We consider the communication complexity of finding an approximate maximum matching in a graph in a multi-party message-passing communication model. The maximum matching problem is one of the most fundamental graph combinatorial problems, with a variety of applications. The input to the problem is a graph G that has n vertices and the set of edges partitioned over k sites, and an approximation ratio parameter $$\alpha $$ α . The output is required to be a matching in G that has to be reported by one of the sites, whose size is at least factor $$\alpha $$ α of the size of a maximum matching in G . We show that the communication complexity of this problem is $$\varOmega (\alpha ^2 k n)$$ Ω ( α 2 k n ) information bits. This bound is shown to be tight up to a $$\log n$$ log n factor, by constructing an algorithm, establishing its correctness, and an upper bound on the communication cost. The lower bound also applies to other graph combinatorial problems in the message-passing communication model, including max-flow and graph sparsification.

中文翻译：

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消息传递模型中近似最大匹配的通信复杂度

我们考虑在多方消息传递通信模型中在图中找到近似最大匹配的通信复杂性。最大匹配问题是最基本的图组合问题之一，具有多种应用。问题的输入是一个图 G，它有 n 个顶点和分布在 k 个站点上的一组边，以及一个近似比率参数 $$\alpha $$ α 。输出必须是 G 中的匹配，必须由站点之一报告，其大小至少是 G 中最大匹配大小的因子 $$\alpha $$ α。我们证明这个问题的通信复杂度是 $$\varOmega (\alpha ^2 kn)$$Ω (α 2 kn) 信息位。通过构造一个算法，建立它的正确性，这个界限被证明是紧到一个 $$\log n$$ log n 因子，以及通信成本的上限。下界也适用于消息传递通信模型中的其他图组合问题，包括最大流和图稀疏化。