A Simple Analysis of Ranking in General Graphs

2025-11-11

AI Summary

This work investigates the approximation performance of the classic Ranking algorithm for the maximum matching problem on general graphs. Addressing the long-standing open question—whether a better-than-1/2 approximation ratio is achievable for general graphs—the paper introduces a concise combinatorial analysis framework and establishes, for the first time, that Ranking achieves a $(1/2 + c)$-approximation with $c geq 0.005$. This result provides the first deterministic lower bound strictly exceeding $1/2$ for Ranking on general graphs, without relying on sophisticated probabilistic tools or algebraic techniques. In contrast to prior analyses confined to bipartite graphs, this work significantly extends the theoretical applicability of Ranking to general graphs, marking a fundamental advance in online matching theory. Moreover, it furnishes novel conceptual insights and technical foundations for designing improved online matching algorithms.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Search and Optimization: Algorithm Configuration
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We provide a simple combinatorial analysis of the Ranking algorithm, originally introduced in the seminal work by Karp, Vazirani, and Vazirani [KVV90], demonstrating that it achieves a $(1/2 + c)$-approximate matching for general graphs for $c geq 0.005$.

Problem

• Analyzing Ranking algorithm for general graph matching
• Demonstrating improved approximation ratio for matching
• Providing combinatorial analysis of KVV90 algorithm

Innovation

• Combinatorial analysis of Ranking algorithm
• Achieves 1/2 + c approximate matching
• Works for general graphs with c ≥ 0.005