Simple Length-Constrained Expander Decompositions

2025-10-11

AI Summary

This work addresses the size optimization of length-constrained expander decompositions, aiming to significantly reduce both the number of edges and vertices in the decomposition graph while preserving quality. We introduce a novel structural property—namely, that the union of sparse length-constrained cuts remains sparse—and integrate combinatorial optimization with graph-theoretic analysis to construct more compact decompositions. Our key innovation is a new sparsity-loss control technique that reduces the joint sparse-cut loss from $log^3 n cdot s^3 cdot n^{O(1/s)}$ to $s cdot n^{O(1/s)}$, eliminating logarithmic factors for the first time. We prove that every graph admits an $(h,s)$-length $phi$-expander decomposition of size $s cdot n^{O(1/s)} cdot phi m$, improving upon prior bounds containing $log n$ factors. This result yields a more efficient foundational graph decomposition tool for approximation algorithms and dynamic graph processing.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

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
• Security and Privacy: Large-scale security measurements

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 $log ^3 ncdot s^3 n^{O(1/s)}$ to $scdot n^{O(1/s)}$.

Problem

• Proving existence of length-constrained expander decompositions with improved size
• Simplifying proof for graph decompositions using sparse length-constrained cuts
• Reducing sparsity loss when taking union of sparse length-constrained cuts

Innovation

• Simple proof for length-constrained expander decompositions
• Improved size bound using union of sparse cuts
• Reduced sparsity loss in union operations