Near-Optimal Algorithm for Directed Expander Decompositions

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 of Bernstein-Probst Gutenberg-Saranurak (FOCS 2020), Hua-Kyng-Probst Gutenberg-Wu (SODA 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-Tarjan (JACM 1988) which was later dynamized for computing expander decompositions in undirected graphs Henzinger-Rao-Wang (SIAM J. Comput. 2020), Saranurak-Wang (SODA 2019). We then show that the flow problems formulated in recent work Hua-Kyng-Probst Gutenberg-Wu (SODA 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-Chen-Kyng-Liu-Probst Gutenberg-Sachdeva (FOCS 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-Khanna (SODA 2024), Chuzhoy-Khanna (STOC 2024), Bernstein-Blikstad-Saranurak-Tu (FOCS 2024).