Approximating Congestion + Dilation in Networks via "Quality of Routing” Games

A classic optimization problem in network routing is to minimize C+D, where C is the maximum edge congestion and D is the maximum path length (also known as dilation). The problem of computing the optimal C^{\ast}+D^{\ast} is NP-complete even when either C^{\ast} or D^{\ast} is a small constant. We study routing games in general networks where each player i selfishly selects a path that minimizes C_i + D_i the sum of congestion and dilation of the player's path. We first show that there are instances of this game without Nash equilibria. We then turn to the related quality of routing (QoR) games which always have Nash equilibria. QoR games represent networks with a small number of service classes where paths in different classes do not interfere with each other (with frequency or time division multiplexing). QoR games have O(\log^4 n) price of anarchy when either C^{\ast} or D^{\ast} is a constant. Thus, Nash equilibria of QoR games give poly-log approximations to hard optimization problems.