Length-Constrained Directed Expander Decomposition and Length-Constrained Vertex-Capacitated Flow Shortcuts

2025-03-29

AI Summary

This paper addresses the length-constrained multicommodity flow problem on directed graphs and undirected graphs with vertex capacities. We introduce the first length-bounded expander decomposition theory for such graphs and establish a corresponding flow-cut equivalence theorem. Methodologically, we extend expander decomposition and flow shortcutting structures—previously limited to undirected edge-capacitated graphs—to directed graphs and vertex-capacitated models; we further devise a constructive algorithm achieving $O(1)$-approximate congestion and $O(1)$-hop routing. Our main contributions are: (1) proving the existence of length-bounded expander decompositions in both directed and undirected vertex-capacitated graphs; (2) constructing the first length-bounded flow shortcutting structure for vertex-capacitated graphs; and (3) establishing extremal theoretical foundations for multicommodity flows under vertex capacities, providing essential tools for designing subsequent approximation algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

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 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.

Problem

• Existence of length-constrained expander decomposition in directed graphs.
• Multi-commodity maxflow-mincut theorems for length-constrained expansion.
• Generalization of flow shortcuts to vertex-capacitated graphs.

Innovation

• Length-constrained expander decomposition in directed graphs
• Multi-commodity maxflow-mincut theorems for length constraints
• Vertex-capacitated flow shortcuts with few edges