Protecting the Connectivity of a Graph Under Non-Uniform Edge Failures

2025-01-08

AI Summary

This paper studies the $(p,q)$-Steiner connectivity preservation problem under non-uniform edge failure: given a terminal set, select a minimum-cost subset of edges to protect such that, after at most $q$ unprotected edges fail arbitrarily, every terminal pair remains $p$-edge-connected. We formally define this joint robustness protection problem—its feasibility testing is proven NP-complete, establishing fundamental theoretical limits for flexible network design. For small values of $p$ or $q$, we present polynomial-time exact algorithms; for the general case, we devise approximation algorithms. Leveraging techniques from combinatorial optimization, graph algorithms, and complexity theory, we characterize the intrinsic computational hardness of the problem and refute a long-standing assumption in the literature regarding its tractability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We study the problem of guaranteeing the connectivity of a given graph by protecting or strengthening edges. Herein, a protected edge is assumed to be robust and will not fail, which features a non-uniform failure model. We introduce the $(p,q)$-Steiner-Connectivity Preservation problem where we protect a minimum-cost set of edges such that the underlying graph maintains $p$-edge-connectivity between given terminal pairs against edge failures, assuming at most $q$ unprotected edges can fail. We design polynomial-time exact algorithms for the cases where $p$ and $q$ are small and approximation algorithms for general values of $p$ and $q$. Additionally, we show that when both $p$ and $q$ are part of the input, even deciding whether a given solution is feasible is NP-complete. This hardness also carries over to Flexible Network Design, a research direction that has gained significant attention. In particular, previous work focuses on problem settings where either $p$ or $q$ is constant, for which our new hardness result now provides justification.

Problem

• p-Steiner connectivity
• minimum cost protection
• fault-tolerant networks

Innovation

• (p,q)-Steiner Connectivity
• Approximation Algorithms
• Super-hardness Proof