Consider a two-stage matching problem, where edges of an input graph are revealed in two stages (batches) and in each stage we have to immediately and irrevocably extend our matching using the edges from that stage. The natural greedy algorithm is half competitive. Even though there is a huge literature on online matching in adversarial vertex arrival model , no positive results were previously known in adversarial edge arrival model . For two-stage bipartite matching problem, we show that the optimal competitive ratio is exactly 2/3 in both the fractional and the randomized-integral models. Furthermore, our algorithm for fractional bipartite matching is instance optimal , i.e., it achieves the best competitive ratio for any given first stage graph. We also study natural extensions of this problem to general graphs and to s stages and present randomized-integral algorithms with competitive ratio ½ + 2− O(s) . Our algorithms use a novel Instance-Optimal-LP and combine graph decomposition techniques with online primal-dual analysis.

中文翻译：

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在线批量到达模型中的最大匹配

考虑一个两阶段匹配问题，其中输入图的边缘在两个阶段（批次）中显示，并且在每个阶段中，我们必须立即且不可撤销地使用该阶段的边缘扩展我们的匹配。自然贪心算法是半竞争性的。尽管有大量关于对抗性在线匹配的文献顶点到达模型, 以前在对抗中没有阳性结果边缘到达模型. 对于两阶段的二分匹配问题，我们表明在分数模型和随机积分模型中，最佳竞争比正好是 2/3。此外，我们的分数二分匹配算法是实例最优，即，它实现了任何产品的最佳竞争比率给定第一阶段图。我们还研究了这个问题对一般图的自然扩展和s阶段并提出具有竞争比 ½ + 2− 的随机积分算法O(s). 我们的算法使用小说实例优化-LP并将图分解技术与在线原始对偶分析相结合。