Scattering and Sparse Partitions, and their Applications

Abstract

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(τ σ 2 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(τ 3 σ 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.

Scattering and Sparse Partitions, and Their Applications

“…However they measured radius rather than diameter. 10 More generally, for a parameter t = Ω(1), [28] Formal statements, and proofs of all our partitions are differed to the full version [31]. The main contribution of this paper is the definition of scattering partition and the finding that good scattering partitions imply low distortion solutions for the SPR problem (Theorem 2).…”

“…Specifically there are graph families that admit (O(1), O(1))-strong sparse cover schemes, while there are no constants σ, τ , such that they admit (σ, τ )-strong sparse partitions. Description of our findings on the connection between sparse partitions and sparse covers, and a classification of various graph families are differed to the full version [31].…”

“…Next we survey the partitions for various graph families. Formal statements and proofs are differed to the full version [31].…”

On Light Spanners, Low-treewidth Embeddings and Efficient Traversing in Minor-free Graphs

“…We do not know whether it is possible to obtain the properties of Theorem 2 while embedding into n-vertex graphs. In this context, the Steiner point removal problem studies whether it is possible to remove all Steiner points while preserving both pairwise distance and topological structure [Fil19b,Fil20]. Unfortunately, in general, even if G is a tree, a multiplicative distortion of 8 is necessary [CXKR06].…”

“…Formally max {u,v∈A} d G (u, v). See[Fil19a,Fil20] for further details on sparse covers and related notions.…”