Minimum number of binary integer variables to handle $AND$ and $OR$ implications in Mixed Integer Linear Programming?

Question

Suppose I want to have an integer program for handling the cases

1. $(x_1>1)\wedge(x_2>1)\wedge(x_3>1)\wedge\dots\wedge(x_n>1)\implies\delta=1$

2. $(x_1>1)\vee(x_2>1)\vee(x_3>1)\vee\dots\vee(x_n>1)\implies\delta=1$

3. $(x_1>1)\wedge(x_2>1)\wedge(x_3>1)\wedge\dots\wedge(x_n>1)\iff\delta=1$

4. $(x_1>1)\vee(x_2>1)\vee(x_3>1)\vee\dots\vee(x_n>1)\iff\delta=1$

how many number of integer variables are needed to handle case?

Is it possible at least one of them needs at most a constant number of binary variables?

Answer 1

The following formulations assume that $\delta \in \{0,1\}$ and $u_i$ is an upper bound on $x_i$.

For #1, introduce $n$ binary variables $y_i$ and linear constraints: \begin{align} x_i - 1 &\le (u_i - 1) y_i &&\text{for $i=1,\dots,n$}\\ \sum_{i=1}^n y_i - n + 1 &\le \delta \end{align}

For #2, you do not need any additional variables: $$x_i - 1 \le (u_i - 1) \delta \quad \text{for $i=1,\dots,n$}$$

For #3 and #4, you need to consider @prubin's question about $\epsilon$.