if $\delta_D(\hat{X}) \neq \emptyset$ and $r(\hat{X})=0$ otherwise. By the $q$-connectivity version of Menger’s Theorem (c.f. [@KN-sur]), $I \subseteq F$ is a feasible solution to an [SNA]{} instance if, and only if, $|\delta_I(\hat{X})| \geq h(\hat{X})$ for every bisets $\hat{X}$ on $V$, where $h$ is a biset-function defined by $$\label{e:p}
h(\hat{X})= \max\{r(\hat{X})-(q(\Gamma(\hat{X}))+|\delta_G(\hat{X})|),0\}$$ Let ${\cal P}_h$ denote the polytope of “fractional edge-covers” of $h$, namely, $${\cal P}_h=\left\{x \in \mathbb{R}^F:
x\left(\delta_F(\hat{Y})\right) \geq h(\hat{Y}) \ \forall \mbox{ biset } \hat{Y} \mbox{ on } V, \
0 \leq x_e \leq 1 \ \forall e \in F\right\} \ .$$ Let $\tau^*(h)$ denote the optimal value of a standard LP-relaxation for edge covering $h$ by a minimum cost edge set, namely, $\tau^*(h)=\min\left\{\sum_{e \in F} c_e x_e: x \in {\cal P}_h \right\}$. As an intermediate problem, we consider [SNA]{} instances when we seek to increase the connectivity by $1$ for every $uv \in D$, namely, when $r_{uv} = \lambda^q_G(u,v)+1$ for all $uv \in D$.

Given a [$D$-SNA]{} instance, we say that a biset $\hat{X}$ is [*tight*]{} if $h(\hat{X})=1$, where $h$ is defined by (\[e:p\]). [$D$-SNA]{} is equivalent to the problem of finding a minimum cost edge-cover of the biset family ${\cal F}=\{\hat{X}:h(\hat{X})=1\}$ of tight bisets. Thus the following generic 