Near-Optimal Algorithm for Directed Expander Decompositions

2024-03-07

AI Summary

This work resolves a decades-old open problem by achieving near-optimal $ ilde{O}(m)$ time complexity for computing and dynamically maintaining expander decompositions in directed graphs. We introduce a novel *push-pull-relabel* flow framework that overcomes fundamental limitations of the classical push-relabel method in directed settings, enabling generalized flow modeling and efficient computation. Theoretically, our approach attains near-linear time for both static construction and dynamic updates—strictly improving upon all prior algorithms with subpolynomial overhead factors. Practically, it yields the fastest known algorithm for minimum-cost flow and provides a new foundational tool for core combinatorial optimization problems, including maximum flow.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this work, we present the first algorithm to compute expander decompositions in an m-edge directed graph with near-optimal time ~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. 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).

Problem

• Develops near-optimal expander decomposition algorithm
• Improves dynamic graph decomposition efficiency
• Enhances speed for combinatorial graph algorithms

Innovation

• Near-optimal directed expander decomposition
• Dynamic graph decomposition maintenance
• Push-pull-relabel flow framework enhancement