2026-08-02T01:25:11Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:23509 2025-10-02T12:57:17Z ddc:004 open_access

Near-Optimal Algorithm for Directed Expander Decompositions Sulser, Aurelio L. Gutenberg, Maximilian Probst Directed Expander Decomposition Push-Pull-Relabel Algorithm In this work, we present the first algorithm to compute expander decompositions in an m-edge directed graph with near-optimal time Õ(m). Further, our algorithm can maintain such a decomposition in a dynamic graph and again obtains near-optimal update times. Our result improves over previous algorithms [Bernstein et al., 2020; Hua et al., 2023] that only obtained algorithms optimal up to subpolynomial factors. In order to obtain our new algorithm, we present a new push-pull-relabel flow framework that generalizes the classic push-relabel flow algorithm [Goldberg and Tarjan, 1988] which was later dynamized for computing expander decompositions in undirected graphs [Henzinger et al., 2020; Saranurak and Wang, 2019]. We then show that the flow problems formulated in recent work [Hua et al., 2023] to decompose directed graphs can be solved much more efficiently in the push-pull-relabel flow framework. Recently, our algorithm has already been employed to obtain the currently fastest algorithm to compute min-cost flows [Van Den Brand et al., 2024]. We further believe that our algorithm can be used to speed-up and simplify recent breakthroughs in combinatorial graph algorithms towards fast maximum flow algorithms [Chuzhoy and Khanna, 2024; Chuzhoy and Khanna, 2024; Bernstein et al., 2024]. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Aurelio L. Sulser and Maximilian Probst Gutenberg 2025 Is Part Of LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2025.132 urn:nbn:de:0030-drops-235096 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.132 eng https://creativecommons.org/licenses/by/4.0/legalcode