树上最短增益路径的紧界 (A Tight Bound for Shortest Augmenting Paths on Trees)

The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree $T=(W \uplus B, E)$ is being revealed online, i.e., in each round one vertex from $B$ with its incident edges arrives. It was conjectured by Chaudhuri et. al. [K. Chaudhuri, C. Daskalakis, R. D. Kleinberg, and H. Lin. Online bipartite perfect matching with augmentations. In INFOCOM 2009] that the total length of all shortest augmenting paths found is $O(n \log n)$. In this paper, we prove a tight $O(n \log n)$ upper bound for the total length of shortest augmenting paths for trees improving over $O(n \log^2 n)$ bound [B. Bosek, D. Leniowski, P. Sankowski, and A. Zych. Shortest augmenting paths for online matchings on trees. In WAOA 2015].

翻译：最短的扩大路径技术是最大匹配和最大流量算法中使用的基本理念之一。自1972年Edmonds和Karp 被Edmonds和Karp 引入以来,它被广泛应用于许多不同的环境。令人惊讶的是,尽管使用如此广泛,但即使在最简单的例子中,它仍然没有被很好地理解:在线双边匹配树木的问题。在这个问题中,双边树$T=(W\uplus B, E)正在在线披露,也就是说,在每圆一面顶端中,从美元到事件发生边缘的美元每圆一个顶端中,它被Chaudhuri et al. [C. Chaudhuuri, C. Dskalakis, R. D. Kleinberg, H. Lin. 在线双边匹配增强。在INFOCOM 2009 中,发现所有最短的扩大路径的总长度为$O(n\log n) $。在本文中,我们证明2015年的红树、红树、红树、红树、红树、红树、红树、红树、红树、红树、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、红、

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