In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one-by-one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O (log 2 n ) replacements per insertion, where n is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for any replacement strategy, almost matching the Ω (log n ) lower bound. The previous best strategy known achieved amortized O (√ n ) replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than the trivial O ( n ) bound was known except in special cases. Our analysis immediately implies the same upper bound of O (log 2 n ) reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices. We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load L , then the SAP protocol makes amortized O (min { L log 2 n , √ n log n }) reassignments. We also show that this is close to tight, because Ω (min { L , √ n }) reassignments can be necessary.

中文翻译：

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使用摊销 O (log 2 n ) 替换的在线二分匹配

在带替换的在线二分匹配问题中，给定二分一侧的所有顶点，另一侧的顶点与其所有的入射边一一到达。目标是保持最大匹配，同时最小化匹配的更改（替换）数量。我们表明，总是从新插入的顶点（表示为 SAP 协议）采用最短增广路径的贪心算法最多使用摊销○（日志2 n) 每次插入的替换，其中n是插入的顶点总数。这是第一个实现多对数替换的分析任何替换策略，几乎匹配 Ω (logn) 下限。以前已知的最佳策略实现了摊销○(√n) 替换 [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]。特别是对于 SAP 协议，没有什么比琐碎更好的了○(n) bound 是已知的，除非在特殊情况下。我们的分析立即暗示了相同的上限○（日志2 n) 对容量分配问题进行重新分配，其中二分法静态侧的每个顶点都被初始化为具有服务多个顶点的能力。我们还分析了最小化最大服务器负载的问题。我们表明，如果最终图具有最大服务器负载大号，则 SAP 协议使摊销○（分钟{大号日志2 n, √n日志n}) 重新分配。我们还表明这接近于紧，因为 Ω (min {大号, √n}) 重新分配可能是必要的。