二选三：三向在线相关选择
Making Three Out of Two: Three-Way Online Correlated Selection
摘要：
双向在线相关选择（英语：Two-way online correlated selection，缩写：OCS）是一种在线算法，在每个时间步，从基集中取一对元素，并可选择两个元素中的一个，同时确保算法选择的负相关性。虽然OCS最初是由Fahrbach，Huang，Tao和Zadimoghousehold发明的，用于解决边加权在线二分匹配问题，但由于其能够将强大的算法工具（即负相关）引入在线算法，因此它本身就是一种有趣的技术。因此，Fahrbach等人在他们的论文中提出了两个诱人的开放问题，其中一个是：我们能否获得n路OCS（n>2），其中算法可以在每个时间步中选择n>2个元素？在本文中，我们肯定地回答了这个开放的问题，提出了一个三路OCS。我们的算法使用双向OCS作为其构建块，并且很容易描述;然而，由于它在内部运行两个双向OCS实例，其中一个实例是由另一个实例的输出提供的，因此最终的输出概率分布变得非常难以捉摸。我们解决这个困难的近似输出分布的OCS由一个平坦的，相关性较低的功能，并使用它作为一个安全的“代理”的真实分布。我们的三路OCS也产生了0.5093的边缘加权在线匹配的竞争力的算法，证明了它的实用性。
英文摘要：
Two-way online correlated selection (two-way OCS) is an online algorithm that, at each timestep, takes a pair of elements from the ground set and irrevocably chooses one of the two elements, while ensuring negative correlation in the algorithm's choices. Whilst OCS was initially invented by Fahrbach, Huang, Tao, and Zadimoghaddam to solve the edge-weighted online bipartite matching problem, it is an interesting technique on its own due to its capability of introducing a powerful algorithmic tool, namely negative correlation, to online algorithms. As such, Fahrbach et al. posed two tantalizing open questions in their paper, one of which was the following: Can we obtain n-way OCS for n>2, in which the algorithm can be given n>2 elements to choose from at each timestep? In this paper, we affirmatively answer this open question by presenting a three-way OCS. Our algorithm uses two-way OCS as its building block and is simple to describe; however, as it internally runs two instances of two-way OCS, one of which is fed with the output of the other, the final output probability distribution becomes highly elusive. We tackle this difficulty by approximating the output distribution of OCS by a flat, less correlated function and using it as a safe "surrogate" of the real distribution. Our three-way OCS also yields a 0.5093-competitive algorithm for edge-weighted online matching, demonstrating its usefulness.
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