Finding a special cut-set in an weighted undirected graph
This problem is NP-complete.

Here's a simple reduction from MAXCUT:

Given an input graph $G$, we create a new graph $G'=G\cup K_{n^2}$ (i.w. add a separated clique over $n^2$ vertices).

We can now set the weight to be $-1$ for the original graph edges, and $1$ for the edges of the clique.

Claim:
If $G$ has a max cut $(S,\bar S  = V\setminus S)$ such that $|E\cap (S\times \bar S)| = k$, then $G'$ has a minimal "special cut-set" whose value is $|n^2-k-1|$.

Proof:
Since the special cut set's value is required to be positive, $S\cap K_{n^2}$ is neither empty or the entire clique. Since the contribution from the negatively weighted edges of $G$ less than $n^2-1$, we can conclude that $|S\cap K_{n^2}|\in \{1,n-1\}$, and it's contribution to the special cut value is $n^2-1$.

Now the set intersection with $G$ has to be the max cut of $G$, as all of the weights are negative, which contributes another $k$ to the special cut value.