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

We settle the complexity of 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) showing that it is PPAD-complete. To prove that the problem is contained in PPAD, we develop a homotopy method that traces an equilibrium for varying flow demands of the players. A key technique is to describe the evolution of the equilibrium locally by a novel block Laplacian matrix. This leads to a path following formulation where states correspond to supports that are feasible for some demands and neighboring supports are feasible for increased or decreased flow demands. A closer investigation of the block Laplacian system allows to orient the states giving rise to unique predecessor and successor states thus putting the problem into PPAD. For the 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 PPAD-complete as well. As a byproduct of our PPAD-completeness proof, we obtain an algorithm that computes all equilibria parametrized by the players' flow demands. For player-specific costs, this computation may require several increases and decreases of the demands leading to an algorithm that runs in polynomial space but exponential time. For player-independent costs only demand increases are necessary. If the coefficients b(e,i) are in general position, this yields an algorithm computing all equilibria as a function of the flow demand running in time polynomial in the output.