Reachability in Vector Addition Systems is Ackermann-complete

Vector Addition Systems and equivalent Petri nets are a well established models of concurrency. The central algorithmic problem for Vector Addition Systems with a long research history is the reachability problem asking whether there exists a run from one given configuration to another. We settle its complexity to be Ackermann-complete thus closing the problem open for 45 years. In particular we prove that the problem is
$\mathcal{F}_{k}$
-hard for Vector Addition Systems with States in dimension 6k, where
$\mathcal{F}_{k}$
is the
$k$
-th complexity class from the hierarchy of fast-growing complexity classes.