I have a min cut problem formulated below.
\begin{aligned} \min_{b\in \mathbb{R}^{|E|}, x\in \mathbb{R}^{|V|}} \quad & \sum_{(i,j)\in E}b_{ij}c_{ij} \\ \text{subject to} \quad \quad & b_{ij} \geq x_i - x_j \quad \text{for all} \; (i,j) \in E \\ & b \geq 0 \\ & x_s - x_t \geq 1 \\ \end{aligned}
Now I am deriving its dual, I've introduced three slack variables $f,y,w$ that corresponding to three constraints in the primal.
What I have achieved so far. \begin{align*} \max_{f,y,w}\min_{b,x}\left( \sum_{(i,j)\in E}b_{ij}c_{ij}+\sum_{(i,j)\in E}f_{ij}(x_i-x_j-b_{ij})-\sum_{(i,j)\in E}y_{ij}b_{ij}+w(1-x_s+x_t)\right)\\ f,y,w \geq 0\\ \max_{f,y,w}\min_{b,x}\left( w+\sum_{(i,j)\in E}b_{ij}c_{ij}+\sum_{(i,j)\in E}f_{ij}(x_i-x_j-b_{ij})-\sum_{(i,j)\in E}y_{ij}b_{ij}+w(-x_s+x_t)\right)\\ =\max_{f,y,w}\min_{b,x}\left( w+\sum_{(i,j)\in E}b_{ij}\left(c_{ij}-f_{ij}-y_{ij}\right)+\sum_{j:(j,i)\in E}x_jf_{ij}-\sum_{j:(i,j)\in E}x_jf_{ij}+(-x_s+x_t)w\right)\\ =\max_{f,y,w}\min_{b,x}\left( w+\sum_{(i,j)\in E}b_{ij}\left(c_{ij}-f_{ij}-y_{ij}\right)+\sum_{j:(j,i)\in E}x_j(f_{ij}-f_{ji})+(-x_s+x_t)w\right)\\ \end{align*}
However I have some difficulty deriving the constraints, what would that be? I've extract all the dual variables at front.
Typically you derive the dual constraints by finding for which f,y,w the $\min_{b,x}$ is $-\infty$. However, since $x$ is binary, that is far from trivial. Additionally, since the problem is not convex (due to the domain of $x$ and $b$), the duality gap is not 0. For these reasons, the dual of an integer linear program is never derived.