= \max\limits_{e \in F} \sum_{uv \in D} [f_{uv}(\{e\})-f_{uv}(\emptyset)]$ in the case of edge costs, and $\alpha = \max\limits_{z \in V} \sum_{uv \in D}[f_{uv}(\delta_F(z))-f_{uv}(\emptyset)]$ in the case of node costs.

In Section \[s:a\] we observe that for [$a$-Based SNA]{} instances, the set-function on $F$ defined by $f_{uv}(I)=\lambda^q_{G+I}(u,v)$ is submodular and non-decreasing, and that $\alpha \leq |D|$ in the case of edge-costs and $\alpha \leq \min\{r(D),|D| \cdot p_{\max}\}$ in the case of node-costs, where $p_{\max}$ is the maximum number of parallel edges in $F$.

\[t:a\] Directed [$a$-Based SNA]{} admits approximation ratios $H(|D|)$ for edge-costs and $H(\min\{r(D),|D| \cdot p_{\max}\})$ for node-costs. Thus directed [SSL]{} admits approximation ratio $H(\min\{r(D),|D| \cdot p_{\max}\})$.

For both [SSL]{} and [SNA]{}, an approximation ratio $\rho$ for directed graphs implies approximation ratio $2\rho$ for undirected graphs. Usually, undirected connectivity problems are easier to approximate than the directed ones. Directed [SSL]{} is already [Set-Cover]{} hard even for $k=1$ and unit costs, but for undirected [SSL]{} with $k=1$ (and even with $k=2$) it is not hard to obtain a polynomial time algorithm. Hence a natural question is whether undirected [SSL]{} admits an approximation ratio that depends on $k$ only. As was mentioned, Fukunaga [@Fuk] observed that $Q$-connectivity [SSL]{} with $Q=V$ is equivalent 