🤖 AI Summary
This paper studies the *F*-augmentation problem in network design—i.e., augmenting a graph to increase the connectivity of a given family *F* of cuts. Building upon Williamson et al.’s 1993 primal-dual 2-approximation algorithm—which applies only to uncrossable cut families—we generalize it to the broader class of *pliable* cut families and introduce the novel concept of *crossing density*, enabling a unified primal-dual analysis framework. Our main contributions are: (1) the first constant-factor approximation algorithm for *F*-augmentation over arbitrary pliable cut families; (2) an improved approximation ratio of 5 for *F*-augmentation when *F* consists of near-minimum cuts, significantly tightening the prior *O*(1/ε) bound; and (3) the first 11-approximation algorithm for the (*p*,3)-flexible graph connectivity problem. All results rely solely on combinatorial optimization techniques and structural characterizations of cuts, yielding theoretical breakthroughs for three fundamental classes of cut-augmentation problems.
📝 Abstract
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.
