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 R\"{a}cke 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 $k$.

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关于鲁棒网络设计的逼近性

考虑到流量的动态特性，稳健的网络设计问题在于计算每个网络链路上要保留的容量，以便可以路由属于多面体集的任何需求向量。目标是最小化拥塞或线性成本。自由路由取决于需求。我们首先证明具有最小拥塞的稳健网络设计问题不能在任何常数因子内近似。然后，使用 ETH 猜想，我们得到这个问题的近似性的 $\Omega(\frac{\log n}{\log \log n})$ 下界。这意味着 R\"{a}cke 在 2008 年建立的众所周知的 $O(\log n)$ 近似比率是紧的。使用拉格朗日松弛，我们获得了 $O(\log n)$ 的新证明近似。基于拉格朗日的归约和我们的不可逼近性结果的一个重要结果是，具有线性保留成本的稳健网络设计问题无法在任何恒定比率内近似。这回答了 Chekuri 的一个长期悬而未决的问题。最后，我们表明，即使每种商品只允许两条给定的路径，具有最小拥塞或线性成本的稳健网络设计问题也很难在某个常数 $k$ 内近似。