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

TL;DR

This work extends length-constrained expander decomposition (LC-ED) to directed graphs and undirected vertex-capacitated graphs, generalizing prior LC-ED results for undirected edge-capacitated graphs. It establishes the routing equivalence between LC-expansion and LC-flow frameworks via a directed and vertex-capacitated setting, and provides existential LC-ED results with explicit slack bounds using a potential/expansion framework. A central technical contribution is a vertex-capacitated LC-flow shortcut: for any undirected vertex-capacitated graph and ε>0, there exists a $2^{O(1/ obreak ext{varepsilon})}$-step flow shortcut of size $|E'|=O(n^{1+O( obreak ext{varepsilon})} ext{polylog}(n))$ with length slack $O(1/ obreak ext{varepsilon}^{3})$ and congestion slack $n^{O( obreak ext{varepsilon})}$, generalizing earlier edge-capacitated results. The approach combines a top-down analysis that avoids boundary-linkedness with exponential-demand gadgets to control cut sizes, laying groundwork toward near-linear-time constructions and broader applicability to minimum-cost multi-commodity flow in vertex-capacitated and directed graphs.

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-Räcke-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.

Figures (5)

• Figure 1: An LC-flow shortcut of a low-diameter LC-expander.
• Figure 2: A toy example of forward mapping given we have 2 levels in total. Crossings represent cut vertices in $C_{1}$ along the witness path $P_{u,v}$.
• Figure 3: (Left) A vertex-capacitated graph. (Right) The corresponding directed edge-capacitated graph.
• Figure 4: Suppose there is a path $\{u, v, w\}$ in the vertex-capacitated graph $G_{\mathrm{vc}}$(Left), then we can follow it and create a path $\{u_{\mathrm{mid}}, u_{\mathrm{out}}, v_{\mathrm{in}}, v_{\mathrm{out}}, w_{\mathrm{in}}, w_{\mathrm{mid}}\}$ in $G_{\mathrm{ec}}$(Right). We note that both paths have the same length.
• Figure 5: The concurrent flow LP relaxation and its dual.

Theorems & Definitions (72)

• Definition 1.1: Length-Constrained Flow Shortcut
• Theorem 2.1: peleg2000distributed
• Definition 3.1: Length-Constrained Cut
• Definition 3.2: $h$-Length Separated Demand
• Definition 3.3: $h$-Length Sparsity of a Cut $C$ for Demand $D$
• Definition 3.4: $(h,s)$-Length Sparsity of a Cut w.r.t. a Node-Weighting
• Definition 3.5: $(h,s)$-Length Conductance of a Node-Weighting
• Definition 3.6: $(h,s)$-Length $\phi$-Expanding Node-Weightings
• Definition 3.7: Length-Constrained Directed Expander Decomposition
• Theorem 3.8: Routing Characterization of Length-Constrained Directed Expanders