Scattering and Sparse Partitions, and their Applications

TL;DR

A solution for the Steiner Point Removal problem with stretch $O(\tau^3\sigma^3)$ is constructed and sparse and scattering partitions for various different graph families are constructed.

Abstract

A partition $\mathcal{P}$ of a weighted graph $G$ is $(σ,τ,Δ)$-sparse if every cluster has diameter at most $Δ$, and every ball of radius $Δ/σ$ intersects at most $τ$ clusters. Similarly, $\mathcal{P}$ is $(σ,τ,Δ)$-scattering if instead for balls we require that every shortest path of length at most $Δ/σ$ intersects at most $τ$ clusters. Given a graph $G$ that admits a $(σ,τ,Δ)$-sparse partition for all $Δ>0$, Jia et al. [STOC05] constructed a solution for the Universal Steiner Tree problem (and also Universal TSP) with stretch $O(τσ^2\log_τn)$. Given a graph $G$ that admits a $(σ,τ,Δ)$-scattering partition for all $Δ>0$, we construct a solution for the Steiner Point Removal problem with stretch $O(τ^3σ^3)$. 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.

Figures (6)

• Figure 1: Classification of various graph families according to the possibility of construction different partitions. Graphs with bounded doubling dimension or SPD$^{}$ (pathwidth) admit strong sparse partitions with parameters depending only on the dimension/SPDdepth. Trees, Chordal and Cactus graphs admit both $(O(1),O(1))$-weak sparse and scattering partitions, while similar strong partitions are impossible. $\mathbb{R}^d$ with norm $2$ admit $(1,2d)$ scattering partition while weak sparse partition with constant padding will have an exponential number of intersections. Planar graphs admit $(O(1),O(1))$-weak sparse partitions, while it is an open question whether similar scattering partitions exist. Finally, while sparse partitions for general graphs are well understood, we lack a lower bound for scattering partitions.
• Figure 2: The left side of the figure contains a weighted graph $G=(V,E)$, with weights specified in red, and four terminals $\{t_1,t_2,t_3,t_4\}$. The dashed black curves represent a terminal partition of the vertex set $V$ into the subsets $V_1,V_2,V_3,V_4$. The right side of the figure represents the minor $M$ induced by the terminal partition. The distortion is realized between $t_1$ and $t_3$, and is $\frac{d_M(t_1,t_3)}{d_G(t_1,t_3)}=\frac{12}{4}=3$.
• Figure 3: The Venn diagram demonstrates the containment relations between the set of graphs admitting weak/strong sparse covers/partitions. Graphs with constant doubling dimension or SPDdepth (or pathwidth) admit strong sparse partitions scheme with constant parameters (\ref{['cor:coversFromPartitions']}). All graph families excluding a fixed minor admit strong sparse covers with constant parameters AGMW10, while no such strong sparse partitions exist (\ref{['thm:treeLBStrong']}). The family of general graphs do not admit weak sparse partitions with constant parameters (\ref{['thm:GeneralWeakLB']}). We currently lack an example of a graph family that admit weak sparse covers but do not admit strong sparse covers. Finding such an example, or alternatively proving that weak sparse covers imply the existence of strong sparse covers with similar parameters remains an open question.
• Figure 4: The figure illustrates two partitions of the unweighted full binary tree of depth $7$. On the left side is the $(2,3,6)$-scattering partition from \ref{['thm:tressScat']} for parameter $\Delta=6$, while on the right side is the $(4,3,8)$-weak sparse partition from \ref{['thm:treeWeak']} for parameter $\Delta=8$.
• Figure :

Theorems & Definitions (72)

• Definition 1: Scattering Partition
• Theorem 1: Scattering Partitions imply SPR
• Definition 2: Strong/Weak Sparse Partition
• Theorem 2: Sparse Partitions imply UST, JLNRS05
• Definition 3: Strong/Weak Sparse cover
• Lemma 3: Packing Property
• Claim 1
• proof
• Claim 2
• proof