  We present a deterministic $(1+\varepsilon)$-approximate maximum matching
algorithm in $\mathsf{poly} 1/\varepsilon$ passes in the semi-streaming model,
solving the long-standing open problem of breaking the exponential barrier in
the dependence on $1/\varepsilon$. Our algorithm exponentially improves on the
well-known randomized $(1/\varepsilon)^{O(1/\varepsilon)}$-pass algorithm from
the seminal work by McGregor~[APPROX05], the recent deterministic algorithm by
Tirodkar with the same pass complexity~[FSTTCS18]. Up to polynomial factors in
$1/\varepsilon$, our work matches the state-of-the-art deterministic $(\log n /
\log \log n) \cdot (1/\varepsilon)$-pass algorithm by Ahn and Guha~[TOPC18],
that is allowed a dependence on the number of nodes $n$. Our result also makes
progress on the Open Problem 60 at sublinear.info.
  Moreover, we design a general framework that simulates our approach for the
streaming setting in other models of computation. This framework requires
access to an algorithm computing an $O(1)$-approximate maximum matching and an
algorithm for processing disjoint $(\mathsf{poly} 1 / \varepsilon)$-size
connected components. Instantiating our framework in $\mathsf{CONGEST}$ yields
a $\mathsf{poly}(\log{n}, 1/\varepsilon)$ round algorithm for computing
$(1+\varepsilon$)-approximate maximum matching. In terms of the dependence on
$1/\varepsilon$, this result improves exponentially state-of-the-art result by
Lotker, Patt-Shamir, and Pettie~[LPSP15]. Our framework leads to the same
quality of improvement in the context of the Massively Parallel Computation
model as well.
