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

TL;DR

The paper addresses preserving $p$-edge-connectivity between terminal pairs under up to $q$ unprotected edge failures by protecting a minimum-cost edge set, formalizing the $(p,q)$-Steiner-Connectivity Preservation problem. It develops polynomial-time exact algorithms for small parameters $(p,q)$, and broad, scalable approximation algorithms for general values, including an $O((p+q)\, ext{log}\,p)$-approximation when $p$ is constant and an $O( ext{log}\,p\, ext{min}\{p+q, ext{log} ext{ } ight ext{)}$-approximation for the global variant. The work also establishes hardness results when both $p$ and $q$ are part of the input, and connects these problems to related frameworks such as Flexible Network Design and bulk-robust survivability. Methodologically, it leverages cut-based formulations, cycle and tree decompositions, and primal-dual augmentation to derive both exact and approximate solutions. The findings advance survivable network design under non-uniform failures and provide practical augmentation strategies with broad implications for infrastructure resilience and network design research.

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.

Figures (4)

• Figure 1: Illustration of the decomposition (\ref{['decomposition']}, \ref{['lemma: decomposition']}).
• Figure 2: Illustration of \ref{['lem:global22:3-connected']}: Equivalence to solving two new instances on $G_1 \cup \{(u_1,v_1)\}$ where $(u_1,v_1)$ is an edge with zero cost and $G_2 \cup \{(u_2,v_2)\}$ where $\{(u_2,v_2)\}$ has zero cost.
• Figure 3: Illustration of subproblems: consider the red path $(v_1,v_2,v_3=z_i,v_4=v,v_5=z_j,v_6,v_7)$ through $v$. If we select the red path, the subtrees $T_{z_i}$ and $T_{z_j}$ break into $T_{v_1}, T_{(v_1,v_2)} = T_{v_2}\setminus T_{v_1}, T_{(z_i, v_2)} = T_{z_i}\setminus T_{v_2}$ and $T_{(z_j,v_6)}=T_{z_j} \setminus T_{v_6}, T_{(v_6,v_7)}=T_{v_6} \setminus T_{v_7}, T_{v_7}$, respectively. They define independent subproblems and their optimal solutions have been computed before we compute $f(T_v)$.
• Figure 4: The reduction: the blue edges are protected edges and the black edges are unprotected.

Theorems & Definitions (41)

• Theorem 1: summarized
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 4
• proof
• Theorem 5
• proof
• Theorem 6
• proof