problem. Interestingly, the algorithms of [@SMF] for [SL]{} with connectivity functions $\lambda$,$\kappa,\hat{\kappa}$, and the algorithm of [@KN-aug] for [$a$-Based Connectivity Augmentation]{} both use the same method of reducing the problem to a [Submodular Cover]{} problem. Now we see that this is not a coincidence, since by Observations \[o:aug\] and \[o:reduction\], both problems are particular cases of the [$a$-Based SNA]{} problem. Furthermore, we will show by a simple proof that such a reduction is possible whenever the connectivity function is submodular and non-decreasing. A set function $f$ on $U$ is submodular if $f(X)+f(Y) \geq f(X \cap Y)+f(X \cup Y)$ for all $X,Y \subseteq U$, and $f$ is non-decreasing if $f(X) \leq f(Y)$ for all $X \subseteq Y \subseteq U$. Our next result is obtained by observing that the algorithm of [@KN-aug] for [$a$-Based Connectivity Augmentation]{} extends to [$a$-Based NA]{} with arbitrary submodular non-decreasing connectivity function, for both edge-costs and node-costs, as follows. Recall that for an [$a$-Based SNA]{} instance, $p_{\max}$ denotes the maximum number of parallel edges in $F$, and that $p_{\max} \leq k$.

\[t:aug\] If for a directed [$a$-Based SNA]{} instance each function $f_{uv}(I)$ is submodular and non-decreasing, then the problem admits an approximation algorithm with ratio $H(\alpha)$, where $\alpha 