Expander Decomposition with Almost Optimal Overhead

2026-02-16

AI Summary

This work addresses the problem of decomposing a graph into highly expanding subgraphs in polynomial time while minimizing the fraction of edges removed. We present the first polynomial-time algorithm that, by deleting at most a φ·log^{1+o(1)}n fraction of edges, ensures every remaining connected component is a φ-spectral expander. This result achieves a log^{1+o(1)}n overhead, approaching the theoretical lower bound of Ω(log n) and significantly improving upon the previous best guarantee of O(φ log²n). The key innovation lies in the integration of graph decomposition techniques, spectral expander theory, and a refined edge-pruning strategy, which together yield near-optimal edge deletion while preserving strong expansion properties and closely approaching fundamental graph-theoretic limits.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial 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: Models for Web evolution

Abstract

We present the first polynomial-time algorithm for computing a near-optimal \emph{flow}-expander decomposition. Given a graph $G$ and a parameter $\phi$, our algorithm removes at most a $\phi\log^{1+o(1)}n$ fraction of edges so that every remaining connected component is a $\phi$-\emph{flow}-expander (a stronger guarantee than being a $\phi$-\emph{cut}-expander). This achieves overhead $\log^{1+o(1)}n$, nearly matching the $\Omega(\log n)$ graph-theoretic lower bound that already holds for cut-expander decompositions, up to a $\log^{o(1)}n$ factor. Prior polynomial-time algorithms required removing $O(\phi\log^{1.5}n)$ and $O(\phi\log^{2}n)$ fractions of edges to guarantee $\phi$-cut-expander and $\phi$-flow-expander components, respectively.

Problem

• expander decomposition
• flow expander
• graph partitioning
• edge removal
• near-optimal overhead

Innovation

• flow-expander decomposition
• nearly optimal overhead
• polynomial-time algorithm
• graph partitioning
• expansion