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Simple Length-Constrained Expander Decompositions

Greg Bodwin, Bernhard Haeupler, D Ellis Hershkowitz, Zihan Tan

TL;DR

This work addresses the existence of length-constrained expander decompositions by proving a simplified existence result and tightening the sparsity loss. The authors develop a dispersion/counting framework to bound the arboricity of $s$-parallel-greedy graphs, enabling a tighter union-of-cuts sparsity bound and removing reliance on exponential demand structures. They show that for any $h\ge1$, $s\ge2$, $\phi>0$ and graph $G$, there exists an $(h,s)$-length $\phi$-expander decomposition with cut slack $O\left(s\cdot n^{O(1/s)}\right)$ (and length slack $s$), with a generalized form for edge-capacitated graphs and arbitrary node-weightings yielding $O\left(s\cdot|A|^{2/s}\right)$ cut slack. These results enhance the toolkit for fast graph algorithms by enabling near-linear-time constructions of expanders and related decompositions, with direct implications for fast min-cost flow, distance oracles, and dynamic graph problems.

Abstract

Length-constrained expander decompositions are a new graph decomposition that has led to several recent breakthroughs in fast graph algorithms. 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 $hs$ while using each edge to an extent at most $1/φ$. Prior work showed that every $n$-node and $m$-edge graph admits an $(h, s)$-length $φ$-expander decomposition of size $\log n \cdot s n^{O(1/s)} \cdot φm$. In this work, we give a simple proof of the existence of $(h, s)$-length $φ$-expander decompositions with an improved size of $s n^{O(1/s)}\cdot φm$. Our proof is a straightforward application of the fact that the union of sparse length-constrained cuts is itself a sparse length-constrained cut. In deriving our result, we improve the loss in sparsity when taking the union of sparse length-constrained cuts from $\log ^3 n\cdot s^3 n^{O(1/s)}$ to $s\cdot n^{O(1/s)}$.

Simple Length-Constrained Expander Decompositions

TL;DR

This work addresses the existence of length-constrained expander decompositions by proving a simplified existence result and tightening the sparsity loss. The authors develop a dispersion/counting framework to bound the arboricity of -parallel-greedy graphs, enabling a tighter union-of-cuts sparsity bound and removing reliance on exponential demand structures. They show that for any , , and graph , there exists an -length -expander decomposition with cut slack (and length slack ), with a generalized form for edge-capacitated graphs and arbitrary node-weightings yielding cut slack. These results enhance the toolkit for fast graph algorithms by enabling near-linear-time constructions of expanders and related decompositions, with direct implications for fast min-cost flow, distance oracles, and dynamic graph problems.

Abstract

Length-constrained expander decompositions are a new graph decomposition that has led to several recent breakthroughs in fast graph algorithms. Roughly, an -length -expander decomposition is a small collection of length increases to a graph so that nodes within distance can route flow over paths of length while using each edge to an extent at most . Prior work showed that every -node and -edge graph admits an -length -expander decomposition of size . In this work, we give a simple proof of the existence of -length -expander decompositions with an improved size of . Our proof is a straightforward application of the fact that the union of sparse length-constrained cuts is itself a sparse length-constrained cut. In deriving our result, we improve the loss in sparsity when taking the union of sparse length-constrained cuts from to .

Paper Structure

This paper contains 19 sections, 22 theorems, 36 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Preliminaries
  4. Graphs and Arboricity
  5. Graph Arboricity.
  6. Background on Length-Constrained Expander Decompositions
  7. Demands and Node-Weightings
  8. Length-Constrained Cuts and Length-Constrained Cut Sparsity
  9. Length-Constrained Expanders and Length-Constrained Expander Decompositions
  10. New Arboricity Bounds for Parallel-Greedy Graphs
  11. Dispersion Lemma
  12. Counting Lemma
  13. Completing Our Arboricity Bound
  14. The Union of Sparse Length-Constrained Cuts is Sparse
  15. Existence of Length-Constrained Expander Decompositions
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any $h \geq 1$, $s \geq 2$, $\phi > 0$ and any graph $G = (V,E)$ with edge lengths, there is an $(h,s)$-length $\phi$-expander decomposition with cut slack $s \cdot n^{O(1/s)}$.

Figures (3)

  • Figure 1: A $12$-parallel-greedy graph $G = (V, M_1 \sqcup M_2 \sqcup M_3)$. \ref{['sfig:pg3']} gives the final graph $G$.
  • Figure 2: The proof of our dispersion lemma (\ref{['lem:disp']}) for $s=10$. \ref{['sfig:dispLem1']} gives paths $P_a$ (solid) and $P_b$ (dashed) from $u$ to $v$. \ref{['sfig:dispLem2']} gives $Q_a$, $Q_b$, $e_a^*$ and $e_b^*$. \ref{['sfig:dispLem3']} labels each edge with the index of its matching and cycle $C$ which contradicts \ref{['lem:PGAlt']}.
  • 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 $2$.

Theorems & Definitions (53)

  • Definition 1.0: $s$-Parallel-Greedy Graph
  • Theorem 1.1: Existence of Length-Constrained Expander Decompositions---Simplified
  • Theorem 1.2: Parallel-Greedy Graph Arboricity
  • Theorem 1.3: Union of Sparse Length-Constrained Cuts---Simplified
  • Theorem 2.1: nash1961edgenash1964decompositionchen1994short
  • Definition 2.2: Demand
  • Definition 2.3: Node-Weighting
  • Definition 2.4: Demand Respecting a Node-Weighting
  • Definition 2.5: Length-Constrained Cut
  • Definition 2.6: $G-C_h$
  • ...and 43 more