We show the existence of length-constrained expander decomposition in directed graphs and undirected vertex-capacitated graphs. Previously, its existence was shown only in undirected edge-capacitated graphs [Haeupler-R\"acke-Ghaffari, STOC 2022; Haeupler-Hershkowitz-Tan, FOCS 2024]. Along the way, we prove the multi-commodity maxflow-mincut theorems for length-constrained expansion in both directed and undirected vertex-capacitated graphs. Based on our decomposition, we build a length-constrained flow shortcut for undirected vertex-capacitated graphs, which roughly speaking is a set of edges and vertices added to the graph so that every multi-commodity flow demand can be routed with approximately the same vertex-congestion and length, but all flow paths only contain few edges. This generalizes the shortcut for undirected edge-capacitated graphs from [Haeupler-Hershkowitz-Li-Roeyskoe-Saranurak, STOC 2024]. Length-constrained expander decomposition and flow shortcuts have been crucial in the recent algorithms in undirected edge-capacitated graphs [Haeupler-Hershkowitz-Li-Roeyskoe-Saranurak, STOC 2024; Haeupler-Long-Saranurak, FOCS 2024]. Our work thus serves as a foundation to generalize these concepts to directed and vertex-capacitated graphs.

翻译：我们证明了在有向图和无向顶点容量图中存在长度约束扩展图分解。此前，其存在性仅在无向边容量图中得到证明[Haeupler-Räcke-Ghaffari, STOC 2022; Haeupler-Hershkowitz-Tan, FOCS 2024]。在此过程中，我们证明了有向图和无向顶点容量图中长度约束扩展性的多商品最大流-最小割定理。基于我们的分解，我们为无向顶点容量图构建了长度约束流捷径，粗略地说，这是添加到图中的一组边和顶点，使得每个多商品流需求能够以近似相同的顶点拥塞和长度进行路由，但所有流路径仅包含少量边。这推广了[Haeupler-Hershkowitz-Li-Roeyskoe-Saranurak, STOC 2024]中针对无向边容量图的捷径。长度约束扩展图分解和流捷径在近期无向边容量图的算法中至关重要[Haeupler-Hershkowitz-Li-Roeyskoe-Saranurak, STOC 2024; Haeupler-Long-Saranurak, FOCS 2024]。因此，我们的工作为将这些概念推广到有向图和顶点容量图奠定了基础。