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

TL;DR

This paper improves the approximation ratio for the Cover Small Cuts problem from 6 to 5 by analyzing the primal-dual algorithm with a stronger set family notion, enhancing theoretical bounds.

Contribution

It presents a tighter analysis of the primal-dual algorithm, achieving a 5-approximation for the Cover Small Cuts problem using symmetry and structural submodularity.

Findings

Approximation ratio improved from 6 to 5.

Analysis introduces stronger notions of pliable set families.

Provides theoretical bounds for the primal-dual algorithm.

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.