A Global Analysis of the Primal-Dual Method for Pliable Families

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.