在线二分匹配的无损在线舍入（尽管不可能）
Lossless Online Rounding for Online Bipartite Matching (Despite its Impossibility)
摘要：
对于许多在线二分匹配问题，如边加权匹配和双边顶点到达下的匹配，最先进的分数算法优于其随机积分对应。这个差距是令人惊讶的，因为二分分数匹配多面体是整数，所以无损舍入是可能的。Devanur等人解释了这一差距。(SODA'13），他证明了\emph{online}无损舍入是不可能的。尽管如此，我们开始研究无损在线四舍五入的在线二分匹配问题。我们的主要观察是，虽然无损在线舍入是不可能的\emph{in general}，随机算法诱导分数算法的相同的竞争比，根据定义是lossslightroundable在线。这促使增加限制，减少“在线完整性差距”，从而允许无损在线四舍五入。我们描述了一组非凸约束，允许这样的无损在线舍入，和更好的竞争比产生的确定性算法。作为我们的无损在线舍入方法的应用，我们得到了两个独立感兴趣的结果：（i）双指数改进，以及优于确定性在线（顶点加权）二分匹配算法所需的随机性（或建议）量的尖锐阈值，以及（ii）最佳半OCS，匹配Gao等人的最新结果。(FOCS'21）回答Fahrbach等人的一个问题~ (FOCS'20）。
英文摘要：
For numerous online bipartite matching problems, such as edge-weighted matching and matching under two-sided vertex arrivals, the state-of-the-art fractional algorithms outperform their randomized integral counterparts. This gap is surprising, given that the bipartite fractional matching polytope is integral, and so lossless rounding is possible. This gap was explained by Devanur et al.~(SODA'13), who showed that \emph{online} lossless rounding is impossible. Despite the above, we initiate the study of lossless online rounding for online bipartite matching problems. Our key observation is that while lossless online rounding is impossible \emph{in general}, randomized algorithms induce fractional algorithms of the same competitive ratio which by definition are losslessly roundable online. This motivates the addition of constraints that decrease the ``online integrality gap'', thus allowing for lossless online rounding. We characterize a set of non-convex constraints which allow for such lossless online rounding, and better competitive ratios than yielded by deterministic algorithms. As applications of our lossless online rounding approach, we obtain two results of independent interest: (i) a doubly-exponential improvement, and a sharp threshold for the amount of randomness (or advice) needed to outperform deterministic online (vertex-weighted) bipartite matching algorithms, and (ii) an optimal semi-OCS, matching a recent result of Gao et al.~(FOCS'21) answering a question of Fahrbach et al.~(FOCS'20).
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