We address long-standing open questions raised by Williamson, Goemans, Vazirani and Mihail pertaining to the design of approximation algorithms for problems in network design via the primal-dual method (Combinatorica 15(3):435-454, 1995). Williamson et al. prove an approximation guarantee of two for connectivity augmentation problems where the connectivity requirements can be specified by so-called uncrossable functions. They state: ``Extending our algorithm to handle non-uncrossable functions remains a challenging open problem. The key feature of uncrossable functions is that there exists an optimal dual solution which is laminar. This property characterizes uncrossable functions\dots\ A larger open issue is to explore further the power of the primal-dual approach for obtaining approximation algorithms for other combinatorial optimization problems.'' Our main result proves that the primal-dual algorithm of Williamson et al. achieves an approximation ratio of 16 for a class of functions that generalizes the notion of an uncrossable function. There exist instances that can be handled by our methods where none of the optimal dual solutions has a laminar support. We present three applications of our main result. (1) A 16-approximation algorithm for augmenting a family of small cuts of a graph $G$. (2) A $16 \cdot {\lceil k/u_{min} \rceil}$-approximation algorithm for the Cap-$k$-ECSS problem which is as follows: Given an undirected graph $G = (V,E)$ with edge costs $c \in \mathbb{Q}_{\geq 0}^E$ and edge capacities $u \in \mathbb{Z}_{\geq 0}^E$, find a minimum-cost subset of the edges $F\subseteq E$ such that the capacity of any cut in $(V,F)$ is at least $k$; we use $u_{min}$ to denote the minimum capacity of an edge in $E$. (3) An $O(1)$-approximation algorithm for the model of $(p,2)$-Flexible Graph Connectivity.

翻译：我们解决了Williamson、Goemans、Vazirani和Mihail提出的关于通过网络设计问题的原始-对偶方法设计近似算法的长期开放问题（Combinatorica 15(3):435-454, 1995）。Williamson等人证明了对于连通性要求可由所谓不可交叉函数指定的连通性增强问题，近似保证为2。他们指出：“将我们的算法扩展到处理非不可交叉函数仍是一个具有挑战性的开放问题。不可交叉函数的关键特征在于存在一个最优对偶解是层状的。该性质刻画了不可交叉函数……一个更大的开放问题是进一步探索原始-对偶方法在获得其他组合优化问题近似算法方面的能力。”我们的主要结果证明，对于一类推广不可交叉函数概念的函数，Williamson等人的原始-对偶算法可实现16的近似比。存在某些实例可通过我们的方法处理，其中没有一个最优对偶解具有层状支撑集。我们给出主要结果的三个应用：（1）对于增强图$G$中一族小割的16-近似算法。（2）对于Cap-$k$-ECSS问题的$16 \cdot {\lceil k/u_{min} \rceil}$-近似算法，该问题如下：给定无向图$G = (V,E)$，边成本$c \in \mathbb{Q}_{\geq 0}^E$和边容量$u \in \mathbb{Z}_{\geq 0}^E$，寻找边的最小成本子集$F\subseteq E$使得$(V,F)$中任何割的容量至少为$k$；我们用$u_{min}$表示$E$中边的最小容量。（3）对于$(p,2)$-灵活图连通性模型的$O(1)$-近似算法。