Tight analysis of the primal-dual method for edge-covering pliable set families

A classic result of Williamson, Goemans, Mihail, and Vazirani [STOC 1993: 708-717] states that the problem of covering an uncrossable set family by a min-cost edge set admits approximation ratio $2$, by a primal-dual algorithm with a reverse delete phase. Bansal, Cheriyan, Grout, and Ibrahimpur [ICALP 2023: 15:1-15:19] showed that this algorithm achieves approximation ratio $16$ for a larger class of so called $\gamma$-pliable set families, that have much weaker uncrossing properties. The approximation ratio $16$ was improved to $10$ by the author [WAOA 2025: 151-166]. Recently, Bansal [arXiv:2308.15714] stated approximation ratio $8$ for $\gamma$-pliable families and an improved approximation ratio $5$ for an important particular case of the family of cuts of size $<k$ of a graph $H$, but his proof has an error. We will improve the approximation ratio to $7$ for the former case and give a simple proof of approximation ratio $6$ for the latter case. Furthermore, if $H$ is $\lambda$-edge-connected then we will show a slightly better approximation ratio $6-\frac{1}{\beta+1}$, where $\beta=\left\lfloor\frac{k-1}{\lceil(\lambda+1)/2\rceil}\right\rfloor$. Our analysis is supplemented by examples showing that these approximation ratios are tight for the primal-dual algorithm.