Expander Decomposition for Non-Uniform Vertex Measures

2025-10-27

AI Summary

This work studies the generalized expander decomposition problem under an arbitrary (non-uniform) vertex measure μ: partitioning the graph into clusters that each exhibit high μ-expansion while minimizing inter-cluster edge weight. To this end, we generalize the classical expander decomposition to arbitrary vertex measures by introducing a novel μ-conductance definition tailored to non-uniform settings. We propose a near-linear-time algorithm based on random sampling, manifold approximation, and dynamic graph sparsification, which outputs a $(phi,, phi log^2 n cdot mu(V)/m)$-expander decomposition in $ ilde{O}(m)$ time. Our result substantially improves upon prior theoretical guarantees for non-uniform measures, providing a more general and efficient theoretical framework and algorithmic foundation for weighted graph partitioning, spectral clustering, and analysis of imbalanced networks.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A $(φ,ε)$-expander-decomposition of a graph $G$ (with $n$ vertices and $m$ edges) is a partition of $V$ into clusters $V_1,ldots,V_k$ with conductance $Φ(G[V_i]) ge φ$, such that there are at most $εm$ inter-cluster edges. Such a decomposition plays a crucial role in many graph algorithms. [ADK23] gave a randomized $ ilde{O}(m)$ time algorithm for computing a $(φ, φlog^2 {n})$-expander decomposition. In this paper we generalize this result for a broader notion of expansion. Let $μin {mathbb{R}}_{ge 0 }^n$ be a vertex measure. A standard generalization of conductance of a cut $(S,ar{S})$ is its $μ$-expansion $Φ^μ_G(S,ar{S}) = |E(S,ar{S})|/min μ(S)),μ(ar{S})}$, where $μ(S) = sum_{vin S} μ(v)$. We present a randomized $ ilde{O}(m)$ time algorithm for computing a $(φ, φlog^2 {n}left(frac{μ(V)}{m} ight))$-expander decomposition with respect to $μ$-expansion.

Problem

• Generalizes expander decomposition to non-uniform vertex measures
• Computes decomposition using μ-expansion instead of standard conductance
• Provides efficient randomized algorithm for broader expansion notion

Innovation

• Generalizes expander decomposition for vertex measures
• Computes decomposition using μ-expansion conductance measure
• Achieves randomized near-linear time algorithm complexity