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.

翻译：我们针对有向图获得了更快的扩展图分解算法，其性能保证与Saranurak和Wang（SODA 2019）在无向图上的扩展图分解结果相匹配。我们的算法比先前工作更快，并且几乎无损地推广到带容量图。特别地，我们首次实现了在近线性时间内、对容量参数$φ$具有最优依赖性的有向带容量图扩展图分解算法。为实现这一结果，我们首次针对有向带容量图实现了非停止割匹配博弈的算法并进行了分析。现有所有有向扩展图分解算法均需临时添加“伪边”，并在最终清理步骤中将其移除。我们的结果表明，自然的无向图方法同样适用于有向图。难点在于算法分析，该分析具有技术复杂性，且需对原始无向图设定进行显著修改。