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

TL;DR

This paper presents an improved $O( ext{log } n)$-approximation algorithm for the $(p,q)$-Flexible Graph Connectivity problem using independent rounding and a novel LP formulation with knapsack cover inequalities.

Contribution

It introduces a new LP formulation and rounding technique that improves the approximation ratio from $O(q ext{log } n)$ to $O( ext{log } n)$ for the $(p,q)$-FGC problem.

Findings

Achieved an $O( ext{log } n)$-approximation ratio.

Extended the model to multiple safety tiers.

Provided a polynomial-time separation oracle for the LP.

Abstract

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 $\Theta(\log n)$ 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.