  In this paper we revisit the problem of constructing randomized composable
coresets for bipartite matching. In this problem the input graph is randomly
partitioned across $k$ players, each of which sends a single message to a
coordinator, who then must output a good approximation to the maximum matching
in the input graph. Assadi and Khanna gave the first such coreset, achieving a
$1/9$-approximation by having every player send a maximum matching, i.e. at
most $n/2$ words per player. The approximation factor was improved to $1/3$ by
Bernstein et al.
  In this paper, we show that the matching skeleton construction of Goel,
Kapralov and Khanna, which is a carefully chosen (fractional) matching, is a
randomized composable coreset that achieves a $1/2-o(1)$ approximation using at
most $n-1$ words of communication per player. We also show an upper bound of
$2/3+o(1)$ on the approximation ratio achieved by this coreset.
