We give a $0.5368$-competitive algorithm for edge-weighted online bipartite matching. Prior to our work, the best competitive ratio was $0.5086$ due to Fahrbach, Huang, Tao, and Zadimoghaddam (FOCS 2020). They achieved their breakthrough result by developing a subroutine called \emph{online correlated selection} (OCS) which takes as input a sequence of pairs and selects one item from each pair. Importantly, the selections the OCS makes are negatively correlated. We achieve our result by defining \emph{multiway} OCSes which receive arbitrarily many elements at each step, rather than just two. In addition to better competitive ratios, our formulation allows for a simpler reduction from edge-weighted online bipartite matching to OCSes. While Fahrbach et al. used a factor-revealing linear program to optimize the competitive ratio, our analysis directly connects the competitive ratio to the parameters of the multiway OCS. Finally, we show that the formulation of Farhbach et al. can achieve a competitive ratio of at most $0.5239$, confirming that multiway OCSes are strictly more powerful.

中文翻译：

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多路在线相关选择

我们为边缘加权在线二分匹配提供了一个 0.5368 美元的竞争算法。在我们的工作之前，由于 Fahrbach、Huang、Tao 和 Zadimoghaddam（FOCS 2020），最佳竞争比率是 0.5086 美元。他们通过开发一个名为 \emph{online related selection} (OCS) 的子程序实现了他们的突破性成果，该子程序将一系列对作为输入并从每对中选择一个项目。重要的是，OCS 所做的选择是负相关的。我们通过定义 \emph{multiway} OCS 来实现我们的结果，它在每一步接收任意多个元素，而不仅仅是两个。除了更好的竞争比率之外，我们的公式还允许从边缘加权在线二分匹配到 OCS 的更简单的减少。虽然 Fahrbach 等人。使用因子揭示线性程序来优化竞争比率，我们的分析直接将竞争比率与多路 OCS 的参数联系起来。最后，我们表明 Farhbach 等人的公式。可以达到最多 0.5239 美元的竞争比率，证实多路 OCS 更强大。