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

TL;DR

The paper studies $F$-augmentation for pliable cut families, introducing the crossing density concept to analyze primal-dual methods beyond uncrossable families. It proves the WGMV primal-dual algorithm achieves a $(3+\rho)$-approximation for pliable families with crossing density $\rho$, and shows this bound is nearly tight via instances with $\rho=O(|V|)$ that yield a $\Omega(\sqrt{|V|})$ gap, implying the best possible is $O(\sqrt{\rho}). The framework yields improved approximations for concrete problems: a 6-approximation for Small Cut Augmentation under a $\gamma$-pliable, sparse-crossing setting; a 12-approximation for $(p,3)$-FGC when $p$ is even and an $11+\epsilon$-approximation when $p$ is odd; and improved bounds for capacitated $k$-edge-connected spanning subgraphs. Together with refinements of previous pliable-based results (e.g., an 8-approximation for $\mathcal{F}$-augmentation with $\gamma$-pliable families), the work advances constant-factor algorithms for broad network-design problems and highlights the power of crossing-density-aware analysis.

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 6-approximation algorithm for the ${F}$-augmentation problem when ${F}$ is a family of near min-cuts. 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 12-approximation algorithm.

Theorems & Definitions (1)

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