On strong diameter padded decompositions

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Abstract

Given a weighted graph G = (V, E, w), a partition of V is ∆-bounded if the diameter of each cluster is bounded by ∆. A distribution over ∆-bounded partitions is a β-padded decomposition if every ball of radius γ∆ is contained in a single cluster with probability at least eβ·γ. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee. Formerly, it was proven that Kr free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric dG has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong O(r)-padded decompositions for Kr free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim), Õ(ddim))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019
EditorsDimitris Achlioptas, Laszlo A. Vegh
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771252
DOIs
StatePublished - Sep 2019
Externally publishedYes
Event22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019 - Cambridge, United States
Duration: 20 Sep 201922 Sep 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume145
ISSN (Print)1868-8969

Conference

Conference22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019
Country/TerritoryUnited States
CityCambridge
Period20/09/1922/09/19

Bibliographical note

Publisher Copyright:
© Arnold Filtser.

Keywords

  • Distance oracles
  • Doubling dimension
  • Minor free graphs
  • Padded decomposition
  • Spanners
  • Sparse cover
  • Strong diameter
  • Unique games

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