We consider the communication complexity of finding an approximate maximummatching in a graph in a multi-party message-passing communication model. Themaximum matching problem is one of the most fundamental graph combinatorialproblems, with a variety of applications. The input to the problem is a graph $G$ that has $n$ vertices and the set ofedges partitioned over $k$ sites, and an approximation ratio parameter$lpha$. The output is required to be a matching in $G$ that has to bereported by one of the sites, whose size is at least factor $lpha$ of thesize of a maximum matching in $G$. We show that the communication complexity of this problem is $Omega(lpha^2k n)$ information bits. This bound is shown to be tight up to a $log n$factor, by constructing an algorithm, establishing its correctness, and anupper bound on the communication cost. The lower bound also applies to othergraph combinatorial problems in the message-passing communication model,including max-flow and graph sparsification.
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机译：我们考虑在多方消息传播通信模型中在图中找到近似Maximummumumument的通信复杂性。 Thalaximum匹配问题是具有各种应用的最基本的图表组合组合问题之一。问题的输入是一个G $的图形$ n $顶点，并且划分超过$ k $ sites的集合和近似值参数$ alpha $。该输出必须是$ g $的匹配，它必须由其中一个站点进行BEREPERTED，其大小至少为$ alpha $的最大匹配以$ g $。我们表明这个问题的通信复杂性是$ omega（ alpha ^ 2k n）$信息位。通过构建算法，建立其正确性和通信成本绑定的绑定，将此绑定达到$ log n $因子。下限也适用于消息传递通信模型中的其他组合问题，包括MAX-Flow和Graph Sparsification。
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