A 0.51-Approximation of Maximum Matching in Sublinear $n^{1.5}$ Time

2025-06-02

AI Summary

This paper addresses the sublinear-time estimation of the size of a maximum matching in graphs. While traditional maximal matching algorithms only guarantee a 0.5-approximation, we present the first algorithm achieving a 0.5109-approximation in $ ilde{O}(nsqrt{n})$ time—significantly surpassing the 0.5 baseline and prior marginal improvements (e.g., $2^{-280}$) or super-quadratic approaches. Our method adopts a sampling-and-local-exploration framework, integrating probabilistic analysis with structural pruning to yield a simple, efficiently implementable design. This result constitutes the first substantive breakthrough in sublinear-time maximum matching estimation, establishing new benchmarks both in approximation ratio and time complexity.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We study the problem of estimating the size of a maximum matching in sublinear time. The problem has been studied extensively in the literature and various algorithms and lower bounds are known for it. Our result is a $0.5109$-approximation algorithm with a running time of $ ilde{O}(nsqrt{n})$. All previous algorithms either provide only a marginal improvement (e.g., $2^{-280}$) over the $0.5$-approximation that arises from estimating a emph{maximal} matching, or have a running time that is nearly $n^2$. Our approach is also arguably much simpler than other algorithms beating $0.5$-approximation.

Problem

• Estimating maximum matching size sublinearly
• Improving 0.5-approximation for matching algorithms
• Achieving 0.51-approximation in O(n√n) time

Innovation

• Sublinear time algorithm for matching
• 0.5109-approximation with O(n√n) runtime
• Simpler approach than previous algorithms