Complexity and Parametric Computation of Equilibria in Atomic Splittable Congestion Games via Weighted Block Laplacians

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.