Improved Online Correlated Selection

This paper studies the online correlated selection (OCS) problem introduced by Fahrbach, Huang, Tao, and Zadimoghaddam (2020) to get the first edge-weighted online bipartite matching algorithm that breaks the $0.5$ barrier. Suppose that we receive a pair of elements in each round and select one of them. Can we select with negative correlation to be more effective than independent random selections? Our contributions are threefold. For semi-OCS, which considers the probability that an element remains unselected after appearing in $k$ rounds, we give an optimal algorithm that minimizes this probability for all $k$. It leads to $0.536$-competitive unweighted and vertex-weighted online bipartite matching algorithms that randomize over only two options in each round, improving the previous 0.508-competitive ratio by Fahrbach et al. (2020). Further, we give the first multi-way semi-OCS that allows an arbitrary number of elements with arbitrary masses in each round. As an application, it rounds the Balance algorithm in unweighted and vertex-weighted online bipartite matching to get a $0.593$-competitive ratio. This is the first algorithm other than Ranking whose competitive ratio is beyond the $0.5 + \epsilon$ regime. Finally, we study OCS, which further considers the probability that an element is unselected in any subset of rounds. We prove that the optimal "level of negative correlation" is between $0.167$ and $0.25$, improving the previous bounds of $0.109$ and $1$ by Fahrbach et al. (2020). Our OCS gives a $0.519$-competitive edge-weighted online bipartite matching algorithm, improving the previous $0.508$-competitive ratio by Fahrbach et al. (2020).