In this work, we present the first algorithm to compute expander decompositions in an $m$-edge directed graph with near-optimal time $\tilde{O}(m)$. 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. At the same time, our algorithm is much simpler and more accessible than previous work. In order to obtain our new algorithm, we present a new push-pull-relabel flow framework that generalizes the classic push-relabel flow algorithm of Goldberg-Tarjan (JACM 1988), which was later dynamized for computing expander decompositions in undirected graphs by Henzinger-Rao-Wang (SIAM J. Comput. 2020), Saranurak-Wang (SODA 2019). We then show that the flow problems formulated in recent work of Hua-Kyng-Probst Gutenberg-Wu (SODA 2023) to decompose directed graphs can be solved much more efficiently in the push-pull-relabel flow framework.

翻译：本文提出了首个能够在包含$m$条边的有向图上以近乎最优时间$\tilde{O}(m)$计算扩展子图分解的算法。此外，我们的算法能够在动态图中维持这种分解，并同样获得近乎最优的更新时间。这一结果改进了Bernstein-Probst Gutenberg-Saranurak（FOCS 2020）以及Hua-Kyng-Probst Gutenberg-Wu（SODA 2023）等先前算法——这些算法仅在子多项式因子内达到最优。同时，我们的算法比先前工作更简单且更易理解。为了获得这一新算法，我们提出了一种新的推-拉-重标记流框架，该框架推广了Goldberg-Tarjan（JACM 1988）的经典推-重标记流算法——后者后来由Henzinger-Rao-Wang（SIAM J. Comput. 2020）和Saranurak-Wang（SODA 2019）动态化用于计算无向图的扩展子图分解。我们随后证明，最近Hua-Kyng-Probst Gutenberg-Wu（SODA 2023）提出的用于分解有向图的流问题，可以在推-拉-重标记流框架下更高效地求解。