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Near-Optimal Algorithm for Directed Expander Decompositions

Aurelio L. Sulser, Maximilian Probst Gutenberg

TL;DR

The paper presents a near-optimal, randomized algorithm for computing and maintaining expander decompositions in directed graphs, achieving Õ(m) initialization and near-linear dynamic update times. It introduces a novel Push-Pull-Relabel framework that extends the classic push-relabel method to directed settings and dynamic edge deletions, enabling efficient refinement of expander components via witness embeddings and sparsity certificates. The approach reduces the directed problem to maintaining out-expanders and integrates bidirectional pruning to produce a full directed expander decomposition with a refined incidence structure and witness framework. The results subsume and improve upon prior subpolynomial-factor bounds, and the techniques have already influenced fast min-cost flow algorithms, with potential to accelerate broader dynamic and combinatorial graph algorithm breakthroughs. Overall, the work contributes a simpler, faster, and more accessible framework for directed expander decompositions with dynamic maintenance capabilities and broad algorithmic applications.

Abstract

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).

Near-Optimal Algorithm for Directed Expander Decompositions

TL;DR

The paper presents a near-optimal, randomized algorithm for computing and maintaining expander decompositions in directed graphs, achieving Õ(m) initialization and near-linear dynamic update times. It introduces a novel Push-Pull-Relabel framework that extends the classic push-relabel method to directed settings and dynamic edge deletions, enabling efficient refinement of expander components via witness embeddings and sparsity certificates. The approach reduces the directed problem to maintaining out-expanders and integrates bidirectional pruning to produce a full directed expander decomposition with a refined incidence structure and witness framework. The results subsume and improve upon prior subpolynomial-factor bounds, and the techniques have already influenced fast min-cost flow algorithms, with potential to accelerate broader dynamic and combinatorial graph algorithm breakthroughs. Overall, the work contributes a simpler, faster, and more accessible framework for directed expander decompositions with dynamic maintenance capabilities and broad algorithmic applications.

Abstract

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).
Paper Structure (29 sections, 8 theorems, 36 equations, 1 figure, 1 algorithm)

This paper contains 29 sections, 8 theorems, 36 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Applications
  4. Our Techniques
  5. High-Level Strategy.
  6. The (Dynamic) Flow Problem in saranurak2019expander.
  7. The (Dynamic) Flow Problem in Directed Graphs.
  8. The Push-Pull-Relabel Framework.
  9. A Sketch of the Runtime of Push-Pull-Relabel.
  10. Roadmap.
  11. Preliminaries
  12. Graphs.
  13. Flows.
  14. Expanders.
  15. Graph Embeddings.
  16. ...and 14 more sections

Key Result

Theorem 1.2

Given a parameter $\phi \leq c/ \log^{12} m$ for a fixed constant $c > 0$, and a directed $m$-edge graph $G$ undergoing a sequence of edge deletions, there is a randomized data structure that constructs and maintains a $(O(\log^{19} m), \Omega(\phi/\log^{12} m), O(1/\log^8 m))$-expander decompositio

Figures (1)

  • Figure 1: Injection of Commodity due to edges in $E_G(A, \bar{A}) \cup F$

Theorems & Definitions (30)

  • Definition 1.1: Directed Expander Decomposition
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Definition 2.2
  • proof
  • proof
  • Definition 2.5: Augmented Expander Decomposition
  • Definition 3.1
  • Lemma 3.2
  • ...and 20 more