In the weighted bipartite matching problem, the goal is to find a maximum-weight matching in a bipartite graph with nonnegative edge weights. We consider its online version where the first vertex set is known beforehand, but vertices of the second set appear one after another. Vertices of the first set are interpreted as items, and those of the second set as bidders. On arrival, each bidder vertex reveals the weights of all adjacent edges and the algorithm has to decide which of those to add to the matching. We introduce an optimal, $e$-competitive truthful mechanism under the assumption that bidders arrive in random order (secretary model). It has been shown that the upper and lower bound of e for the original secretary problem extends to various other problems even with rich combinatorial structure, one of them being weighted bipartite matching. But truthful mechanisms so far fall short of reasonable competitive ratios once respective algorithms deviate from the original, simple threshold form. The best known mechanism for weighted bipartite matching by Krysta and V\"ocking (ICALP 2012) offers only a ratio logarithmic in the number of online vertices. We close this gap, showing that truthfulness does not impose any additional bounds. The proof technique is new in this surrounding, and based on the observation of an independency inherent to the mechanism. The insights provided hereby are interesting in their own right and appear to offer promising tools for other problems, with or without truthfulness.

中文翻译：

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在线加权二等匹配问题的最优真实机制

在加权二分匹配问题中，目标是在具有非负边权重的二分图中找到最大权重匹配。我们考虑其在线版本，其中第一个顶点集是事先已知的，但是第二个顶点集的顶点彼此接连出现。第一组的顶点解释为项目，第二组的顶点解释为投标人。到达时，每个投标人顶点都会显示所有相邻边的权重，并且算法必须决定将哪些权重添加到匹配中。我们在竞标者以随机顺序到达的假设下（秘书模型）引入了一种最佳的$ e $竞争性真实机制。已经表明，即使具有丰富的组合结构，原始秘书问题的e的上下边界也扩展到其他各种问题，其中之一是加权二分匹配。但是，一旦相应的算法偏离了原始的简单阈值形式，那么真实的机制就无法达到合理的竞争比率。Krysta和V'ocking提出的最著名的加权二分匹配机制（ICALP 2012）仅提供在线顶点数量的对数比率。我们缩小了这一差距，表明真实性并不强加任何其他限制。证明技术是因此，本文所提供的见解本身很有趣，并且似乎为解决其他问题提供了有希望的工具，无论是否具有真实性。