# Integrality gap in bilevel binary linear programming problem

I have a bilevel max-min optimization problem over binary variables, with constraints expressed using linear inequalities. The inner (minimization) problem is \begin{alignat}2 \min\limits_x&\quad c^\top x\\ \text{s.t. }&\quad Ax\geq b\\ &\quad x_i \in \{0,1\}, \end{alignat} with $$c$$ depending linearly on the variables over which maximization happens.

To obtain a single maximization problem, I want to consider the dual of the inner problem. I am unable to verify if the inner program is total dual integral (all I know is that the matrix $$A$$ is not totally unimodular). So I start by considering the relaxation of the inner problem: \begin{align*} \min\limits_x&\quad c^\top x\\ \text{s.t.}&\quad Ax \geq b\\ &\quad x_i \in [0,1], \end{align*} which can be rewritten as \begin{align*} \min\limits_x&\quad c^\top x\\ \text{s.t.}&\quad A'x \geq b'\\ &\quad x_i \geq 0, \end{align*} where $$A'$$ is obtained by appending rows of negated identity matrix to $$A$$, and $$b'$$ by extending $$b$$ with a vector of $$-1$$'s.

Then, I take the dual of the relaxed problem, which is \begin{align*} \max\limits_{\lambda}&\quad b'^\top \lambda\\ \text{s.t.}&\quad A'^\top\lambda \leq c\\ &\quad \lambda \geq 0. \end{align*} Finally, I activate the maximization variables, say $$\mathbf{z}$$, and obtain the program \begin{align*} \max\limits_{\mathbf{z}, \lambda}&\quad b'^\top \lambda\\ \text{s.t.}&\quad A'^\top\lambda \leq c(\mathbf{z})\\ &\quad B z \leq d\\ &\quad \lambda \geq 0, z_i \in \{0,1\}. \end{align*}

Question: Switching to relaxed program introduces integrality gap. How can I check the extent of/minimize its impact on the solution of the final problem?