Expander Decomposition with Fewer Inter-Cluster Edges Using a Spectral Cut Player

Daniel Agassy, Dani Dorfman, Haim Kaplan

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

A (ϕ, ϵ)-expander decomposition of a graph G (with n vertices and m edges) is a partition of V into clusters V1, . . ., Vk with conductance Φ(G[Vi]) ≥ ϕ, such that there are at most ϵm inter-cluster edges. Such a decomposition plays a crucial role in many graph algorithms. We give a randomized Õ(m/ϕ) time algorithm for computing a (ϕ, ϕlog2 n)-expander decomposition. This improves upon the (ϕ, ϕlog3 n)-expander decomposition also obtained in Õ(m/ϕ) time by [Saranurak and Wang, SODA 2019] (SW) and brings the number of inter-cluster edges within logarithmic factor of optimal. One crucial component of SW’s algorithm is a non-stop version of the cut-matching game of [Khandekar, Rao, Vazirani, JACM 2009] (KRV): The cut player does not stop when it gets from the matching player an unbalanced sparse cut, but continues to play on a trimmed part of the large side. The crux of our improvement is the design of a non-stop version of the cleverer cut player of [Orecchia, Schulman, Vazirani, Vishnoi, STOC 2008] (OSVV). The cut player of OSSV uses a more sophisticated random walk, a subtle potential function, and spectral arguments. Designing and analysing a non-stop version of this game was an explicit open question asked by SW.

Original languageEnglish
Title of host publication50th International Colloquium on Automata, Languages, and Programming, ICALP 2023
EditorsKousha Etessami, Uriel Feige, Gabriele Puppis
ISBN (Electronic)9783959772785
DOIs
StatePublished - Jul 2023
Event50th International Colloquium on Automata, Languages, and Programming, ICALP 2023 - Paderborn, Germany
Duration: 10 Jul 202314 Jul 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume261

Conference

Conference50th International Colloquium on Automata, Languages, and Programming, ICALP 2023
Country/TerritoryGermany
CityPaderborn
Period10/07/2314/07/23

Keywords

  • Cut-Matching Game
  • Exapander Decomposition

All Science Journal Classification (ASJC) codes

  • Software

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