Let $X,Y$ be random variables over finite sets $\mathcal{X}, \mathcal{Y}$. In the problem below, $W_{Y|X}$ and $\tilde{W}_{Y|X}$ are conditional probability distribution. The set of all conditional probability distributions is given by $\mathcal{P(Y|X)}$. I have the following linear program.
\begin{aligned} S(W, c)= \inf\ & \gamma \\ {s.t.}\ & p_{XY}(x,y)\geq 0\ \quad \forall(x,y)\in\mathcal{X}\times\mathcal{Y} \\ & p_{X}(x) = \sum_{y\in\mathcal{Y}}p_{XY}(x,y)\leq \gamma \quad \forall x\in\mathcal{X}\\ & p_{XY}(x,y)\geq \tilde{W}_{{Y}|{X}}(y|x) - W_{{Y}|{X}}(y|x) \quad\forall (x,y)\in\mathcal{X}\times\mathcal{Y}\\ & \tilde{W}_{{Y}|{X}}\in\mathcal{P}(\mathcal{Y}|\mathcal{X})\\ & \tilde{W}_{{Y}|{X}}(y|x) \leq \zeta(y) \quad\forall (x,y)\in\mathcal{X}\times\mathcal{Y}\\ & \sum_{y\in\mathcal{Y}} \zeta(y) = c \end{aligned}
I am able to solve this numerically for any conditional probability distribution $W_{Y|X}$ and nonnegative real number $c$. I am now given that $W_{Y|X}$ is of the form
$$\begin{pmatrix} 1-\delta & \delta \\ \delta & 1-\delta \end{pmatrix}^{\otimes n}$$
I am trying to run this numerically for large $n$ but this becomes intractable very quickly since I have exponentially many constraints as a function of $n$.
How can I exploit the symmetry of the problem due to the tensor product structure of $W_{Y|X}$ to simplify this linear program?