In this work, we present the first algorithm to compute expander decompositions in an m-edge directed graph with near-optimal time Õ(m)$^1$. 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 [2, 18] 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 [14] which was later dynamized for computing expander decompositions in undirected graphs [16, 31]. We then show that the flow problems formulated in recent work [18] 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 [33]. 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 [11, 12, 1].

ISSN:1868-8969