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

2025-01-22

AI Summary

This paper studies the $(p,q)$-flexible graph connectivity problem: given a graph with safe and unsafe edges, select a minimum-cost edge set such that the graph remains $p$-edge-connected after the failure of any $q$ unsafe edges. To address this, we introduce knapsack-cover constraints to strengthen the $(p+q)$-connectivity capacity model. Leveraging Karger’s bound on near-minimum cuts, we design an efficient separation oracle, enabling polynomial-time LP solving. We then apply independent randomized rounding to achieve an $O(log n)$ approximation ratio—first established for arbitrary $p$ and $q$, and strictly improving upon the prior best $O(q log n)$ guarantee. This is the first general-purpose, efficiently solvable approximation algorithm for the problem with stronger theoretical guarantees.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $O(qlog n)$. For both separation and rounding, a key insight is to use Karger's bound on the number of near-minimum cuts.

Problem

• (p,q)-resilient connectivity
• minimum cost edge set
• graph connectivity

Innovation

• Independent Rounding
• Logarithmic Efficiency
• Randomized Rounding