Communication complexity of approximate maximum matching in the message-passing model

Abstract

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 $$\alpha $$ . 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 $$\alpha $$ of the size of a maximum matching in G. We show that the communication complexity of this problem is $$\varOmega (\alpha ^2 k n)$$ information bits. This bound is shown to be tight up to a $$\log n$$ 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.