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Scattering and Sparse Partitions, and Their Applications

Author Arnold Filtser

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Arnold Filtser
  • Columbia University, United States, New York, NY, USA

Funding

The research was supported by the Simons Foundation.

Acknowledgements

The author would like to thank Alexandr Andoni, Robert Krauthgamer, Jason Li, Ofer Neiman, Anastasios Sidiropoulos, and Ohad Trabelsi for helpful discussions.

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Arnold Filtser. Scattering and Sparse Partitions, and Their Applications. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 47:1-47:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.47

Abstract

A partition 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, 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(τσ²log_τ n). 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(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.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Scattering partitions
  • sparse partitions
  • sparse covers
  • Steiner point removal
  • Universal Steiner tree
  • Universal TSP

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  • A full version of the paper is available at https://arxiv.org/abs/2001.04447. The reader is encouraged to read the full version of the paper [Arnold Filtser, 2020].

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