The Hardness of Network Design for Unsplittable Flow with Selfish Users

In this paper we consider the network design for selfish users problem, where we assume the more realistic unsplittable model in which the users can have general demands and each user must choose a single path between its source and its destination. This model is also called atomic (weighted) network congestion game. The problem can be presented as follows : given a network, which edges should be removed to minimize the cost of the worst Nash equilibrium? We consider both computational issues and existential issues (i.e. the power of network design). We give inapproximability results and approximation algorithms for this network design problem. For networks with linear edge latency functions we prove that there is no approximation algorithm for this problem with approximation ratio less then $(3+\sqrt{5})/2 \approx 2.618$ unless P=NP. We also show that for networks with polynomials of degree d edge latency functions there is no approximation algorithm for this problem with approximation ratio less then $d^{{\it \Theta}(d)}$ unless P=NP. Moreover, we observe that the trivial algorithm that builds the entire network is optimal for linear edge latency functions and has an approximation ratio of $d^{{\it \Theta}(d)}$ for polynomials of degree d edge latency functions. Finally, we consider general continuous, non-decreasing edge latency functions and show that the approximation ratio of any approximation algorithm for this problem is unbounded, assuming P ≠ NP. In terms of existential issues we show that network design cannot improve the maximum possible bound on the price of anarchy in the worst case. Previous results of Roughgarden for networks with n vertices where each user controls only a negligible fraction of the overall traffic showed optimal inapproximability results of 4/3 for linear edge latency functions, ${\it \Theta}(d /{\rm ln} d)$ for polynomial edge latency functions and n/2 for general continuous non-decreasing edge latency functions. He also showed that the trivial algorithm that builds the entire network is optimal for that case.