An O(log n)-Approximation Algorithm for (p,q)-Flexible Graph Connectivity via Independent Rounding
Sharat Ibrahimpur, László A. Végh
TL;DR
This work studies the $(p,q)$-Flexible Graph Connectivity problem on graphs with safe and unsafe edges, seeking a minimum-cost edge set that remains $p$-edge-connected after the removal of any $q$ unsafe edges. The authors introduce a strengthened integer program using knapsack-cover inequalities, and show the corresponding linear relaxation can be solved in polynomial time via an efficient separation oracle that leverages near-minimum-cut bounds. They then obtain an $O(\log n)$-approximation by independently rounding an optimal LP solution, with a success probability bounded away from zero and a cost factor of $O(\log n)$ times the LP value. This yields an improvement over the prior $O(q \log n)$-approximation for general $p,q$ and establishes the first LP-based framework for this general problem, while highlighting open questions about integrality gaps and potential iterative rounding improvements.
Abstract
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.