We provide online algorithms for secretary matching in general weighted graphs, under the well-studied models of vertex and edge arrivals. In both models, edges are associated with arbitrary weights that are unknown from the outset, and are revealed online. Under vertex arrival, vertices arrive online in a uniformly random order; upon the arrival of a vertex $v$, the weights of edges from $v$ to all previously arriving vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. Under edge arrival, edges arrive online in a uniformly random order; upon the arrival of an edge $e$, its weight is revealed, and the algorithm decides whether to include it in the matching or not. We provide a $5/12$-competitive algorithm for vertex arrival, and show it is tight. For edge arrival, we provide a $1/4$-competitive algorithm. Both results improve upon state of the art bounds for the corresponding settings. Interestingly, for vertex arrival, secretary matching in general graphs outperforms secretary matching in bipartite graphs with 1-sided arrival, where $1/e$ is the best possible guarantee.

中文翻译：

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秘书配对一般来港定居人士

在经过充分研究的顶点和边到达模型下，我们提供了用于一般加权图中秘书匹配的在线算法。在这两种模型中，边都与从一开始就未知的任意权重相关联，并在线显示。在顶点到达下，顶点以一致的随机顺序在线到达；在顶点 $v$ 到达时，显示从 $v$ 到所有先前到达的顶点的边的权重，并且算法决定这些边中的哪些（如果有）包含在匹配中。在边缘到达下，边缘以一致的随机顺序在线到达；当一条边 $e$ 到达时，它的权重被揭示，算法决定是否将它包含在匹配中。我们为顶点到达提供了一个 $5/12$ 的竞争算法，并证明它是紧凑的。对于边缘到达，我们提供了一个 1/4 美元的竞争算法。这两个结果都改进了相应设置的现有技术范围。有趣的是，对于顶点到达，一般图中的秘书匹配优于单边到达的二部图中的秘书匹配，其中 $1/e$ 是最好的保证。