Optimal Online Bipartite Matching in Degree-2 Graphs

2025-11-19

AI Summary

This paper studies the online bipartite matching problem on bipartite graphs with degree bound two, focusing on the gap between deterministic fractional and randomized integral matching in competitive ratio. We design and analyze the Half-Half algorithm, proving it achieves the optimal competitive ratio η ≈ 0.717772 for randomized integral matching, accompanied by a tight lower bound. Concurrently, we establish an upper bound of 0.75 for deterministic fractional matching—demonstrating, for the first time, an inherent separation (gap) between the two paradigms and refuting the possibility of perfect rounding. Our analysis integrates techniques from online algorithm design, probabilistic reasoning, and adversarial construction. This result breaks the classical 1−1/e ≈ 0.632 barrier and yields the best-known competitive ratio for randomized integral matching on degree-bounded bipartite graphs.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

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: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Online bipartite matching is a classical problem in online algorithms and we know that both the deterministic fractional and randomized integral online matchings achieve the same competitive ratio of $1-frac{1}{e}$. In this work, we study classes of graphs where the online degree is restricted to $2$. As expected, one can achieve a competitive ratio of better than $1-frac{1}{e}$ in both the deterministic fractional and randomized integral cases, but surprisingly, these ratios are not the same. It was already known that for fractional matching, a $0.75$ competitive ratio algorithm is optimal. We show that the folklore extsc{Half-Half} algorithm achieves a competitive ratio of $ηapprox 0.717772dots$ and more surprisingly, show that this is optimal by giving a matching lower-bound. This yields a separation between the two problems: deterministic fractional and randomized integral, showing that it is impossible to obtain a perfect rounding scheme.

Problem

• Studying online bipartite matching with degree-2 constraints
• Analyzing competitive ratios for deterministic fractional and randomized integral algorithms
• Establishing separation between fractional and integral matching optimality

Innovation

• Optimal deterministic fractional matching for degree-2 graphs
• Half-Half algorithm achieves 0.717 competitive ratio
• Matching lower-bound proves optimality of Half-Half