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.