  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. Recently, Bansal, Cheriyan, Grout, and Ibrahimpur
[ICALP 2023: 15:1-15:19] showed that this algorithm achieves approximation
ratio $16$ for a larger class of set families, that have much weaker uncrossing
properties. In this paper we will refine their analysis and show an
approximation ratio of $10$. This also improves approximation ratios for
several variants of the Capacitated $k$-Edge Connected Spanning Subgraph
problem.
