# What is the complexity of Parametric Mixed Integer Linear Programming?

We know $$\forall\bf y\in\mathbb Z^n:K\bf y\leq b$$ $$\exists\bf x\in\mathbb Z^m:A\bf x + B\bf y\leq c$$ is in $$\bf P$$ if $$n,m$$ are fixed from Kannan's result (refer page $$1$$ in reference).

What is the complexity of $$\forall\bf y\in\mathbb Z^n\times\mathbb R^a:K\bf y\leq b$$ $$\exists\bf x\in\mathbb Z^m\times\mathbb R^b:A\bf x + B\bf y\leq c$$ where $$n,m$$ are fixed?

1. Is it still $$2^{\mathsf{poly}(m+n)}\mathsf{poly}(mnab)$$?

2. At least when $$n=0$$ is it $$2^{\mathsf{poly}(m)}\mathsf{poly}(mab)$$?