  We study connections between the problem of fully dynamic
$(1-\epsilon)$-approximate maximum bipartite matching, and the dual
$(1+\epsilon)$-approximate vertex cover problem, with the online matrix-vector
($\mathsf{OMv}$) conjecture which has recently been used in several
fine-grained hardness reductions. We prove that there is an online algorithm
that maintains a $(1+\epsilon)$-approximate vertex cover in amortized
$n^{1-c}\epsilon^{-C}$ time for constants $c, C > 0$ for fully dynamic updates
if and only if the $\mathsf{OMv}$ conjecture is false. Similarly, we prove that
there is an online algorithm that maintains a $(1-\epsilon)$-approximate
maximum matching in amortized $n^{1-c}\epsilon^{-C}$ time if and only if there
is a nontrivial algorithm for another dynamic problem, which we call dynamic
approximate $\mathsf{OMv}$, that has seemingly no matching structure. This
provides some evidence against achieving amortized sublinear update times for
approximate fully dynamic matching and vertex cover.
  Leveraging these connections, we obtain faster algorithms for approximate
fully dynamic matching in both the online and offline settings.
  1. We give a randomized algorithm that with high probability maintains a
$(1-\epsilon)$-approximate bipartite matching and $(1+\epsilon)$-approximate
vertex cover in fully dynamic graphs, in amortized $O(\epsilon^{-O(1)}
\frac{n}{2^{\Omega(\sqrt{\log n})}})$ update time. Our algorithm leverages fast
algorithms for $\mathsf{OMv}$ due to Larsen-Williams [SODA 2017].
  2. We give a randomized offline algorithm for $(1-\epsilon)$-approximate
maximum matching with amortized runtime $O(n^{.58}\epsilon^{-O(1)})$ by using
fast matrix multiplication, significantly improving over the runtimes achieved
via online algorithms. We also give an offline algorithm that maintains a
$(1+\epsilon)$-approximate vertex cover in amortized
$O(n^{.723}\epsilon^{-O(1)})$ time.
