New Structures and Algorithms for Length-Constrained Expander Decompositions

TL;DR

This work develops a near-linear-time framework for length-constrained expander decompositions, generalizing classic expander methods to graphs with arbitrary lengths and capacities. The core ideas include a union-sparsity theorem for length-constrained sparse cuts and a novel sparse-flow/blaming-flow approach that enables efficient parallel spirals to construct witnessed length-constrained expanders. By embedding expanders via neighborhood-router demands into local neighborhoods, the authors connect length-constrained expanders to classic expander powers, enabling robust routing guarantees and resilience to edge deletions. The framework yields practical implications for near-linear-time multi-commodity flows, distance oracles, and length-constrained oblivious routing on general graphs, including those with nonuniform lengths and capacities.

Abstract

Expander decompositions form the basis of one of the most flexible paradigms for close-to-linear-time graph algorithms. Length-constrained expander decompositions generalize this paradigm to better work for problems with lengths, distances and costs. Roughly, an $(h,s)$-length $φ$-expander decomposition is a small collection of length increases to a graph so that nodes within distance $h$ can route flow over paths of length $hs$ with congestion at most $1/φ$. In this work, we give a close-to-linear time algorithm for computing length-constrained expander decompositions in graphs with general lengths and capacities. Notably, and unlike previous works, our algorithm allows for one to trade off off between the size of the decomposition and the length of routing paths: for any $ε> 0$ not too small, our algorithm computes in close-to-linear time an $(h,s)$-length $φ$-expander decomposition of size $m \cdot φ\cdot n^ε$ where $s = \exp(\text{poly}(1/ε))$. The key foundations of our algorithm are: (1) a simple yet powerful structural theorem which states that the union of a sequence of sparse length-constrained cuts is itself sparse and (2) new algorithms for efficiently computing sparse length-constrained flows.

Figures (4)

• Figure 1: Our neighborhood router demand for one cluster $S_{i,j}$. \ref{['sfig:NRD1']} gives the cluster $S_{i,j}$ with node $v$ labeled with $A(v)$. \ref{['sfig:NRD2']} gives the graph $H_{i,j}^k$ we want to embed into $S_{i,j}$. \ref{['sfig:NRD3']} gives the demand $D_{i,j}$ corresponding to this embedding (times $\Delta^k$) in dashed blue. Our neighborhood router demand is then computed by doing this for all such clusters and scaling down appropriately.
• Figure 2: A cut sequence (\ref{['sfig:cutTree1']}), the resulting connected components from applying all cuts in the sequence (\ref{['sfig:cutTree2']}) and the corresponding cut sequence tree (\ref{['sfig:cutTree3']}). Cuts colored to correspond to the depth of their nodes in the cut sequence tree. Internal nodes in cut sequence tree labeled according to their corresponding cut and leaves labeled according to their connected component.
• Figure 3: How we disperse demand given a tree $T$ (\ref{['sfig:dispTree1']}). \ref{['sfig:dispTree2']} gives the support of $\mathrm{disperse}_T$ dashed in blue; notice that each vertex has degree at most $2$.
• Figure 4: An overview of the inequalities we show. An arrow from $a$ to $b$ indicates $a \leq b$. Each non-trivial inequality opaque and labeled with the key idea of its proof.

Theorems & Definitions (136)

• Theorem 1.1
• Theorem 1.2: Union of Sparse Moving Cuts is a Sparse Moving Cut
• Theorem 1.3
• Definition 5.1: Classic Cut
• Definition 5.2: (Classic Edge) Cut Sparsity
• Definition 5.3: Classic Expander
• Theorem 5.4: Theorem 1.3 of saranurak2019expander
• Definition 5.5: Length-Constrained Cut (a.k.a. Moving Cut) haeupler2020network
• Definition 5.6: Length-Constrained $S$-$T$ and $\{S_i,T_i\}_i$ Cuts
• Definition 5.7: $G-C$