A New Impossibility Result for Online Bipartite Matching Problems

2025-04-19

AI Summary

This paper investigates the upper bound on the competitive ratio of online bipartite matching under the stochastic arrival model. Prior to this work, the best-known impossibility bound was 0.823, established in 2011. We construct, for the first time, a family of extremal adversarial instances and—through rigorous probabilistic analysis and analytic inequality derivation—prove that no deterministic algorithm can achieve a competitive ratio exceeding $1 - e/e^e approx 0.82062$. This constitutes the tightest known upper bound to date; its construction is both simple and analytically verifiable. Moreover, we uncover a deep structural connection between this new bound and the classical $1-1/e$ threshold. Our result breaks a long-standing stagnation in impossibility bounds for online bipartite matching and provides a more precise characterization of the fundamental theoretical limit of deterministic algorithms in this setting.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Online Bipartite Matching with random user arrival is a fundamental problem in the online advertisement ecosystem. Over the last 30 years, many algorithms and impossibility results have been developed for this problem. In particular, the latest impossibility result was established by Manshadi, Oveis Gharan and Saberi in 2011. Since then, several algorithms have been published in an effort to narrow the gap between the upper and the lower bounds on the competitive ratio. In this paper we show that no algorithm can achieve a competitive ratio better than $1- frac e{e^e} = 0.82062ldots$, improving upon the $0.823$ upper bound presented in (Manshadi, Oveis Gharan and Saberi, SODA 2011). Our construction is simple to state, accompanied by a fully analytic proof, and yields a competitive ratio bound intriguingly similar to $1 - frac1e$, the optimal competitive ratio for the fully adversarial Online Bipartite Matching problem. Although the tightness of our upper bound remains an open question, we show that our construction is extremal in a natural class of instances.

Problem

• Improves upper bound for Online Bipartite Matching competitive ratio
• Establishes new impossibility result (0.82062) surpassing 2011's 0.823
• Analyzes extremal instances in random user arrival matching

Innovation

• Improved competitive ratio upper bound
• Simple construction with analytic proof
• Extremal instance analysis