Improved approximation algorithms for some capacitated $k$ edge connectivity problems

We consider the following two variants of the Capacitated $k$-Edge Connected Subgraph} (Cap-k-ECS) problem. Near Min-Cuts Cover: Given a graph $G=(V,E)$ with edge costs and $E_0 \subseteq E$, find a min-cost edge set $J \subseteq E \setminus E_0$ that covers all cuts with at most $k-1$ edges of the graph $G_0=(V,E_0)$. We obtain approximation ratio $k-\lambda_0+1+\epsilon$, improving the ratio $2\min\{k-\lambda_0,8\}$ of Bansal, Cheriyan, Grout, and Ibrahimpur for $k-\lambda_0 \leq 14$,where $\lambda_0$ is the edge connectivity of $G_0$. $(k,q)$-Flexible Graph Connectivity ($(k,q)$-FGC): Given a graph $G=(V,E)$ with edge costs and a set $U \subseteq E$ of ''unsafe'' edges and integers $k,q$, find a min-cost subgraph $H$ of $G$ such that every cut of $H$ has at least $k$ safe edges or at least $k+q$ edges. We show that $(k,1)$-FGC admits approximation ratio $3.5+\epsilon$ if $k$ is odd (improving the previous ratio $4$), and that $(k,2)$-FGC admits approximation ratio $6$ if $k$ is even and $7+\epsilon$ if $k$ is odd (improving the previous ratio $20$).