We introduce and study the problem Flexible Graph Connectivity, which in contrast to many classical connectivity problems features a non-uniform failure model. We distinguish between safe and unsafe resources and postulate that failures can only occur among the unsafe resources. Given an undirected edge-weighted graph and a set of unsafe edges, the task is to find a minimum-cost subgraph that remains connected after removing at most k unsafe edges. We give constant-factor approximation algorithms for this problem for \(k = 1\) as well as for unit costs and \(k \ge 1\). Our approximation guarantees are close to the known best bounds for special cases, such as the 2-edge-connected spanning subgraph problem and the tree augmentation problem. Our algorithm and analysis combine various techniques including a weight-scaling algorithm, a charging argument that uses a variant of exchange bijections between spanning trees and a factor revealing min–max–min optimization problem.

我们引入并研究了“柔性图连通性”问题，与许多经典连通性问题相反，该问题具有非均匀故障模型。我们区分安全资源和不安全资源，并假定只能在不安全资源之间发生故障。给定一个无向的边加权图和一组不安全的边，任务是要找到一个最小成本的子图，该子图在最多删除k个不安全的边后仍保持连接。对于这个问题，我们针对\（k = 1 \）以及单位成本和\（k \ ge 1 \）给出常数因子近似算法。。对于特殊情况，例如2边连接生成子图问题和树扩充问题，我们的近似保证接近已知的最佳范围。我们的算法和分析结合了多种技术，包括权重缩放算法，使用生成树之间交换双射的变体的计费参数以及揭示最小-最大-最小优化问题的因子。