  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 $\Omega(\alpha^2
k n)$ 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.
