We study a core algorithmic problem in network design called F-augmentation that involves increasing the connectivity of a given family of cuts F. Over 30 years ago, Williamson et al. (STOC `93) provided a 2-approximation primal-dual algorithm when 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 novel 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 F-augmentation of general pliable families. We also improve on the results in Bansal et al. (ICALP `23) by providing a 5-approximation algorithm for the F-augmentation problem when 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 an 11-approximation algorithm.

翻译：我们研究了网络设计中的一个核心算法问题，称为F-增广，该问题涉及提升给定割族F的连通性。三十多年前，Williamson等人（STOC '93）针对F为所谓不可交叉族的情形提出了2-近似原始对偶算法，但将其结果推广至非不可交叉族的问题始终是极具挑战性的课题。本文引入了集合族交叉密度的创新概念，并展示了这一概念如何为分析原始对偶算法开辟全新路径。我们研究了可延展族——这是Bansal等人（ICALP '23）提出的不可交叉族的严格推广形式，并为一般可延展族的F-增广问题提供了首个近似算法。通过运用交叉密度概念，我们还改进了Bansal等人（ICALP '23）的结果，针对F为近最小割族的情形提出了F-增广问题的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，我们研究了$(p,3)$-柔性图连通问题。通过精细分析可行解的结构并运用本文开发的技术，我们为该问题提出了首个常数因子近似算法，展示了11-近似算法的构造。