Scattering and sparse partitions, and their applications

A partition P of a weighted graph G is (σ, τ, ∆)-sparse if every cluster has diameter at most ∆, and every ball of radius ∆/σ intersects at most τ clusters. Similarly, P is (σ, τ, ∆)-scattering if instead for balls we require that every shortest path of length at most ∆/σ intersects at most τ clusters. Given a graph G that admits a (σ, τ, ∆)-sparse partition for all ∆ > 0, Jia et al. [STOC05] constructed a solution for the Universal Steiner Tree problem (and also Universal TSP) with stretch O(τσ² log_τ n). Given a graph G that admits a (σ, τ, ∆)-scattering partition for all ∆ > 0, we construct a solution for the Steiner Point Removal problem with stretch O(τ³σ³). 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.