A $5$-Approximation Analysis for the Cover Small Cuts Problem

2026-02-01

AI Summary

This study addresses the "Cover Small Cuts" problem in undirected capacitated graphs: selecting a minimum-cost set of edges such that every nontrivial cut with capacity less than a given threshold λ is covered. Building upon the primal-dual algorithm of Williamson, Goemans, Mihail, and Vazirani (WGMV), the authors introduce a refined class of flexible set families that simultaneously exhibit symmetry and structural submodularity. This enhanced characterization enables a tighter approximation analysis, improving the best-known approximation ratio from 6 to 5. The result significantly outperforms prior analyses yielding 6- and 16-approximations, demonstrating a powerful synthesis of combinatorial optimization, submodularity theory, and cut-covering techniques.

Technology Category

Constraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionSearch and Optimization: Combinatorial OptimizationReasoning under Uncertainty: Stochastic Optimization

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In the Cover Small Cuts problem, we are given a capacitated (undirected) graph $G=(V,E,u)$ and a threshold value $\lambda$, as well as a set of links $L$ with end-nodes in $V$ and a non-negative cost for each link $\ell\in L$; the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than $\lambda$ is covered by a link. Bansal, Cheriyan, Grout, and Ibrahimpur (arXiv:2209.11209, Algorithmica 2024) showed that the WGMV primal-dual algorithm, due to Williamson, Goemans, Mihail, and Vazirani (Combinatorica, 1995), achieves approximation ratio $16$ for the Cover Small Cuts problem; their analysis uses the notion of a pliable family of sets that satisfies a combinatorial property. Later, Bansal (arXiv:2308.15714v2, IPCO 2025) and then Nutov (arXiv:2504.03910, MFCS 2025) proved that the same algorithm achieves approximation ratio $6$. We show that the same algorithm achieves approximation ratio $5$, by using a stronger notion, namely, a pliable family of sets that satisfies symmetry and structural submodularity.

Problem

• Cover Small Cuts
• approximation algorithm
• graph connectivity
• primal-dual method
• combinatorial optimization

Innovation

• approximation algorithm
• primal-dual method
• pliable family
• structural submodularity
• Cover Small Cuts