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

TL;DR

This paper addresses the problem of maintaining graph connectivity under non-uniform edge failures by designing algorithms for protecting edges to ensure $p$-edge-connectivity, with complexity results and hardness proofs for general cases.

Contribution

It introduces the $(p,q)$-Steiner-Connectivity Preservation problem, providing polynomial algorithms for small parameters and NP-completeness results for general cases.

Findings

Polynomial-time algorithms for small $p$ and $q$

Approximation algorithms for general $p$ and $q$

NP-completeness of feasibility decision when both $p$ and $q$ are input

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.