In the Cover Small Cuts problem, we are given a capacitated (undirected) graph $G=(V,E,u)$ and a threshold value $λ$, 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 $λ$ 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.

翻译：在覆盖小割问题中，给定一个带容量的（无向）图$G=(V,E,u)$和阈值$λ$，以及一组端节点在$V$中的连接$L$，每个连接$\ell\in L$具有非负成本；目标是找到一组成本最小的连接，使得每个容量小于$λ$的非平凡割均被某个连接覆盖。Bansal、Cheriyan、Grout和Ibrahimpur（arXiv:2209.11209，Algorithmica 2024）证明了由Williamson、Goemans、Mihail和Vazirani（Combinatorica，1995）提出的WGMV原始对偶算法对于覆盖小割问题实现了16的近似比；他们的分析使用了满足特定组合性质的柔性集族概念。随后，Bansal（arXiv:2308.15714v2，IPCO 2025）以及Nutov（arXiv:2504.03910，MFCS 2025）证明了同一算法实现了6的近似比。我们通过使用一个更强的概念——即满足对称性和结构次模性的柔性集族——证明了同一算法实现了5的近似比。