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：针对$k$边连通生成子图（$k$-ECSS）问题，我们提出一种多项式时间算法，该算法返回的解成本不超过同一图中最便宜的$(k+10)$-ECSS成本。我们的方法在迭代松弛框架中引入新要素——幽灵值，使得中间问题具有高度稀疏性。当$(k+10)$-ECSS的最优值接近$k$-ECSS时，我们的保证优于$k$-ECSS目前最佳$2$近似因子。该性质同样适用于密切相关的$k$边连通生成多子图（$k$-ECSM）问题——该问题与$k$-ECSS定义相同，但允许边以相同成本被多次选择。由此，我们得到$k$-ECSM的$\left(1+O\left(\frac{1}{k}\right)\right)$近似算法，解决了Pritchard的猜想，并改进了Karlin、Klein、Oveis Gharan和Zhang近期提出的$\left(1+O\left(\frac{1}{\sqrt{k}}\right)\right)$近似算法。此外，我们给出$k$-ECSM的匹配下界，证明除非$P=NP$，否则我们的近似比在$O\left(\frac{1}{k}\right)$的常数因子内是最优的。