An O(log n)-Approximation Algorithm for (p,q)-Flexible Graph Connectivity via Independent Rounding

In the $(p,q)$-Flexible Graph Connectivity problem, the input is a graph $G = (V,E)$ with the edge set $E = \mathscr{S} \cup \mathscr{U}$ partitioned into safe and unsafe edges, and the goal is to find a minimum cost set of edges $F$ such that the subgraph $(V,F)$ remains $p$-edge-connected after removing any $q$ unsafe edges from $F$. We give a new integer programming formulation for the problem, by adding knapsack cover constraints to the $p(p+q)$-connected capacitated edge-connectivity formulation studied in previous work, and show that the corresponding linear relaxation can be solved in polynomial time by giving an efficient separation oracle. Further, we show that independent randomized rounding yields an $O(\log n)$-approximation for arbitrary values of $p$ and $q$, improving the state-of-the-art $O(q\log n)$. For both separation and rounding, a key insight is to use Karger's bound on the number of near-minimum cuts.