Online bipartite matching is a fundamental problem in online optimization, extensively studied both in its integral and fractional forms due to its theoretical significance and practical applications, such as online advertising and resource allocation. Motivated by recent progress in learning-augmented algorithms, we study online bipartite fractional matching when the algorithm is given advice in the form of a suggested matching in each iteration. We develop algorithms for both the vertex-weighted and unweighted variants that provably dominate the naive "coin flip" strategy of randomly choosing between the advice-following and advice-free algorithms. Moreover, our algorithm for the vertex-weighted setting extends to the AdWords problem under the small bids assumption, yielding a significant improvement over the seminal work of Mahdian, Nazerzadeh, and Saberi (EC 2007, TALG 2012). Complementing our positive results, we establish a hardness bound on the robustness-consistency tradeoff that is attainable by any algorithm. We empirically validate our algorithms through experiments on synthetic and real-world data.

翻译：在线二分匹配是在线优化中的一个基本问题，由于其理论意义和实际应用（如在线广告和资源分配），其整数形式和分数形式均得到了广泛研究。受学习增强算法最新进展的启发，我们研究了当算法在每次迭代中获得建议匹配形式指导时的在线二分分数匹配问题。我们为顶点加权和无加权变体开发了算法，这些算法在理论上优于在遵循建议的算法和无建议算法之间随机选择的朴素“抛硬币”策略。此外，我们针对顶点加权设置的算法可扩展至小出价假设下的AdWords问题，相较于Mahdian、Nazerzadeh和Saberi（EC 2007，TALG 2012）的开创性工作取得了显著改进。作为我们积极结果的补充，我们建立了任何算法在鲁棒性与一致性权衡方面可达到的硬度界限。我们通过在合成数据和真实数据上的实验对算法进行了实证验证。