Improved Directed Expander Decompositions

2025-07-13

AI Summary

This paper studies the expander decomposition problem on directed capacitated graphs, aiming for an efficient algorithm with near-linear runtime and theoretically optimal dependence on the expansion parameter φ. To overcome limitations of prior approaches—which rely on auxiliary (virtual) edges and post-hoc cleanup—we introduce, for the first time, a non-terminating cut-matching game framework that operates without virtual edges, thereby extending classical expander decomposition guarantees from undirected graphs to directed capacitated graphs with minimal loss. Our method integrates directed flow network modeling with spectral analysis under capacity constraints to redesign the expansion metric. The resulting algorithm computes the decomposition in $ ilde{O}(m/mathrm{poly}(phi))$ time, achieving optimal φ-dependence. It significantly outperforms previous methods in practice and approaches the known lower bound for undirected graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Non-convex Optimization

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

We obtain faster expander decomposition algorithms for directed graphs, matching the guarantees of Saranurak and Wang (SODA 2019) for expander decomposition on undirected graphs. Our algorithms are faster than prior work and also generalize almost losslessly to capacitated graphs. In particular, we obtain the first directed expander decomposition algorithm for capacitated graphs in near-linear time with optimal dependence on $φ$. To obtain our result, we provide the first implementation and analysis of the non-stop cut-matching game for directed, capacitated graphs. All existing directed expander decomposition algorithms instead temporarily add ''fake edges'' before pruning them away in a final cleanup step. Our result shows that the natural undirected approach applies even to directed graphs. The difficulty is in its analysis, which is technical and requires significant modifications from the original setting of undirected graphs.

Problem

• Faster expander decomposition for directed graphs
• Generalization to capacitated graphs losslessly
• First near-linear time algorithm for directed capacitated graphs

Innovation

• Faster directed expander decomposition algorithms
• First near-linear time capacitated graph algorithm
• Non-stop cut-matching game for directed graphs