to [$s$-Based Rooted SNA]{} with edge-costs. Since undirected $Q$-connectivity [Rooted SNA]{} with edge-costs admits ratio $O(k\ln k)$ [@N-focs], so is $Q$-connectivity [SSL]{} with $Q=V$. However, [Connectivity Augmentation]{} admits ratio $O(\ln^2k)$ for rooted requirements [@N-aug], and a natural question to ask is whether the algorithm of [@N-aug] extends to [$s$-Based Rooted SNA]{}.

\[t:main’\] Undirected [$a$-Based SNA]{} admits the following approximation ratios, where $H(k)=\sum_{i=1}^k 1/i$ denotes the $k$th Harmonic number.

-   For edge-costs, $\sum_{\ell=1}^k \frac{H(\Delta_\ell)}{k-\ell+1} \leq H(k) \cdot H(\Delta_k)=O(\ln^2k)$, where $\Delta_\ell={(4\ell-3)}^2$; [$s$-Based Rooted SNA]{} admits ratio $\sum_{\ell=1}^k \frac{H(2\ell-1)}{k-\ell+1} \leq H(k) \cdot H(2k-1)$.

-   For node-costs, $\sum_{\ell=1}^k H(\Delta_\ell) \min\left\{\frac{p_{\max}}{k-\ell+1},1\right\} \leq
    \min\{k,p_{\max} H(k)\} \cdot H(\Delta_k)$; [$s$-Based Rooted SNA]{}, and thus also undirected [SSL]{}, admits ratio\
    $\sum_{\ell=1}^k H(2\ell-1) \min\left\{\frac{p_{\max}}{k-\ell+1},1\right\} \leq 
    \min\{k,p_{\max} \cdot H(k)\} \cdot H(2k-1)$.

Part (ii) implies that undirected [SSL]{} admits approximation ratio that depends on $k$ only; this ratio is better than the one implied by Theorem \[t:a\] for $k=o\left(\frac{\ln n}{\ln \ln n}\right)$. Furthermore, it improves the ratio $O(k \ln k)$ of Fukunaga [@Fuk] for $\kappa'$-[SSL]{} to $O(\ln^2 k)$, since we have $p_{\max}=1$ in this case (for $\hat{\kappa}$-[SSL]{} we get the same ratio $O(k \ln k)$ as in [@Fuk], since $p_{\max}=k$ in this case).

Finally, we consider the [SSL with Flow-Cost Bounds]{} problem.

