Improved Directed Expander Decompositions

TL;DR

This paper introduces faster algorithms for directed expander decompositions that match undirected guarantees, extend to capacitated graphs, and utilize a novel analysis of the cut-matching game.

Contribution

It provides the first near-linear time directed expander decomposition algorithm for capacitated graphs with optimal dependence on conductance.

Findings

Faster algorithms matching undirected guarantees

Extension to capacitated graphs with near-linear time complexity

First implementation and analysis of non-stop cut-matching game for directed graphs

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 $\phi$. 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.