On the Approximability of Robust Network Design.

Considering the dynamic nature of traffic, the robust network design problem consists in computing the capacity to be reserved on each network link such that any demand vector belonging to a polyhedral set can be routed. The objective is either to minimize congestion or a linear cost. And routing freely depends on the demand. We first prove that the robust network design problem with minimum congestion cannot be approximated within any constant factor. Then, using the ETH conjecture, we get a $\Omega(\frac{\log n}{\log \log n})$ lower bound for the approximability of this problem. This implies that the well-known $O(\log n)$ approximation ratio established by Racke in 2008 is tight. Using Lagrange relaxation, we obtain a new proof of the $O(\log n)$ approximation. An important consequence of the Lagrange-based reduction and our inapproximability results is that the robust network design problem with linear reservation cost cannot be approximated within any constant ratio. This answers a long-standing open question of Chekuri. Finally, we show that even if only two given paths are allowed for each commodity, the robust network design problem with minimum congestion or linear costs is hard to approximate within some constant $k$.