We study a core algorithmic problem in network design called $\mathcal{F}$-augmentation that involves increasing the connectivity of a given family of cuts $\mathcal{F}$. Over 30 years ago, Williamson et al. (STOC `93) provided a 2-approximation primal-dual algorithm when $\mathcal{F}$ is a so-called uncrossable family but extending their results to families that are non-uncrossable has remained a challenging question. In this paper, we introduce the concept of the crossing density of a set family and show how this opens up a completely new approach to analyzing primal-dual algorithms. We study pliable families, a strict generalization of uncrossable families introduced by Bansal et al. (ICALP `23), and provide the first approximation algorithm for $\mathcal{F}$-augmentation of such families based on the crossing density. We also improve on the results in Bansal et al. (ICALP `23) by providing a 5-approximation algorithm for the $\mathcal{F}$-augmentation problem when $\mathcal{F}$ is a family of near min-cuts using the concept of crossing densities. This immediately improves approximation factors for the Capacitated Network Design Problem. Finally, we study the $(p,3)$-flexible graph connectivity problem. By carefully analyzing the structure of feasible solutions and using the techniques developed in this paper, we provide the first constant factor approximation algorithm for this problem exhibiting a 12-approximation algorithm.

翻译：我们研究网络设计中的一个核心算法问题——$\mathcal{F}$-增广问题，该问题涉及提升给定割族$\mathcal{F}$的连通性。三十多年前，Williamson等人（STOC `93）在$\mathcal{F}$为所谓不可交族时提出了2-近似原始对偶算法，但将其结果推广至非不可交族一直是悬而未决的难题。本文引入集合族交叉密度的概念，并展示该概念如何为分析原始对偶算法开辟全新路径。我们研究可延展族——这是Bansal等人（ICALP `23）提出的不可交族的严格推广，并基于交叉密度为此类族的$\mathcal{F}$-增广问题首次给出近似算法。通过运用交叉密度概念，我们改进了Bansal等人（ICALP `23）的结果，为$\mathcal{F}$是近最小割族时的$\mathcal{F}$-增广问题提供了5-近似算法。该结果直接提升了容量网络设计问题的近似因子。最后，我们研究$(p,3)$-柔性图连通问题。通过精细分析可行解结构并运用本文开发的技术，我们首次为该问题给出常数因子近似算法，展示了12-近似算法。