Scattering and Sparse Partitions, and Their Applications

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