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提出的长期悬而未决的开放性问题，这些问题涉及通过网络设计问题中的原始-对偶方法设计近似算法（Combinatorica 15(3):435-454, 1995）。Williamson等人证明了对于连通性增强问题的近似保证为二，其中连通性需求可以由所谓的不可交叉函数规定。他们指出：“将我们的算法扩展到处理非不可交叉函数仍然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。这一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的近似算法方面的能力。”我们的主要结果证明，Williamson等人的原始-对偶算法对于一类推广了不可交叉函数概念的函数实现了16的近似比。存在我们的方法能够处理的实例，其中没有一个最优对偶解具有层状支撑。我们给出了主要结果的三个应用。（1）一个16-近似算法，用于增强图$G$的一族小割集。（2）一个$16 \cdot {\lceil k/u_{min} \rceil}$-近似算法，用于Cap-$k$-ECSS问题，该问题如下：给定一个无向图$G = (V,E)$，边成本$c \in \mathbb{Q}_{\geq 0}^E$和边容量$u \in \mathbb{Z}_{\geq 0}^E$，找到边集$F\subseteq E$的一个最小成本子集，使得$(V,F)$中任何割的容量至少为$k$；我们用$u_{min}$表示$E$中一条边的最小容量。（3）一个$O(1)$-近似算法，用于$(p,2)$-灵活图连通性模型。