In the Flexible Graph Connectivity (FGC) problem, we are given an undirected multigraph on $n$ vertices with nonnegative edge costs, where each edge is classified as either safe or unsafe. Given integer parameters $p$ and $q$, the goal in $(p,q)$-FGC is to purchase a minimum-cost set of edges such that the resulting spanning subgraph remains $p$-edge-connected after the removal of any set of up to $q$ unsafe edges. Our main contribution is an $O(\log n)$-approximation algorithm based on independent rounding, improving the previous best approximation ratio of $O(q \log n)$. Central to our approach is a new linear programming formulation of feasible solutions that encodes knapsack cover inequalities as cut-capacity constraints. Unlike prior work, the capacity of an edge in a cut may depend on the partially purchased solution for this cut. We show that the resulting linear program admits a polynomial-time separation oracle. Scaling the fractional solution by $Θ(\log n)$ and applying independent rounding yields a feasible integral solution with constant probability; here, we leverage the knapsack cover inequalities to obtain strong concentration bounds for the rounded solution relative to any given partial solution. A key ingredient in both separation and rounding is the use of Karger's bound on the number of near-minimum cuts. We also extend the $(p,q)$-FGC problem to model more than two safety tiers and show that our results and techniques extend naturally to this setting, albeit with increased approximation ratios and running times that scale with the number of tiers.

翻译：在灵活图连通性（FGC）问题中，我们给定一个包含n个顶点、边具有非负代价的无向多重图，其中每条边被分类为安全或不安全。给定整数参数p和q，(p,q)-FGC的目标是购买一组最小代价的边，使得生成的生成子图在移除任意至多q条不安全边后仍保持p边连通。我们的主要贡献是一种基于独立舍入的O(log n)近似算法，将先前的最佳近似比O(q log n)进行了改进。我们方法的核心是一种新的可行解线性规划公式，该公式将背包覆盖不等式编码为割容量约束。与先前工作不同，割中边的容量可能取决于该割的局部购买解。我们证明所得线性规划存在多项式时间分离预言机。将分数解缩放Θ(log n)倍并应用独立舍入，能以常数概率得到可行的整数解；在此过程中，我们利用背包覆盖不等式来获得舍入解相对于任意给定局部解的强集中界。分离和舍入过程中的关键要素是使用Karger关于近最小割数量的界。我们还将(p,q)-FGC问题扩展到超过两个安全层级的模型，并证明我们的结果和技术可自然推广到该设定，尽管近似比和运行时间会随层级数量增加而增大。