  We show that computing an equilibrium in atomic splittable congestion games
with player-specific affine cost functions $l_{e,i}(x) = a_{e,i} x + b_{e,i}$
is $\mathsf{PPAD}$-complete. To prove that the problem is contained in
$\mathsf{PPAD}$, we develop a homotopy method that traces an equilibrium for
varying flow demands of the players. A key technique for this method is to
describe the evolution of the equilibrium locally by a novel block Laplacian
matrix. Using the properties of this matrix give rise to a path following
formulation for computing an equilibrium where states correspond to supports
that are feasible for some demands. A closer investigation of the block
Laplacian system further allows to orient the states giving rise to unique
predecessor and successor states thus putting the problem into $\mathsf{PPAD}$.
For the $\mathsf{PPAD}$-hardness, we reduce from computing an approximate
equilibrium of a bimatrix win-lose game. As a byproduct of our reduction we
further show that computing a multi-class Wardrop equilibrium with class
dependent affine cost functions is $\mathsf{PPAD}$-complete as well.
  As another byproduct of our $\mathsf{PPAD}$-completeness proof, we obtain an
algorithm that computes a continuum of equilibria parametrized by the players'
flow demand. For player-specific costs, the algorithm runs in polynomial space.
For games with player-independent costs, we obtain an algorithm computing all
equilibria as a function of the flow demand that runs in time polynomial in the
output.
