We give a poly-time algorithm for the $k$-edge-connected spanning subgraph ($k$-ECSS) problem that returns a solution of cost no greater than the cheapest $(k+10)$-ECSS on the same graph. Our approach enhances the iterative relaxation framework with a new ingredient, which we call ghost values, that allows for high sparsity in intermediate problems. Our guarantees improve upon the best-known approximation factor of $2$ for $k$-ECSS whenever the optimal value of $(k+10)$-ECSS is close to that of $k$-ECSS. This is a property that holds for the closely related problem $k$-edge-connected spanning multi-subgraph ($k$-ECSM), which is identical to $k$-ECSS except edges can be selected multiple times at the same cost. As a consequence, we obtain a $\left(1+O\left(\frac{1}{k}\right)\right)$-approximation algorithm for $k$-ECSM, which resolves a conjecture of Pritchard and improves upon a recent $\left(1+O\left(\frac{1}{\sqrt{k}}\right)\right)$-approximation algorithm of Karlin, Klein, Oveis Gharan, and Zhang. Moreover, we present a matching lower bound for $k$-ECSM, showing that our approximation ratio is tight up to the constant factor in $O\left(\frac{1}{k}\right)$, unless $P=NP$.

翻译：我们针对$k$-边连通生成子图（$k$-ECSS）问题提出一个多项式时间算法，其返回解的成本不超过同一图上最廉价$(k+10)$-ECSS的成本。我们的方法在迭代松弛框架中引入了一个新要素——称为幽灵值——从而允许中间问题具有高度稀疏性。当$(k+10)$-ECSS的最优值接近$k$-ECSS的最优值时，我们的保证改进了$k$-ECSS已知最佳近似因子$2$。这一性质同样适用于密切相关的$k$-边连通生成多重子图（$k$-ECSM）问题，该问题与$k$-ECSS完全相同，只是边可以以相同成本多次选择。由此，我们得到$k$-ECSM的一个$\left(1+O\left(\frac{1}{k}\right)\right)$-近似算法，这解决了Pritchard的一个猜想，并改进了Karlin、Klein、Oveis Gharan和Zhang最近提出的$\left(1+O\left(\frac{1}{\sqrt{k}}\right)\right)$-近似算法。此外，我们给出了$k$-ECSM的一个匹配下界，表明除非$P=NP$，否则我们的近似比在$O\left(\frac{1}{k}\right)$的常数因子意义下是紧的。