  Atomic congestion games are a classic topic in network design, routing, and
algorithmic game theory, and are capable of modeling congestion and flow
optimization tasks in various application areas. While both the price of
anarchy for such games as well as the computational complexity of computing
their Nash equilibria are by now well-understood, the computational complexity
of computing a system-optimal set of strategies -- that is, a centrally planned
routing that minimizes the average cost of agents -- is severely understudied
in the literature. We close this gap by identifying the exact boundaries of
tractability for the problem through the lens of the parameterized complexity
paradigm. After showing that the problem remains highly intractable even on
extremely simple networks, we obtain a set of results which demonstrate that
the structural parameters which control the computational (in)tractability of
the problem are not vertex-separator based in nature (such as, e.g.,
treewidth), but rather based on edge separators. We conclude by extending our
analysis towards the (even more challenging) min-max variant of the problem.
