problem includes the [$D$-SNA]{} problem.

For a biset-family ${\cal F}$ let $\tau^*({\cal F})$ denote the optimal value of a standard LP-relaxation for edge covering ${\cal F}$ by a minimum cost edge set, namely, $\tau^*({\cal F})=\tau^*(h)$ for $h(\hat{X})=1$ if $\hat{X} \in {\cal F}$ and $h(\hat{X})=0$ otherwise.

\[p:scale\] Suppose that [$a$-Based $D$-SNA]{} with edge-costs admits a polynomial time algorithm that computes a solution of cost at most $\rho(k) \tau^*({\cal F})$, where ${\cal F}$ is the family of tight bisets. Then [$a$-Based SNA]{} admits a polynomial time algorithm that computes a solution $I$ such that:

-   For edge-costs, $c(I) \leq \tau^*(h) \cdot \sum_{\ell=1}^k \frac{\rho(\ell)}{k-\ell+1}$, where $h$ is defined by (\[e:p\]).

-   For node-costs, $c(I) \leq {\sf opt} \cdot \sum_{\ell=1}^k \rho(\ell) \cdot \min\left\{\frac{p_{\max}}{k-\ell+1},1\right\}$.

We start with the edge-costs case. Consider the following sequential algorithm. Start with $I=\emptyset$. At iteration $\ell=1, \ldots,k$, add to $I$ and remove from $F$ an edge-set $I_\ell \subseteq F$ that increases by $1$ the $q$-connectivity of $G+I$ on the set of demand edges $D_\ell=\{sv:\lambda_{G+I}^q(s,v)=r(s,v)-k+\ell-1, sv \in D\}$, by covering the corresponding biset-family ${\cal F}_\ell$ using the $\rho$-approximation algorithm. After iteration $\ell$, we have $\lambda_{G+I}^q(s,v) \geq r(s,v)-k+\ell$ for all $sv \in D$. Consequently, after $k$ iterations $\lambda_{G+I}^q(s,v) \geq r(s,v)$ holds for 