We study the problem of online unweighted bipartite matching with $n$ offline vertices and $n$ online vertices where one wishes to be competitive against the optimal offline algorithm. While the classic RANKING algorithm of Karp et al. [1990] provably attains competitive ratio of $1-1/e > 1/2$, we show that no learning-augmented method can be both 1-consistent and strictly better than $1/2$-robust under the adversarial arrival model. Meanwhile, under the random arrival model, we show how one can utilize methods from distribution testing to design an algorithm that takes in external advice about the online vertices and provably achieves competitive ratio interpolating between any ratio attainable by advice-free methods and the optimal ratio of 1, depending on the advice quality.

翻译：我们研究具有$n$个离线顶点和$n$个在线顶点的在线无权二分图匹配问题，目标是与最优离线算法竞争。虽然Karp等人[1990]的经典RANKING算法可证明达到$1-1/e > 1/2$的竞争比，但我们证明在对抗到达模型下，任何学习增强方法都无法同时实现1-一致性且严格优于1/2-鲁棒性。同时，在随机到达模型下，我们展示如何利用分布测试方法设计算法：该算法接收关于在线顶点的外部建议，并能根据建议质量，可证明地实现介于无建议方法所能达到的任意竞争比与最优比1之间的竞争比。