TY - GEN
T1 - Expander Decomposition with Fewer Inter-Cluster Edges Using a Spectral Cut Player
AU - Agassy, Daniel
AU - Dorfman, Dani
AU - Kaplan, Haim
N1 - Publisher Copyright:
© Daniel Agassy, Dani Dorfman, and Haim Kaplan.
PY - 2023/7
Y1 - 2023/7
N2 - 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.
AB - 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.
KW - Cut-Matching Game
KW - Exapander Decomposition
UR - http://www.scopus.com/inward/record.url?scp=85167353580&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ICALP.2023.9
DO - 10.4230/LIPIcs.ICALP.2023.9
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AN - SCOPUS:85167353580
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023
A2 - Etessami, Kousha
A2 - Feige, Uriel
A2 - Puppis, Gabriele
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023
Y2 - 10 July 2023 through 14 July 2023
ER -