The structure and complexity of Nash equilibria for a selfish routing game

We embark on a systematic study of several algorithmic problems related to the computation of Nash equilibria for the selfish routing game we consider. In a nutshell, these problems relate to deciding the existence of a pure Nash equilibrium, constructing a Nash equilibrium, constructing the pure Nash equilibria of minimum and maximum social cost, and computing the social cost of a given mixed Nash equilibrium. Our work provides a comprehensive collection of efficient algorithms, hardness results, and structural results for these algorithmic problems. Our results span and contrast a wide range of assumptions on the syntax of the Nash equilibria and on the parameters of the system.