Scattering and Sparse Partitions, and their Applications

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 \(\Delta \gt 0\), 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 \(\Delta \gt 0\), 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.