定一个对所有$\Delta>0$均存在$(\sigma,\tau,\Delta)$-散射划分的图$G$，我们构造了斯坦纳点移除问题的一个解，其拉伸比为$O(\tau^3\sigma^3)$。随后，我们为多种不同的图族构造了稀疏和散射划分，从而为通用斯坦纳树与斯坦纳点移除问题获得了多项新成果。