  We present an algorithm for the maximum matching problem in dynamic
(insertion-deletions) streams with *asymptotically optimal* space complexity:
for any $n$-vertex graph, our algorithm with high probability outputs an
$\alpha$-approximate matching in a single pass using $O(n^2/\alpha^3)$ bits of
space.
  A long line of work on the dynamic streaming matching problem has reduced the
gap between space upper and lower bounds first to $n^{o(1)}$ factors
[Assadi-Khanna-Li-Yaroslavtsev; SODA 2016] and subsequently to
$\text{polylog}{(n)}$ factors [Dark-Konrad; CCC 2020]. Our upper bound now
matches the Dark-Konrad lower bound up to $O(1)$ factors, thus completing this
research direction.
  Our approach consists of two main steps: we first (provably) identify a
family of graphs, similar to the instances used in prior work to establish the
lower bounds for this problem, as the only "hard" instances to focus on. These
graphs include an induced subgraph which is both sparse and contains a large
matching. We then design a dynamic streaming algorithm for this family of
graphs which is more efficient than prior work. The key to this efficiency is a
novel sketching method, which bypasses the typical loss of
$\text{polylog}{(n)}$-factors in space compared to standard $L_0$-sampling
primitives, and can be of independent interest in designing optimal algorithms
for other streaming problems.
