On the approximability of robust network design

Given the dynamic nature of traffic, we investigate variant robust network design where have to determine capacity reserve on each link so that demand vector belonging a polyhedral set can be routed. The objective is either minimize congestion or linear cost. Routing assumed fractional and (i.e., dependent current traffic vector). We first prove problem with minimum cannot approximated within any constant factor. Then, using ETH conjecture, get Ω(log⁡nlog⁡log⁡n) lower bound for approximability this problem. This implies well-known O(log⁡n) approximation ratio established by Räcke in 2008 tight. Using Lagrange relaxation, obtain new proof approximation. An important consequence Lagrange-based reduction our inapproximability results reservation cost ratio. answers long-standing open question Chekuri (2007). also give another result Goyal et al. (2009) stating optimal under static routing Ω(log⁡n) more expensive than obtained routing. Finally, show even if only two given paths are allowed commodity, hard approximate some constant.