Tight Guarantees for Cut-Relative Survivable Network Design via a Decomposition Technique

2025-07-06

AI Summary

Traditional Survivable Network Design Problems (SNDPs) suffer from limitations in modeling fault tolerance, particularly in capturing relative robustness against structural failures. To address this, we introduce the Cut-Relative SNDP (CR-SNDP), which requires that for every cut, the number of selected edges does not exceed the number of edges crossing that cut in the original graph, while satisfying a given weakly supermodular cut-demand function. Crucially, under this constraint, the effective demand function ceases to be weakly supermodular, rendering standard approximation techniques inapplicable. Method: We propose a novel algorithmic framework based on linear programming relaxation and decomposition over tight constraint families. Contribution/Results: Our approach yields the first tight 2-approximation algorithm for CR-SNDP. We further establish NP-hardness of CR-SNDP via a new reduction and prove a matching lower bound of 2 on the approximability ratio. These results collectively extend the theoretical boundaries of SNDP and introduce a paradigm shift toward relative robustness modeling for fault-tolerant networks.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In the classical emph{survivable-network-design problem} (SNDP), we are given an undirected graph $G = (V, E)$, non-negative edge costs, and some $(s_i,t_i,r_i)$ tuples, where $s_i,t_iin V$ and $r_iinmathbb{Z}_+$. We seek a minimum-cost subset $H subseteq E$ such that each $s_i$-$t_i$ pair remains connected even if any $r_i-1$ edges fail. It is well-known that SNDP can be equivalently modeled using a weakly-supermodular emph{cut-requirement function} $f$, where we seek a minimum-cost edge-set containing at least $f(S)$ edges across every cut $S subseteq V$. Recently, Dinitz et al. proposed a variant of SNDP that enforces a emph{relative} level of fault tolerance with respect to $G$, where the goal is to find a solution $H$ that is at least as fault-tolerant as $G$ itself. They formalize this in terms of paths and fault-sets, which gives rise to emph{path-relative SNDP}. Along these lines, we introduce a new model of relative network design, called emph{cut-relative SNDP} (CR-SNDP), where the goal is to select a minimum-cost subset of edges that satisfies the given (weakly-supermodular) cut-requirement function to the maximum extent possible, i.e., by picking $min{f(S),|δ_G(S)|}$ edges across every cut $Ssubseteq V$. Unlike SNDP, the cut-relative and path-relative versions of SNDP are not equivalent. The resulting cut-requirement function for CR-SNDP (as also path-relative SNDP) is not weakly supermodular, and extreme-point solutions to the natural LP-relaxation need not correspond to a laminar family of tight cut constraints. Consequently, standard techniques cannot be used directly to design approximation algorithms for this problem. We develop a emph{novel decomposition technique} to circumvent this difficulty and use it to give a emph{tight $2$-approximation algorithm for CR-SNDP}. We also show new hardness results for these relative-SNDP problems.

Problem

• Develops cut-relative SNDP model for fault-tolerant networks
• Addresses non-weakly-supermodular cut-requirement function challenges
• Provides tight 2-approximation algorithm via decomposition technique

Innovation

• Decomposition technique for cut-relative SNDP
• Tight 2-approximation algorithm developed
• Handles non-weakly-supermodular cut-requirements