  Maximum matching is one of the most fundamental combinatorial optimization
problems with applications in various contexts such as balanced clustering,
data mining, resource allocation, and online advertisement. In many of these
applications, the input graph is massive. The sheer size of these inputs makes
it impossible to store the whole graph in the memory of a single machine and
process it there. Graph sparsification has been an extremely powerful tool to
alleviate this problem. In this paper, we study a highly successful and
versatile sparsifier for the matching problem: the *edge-degree constrained
subgraph (EDCS)* introduced first by Bernstein and Stein [ICALP'15].
  The EDCS has a parameter $\beta \geq 2$ which controls the density of the
sparsifier. It has been shown through various proofs in the literature that by
picking a subgraph with $O(n\beta)$ edges, the EDCS includes a matching of size
at least $2/3-O(1/\beta)$ times the maximum matching size. As such, by
increasing $\beta$ the approximation ratio of EDCS gets closer and closer to
$2/3$.
  In this paper, we propose a new approach for analyzing the approximation
ratio of EDCS. Our analysis is *tight* for any value of $\beta$. Namely, we
pinpoint the precise approximation ratio of EDCS for any sparsity parameter
$\beta$. Our analysis reveals that one does not necessarily need to increase
$\beta$ to improve approximation, as suggested by previous analysis. In
particular, the best choice turns out to be $\beta = 6$, which achieves an
approximation ratio of $.677$! This is arguably surprising as it is even better
than $2/3 \sim .666$, the bound that was widely believed to be the limit for
EDCS.
