Boolean expressions are found in many MIP models. AND and OR are the most common. When we have an expression like \(x {\bf{\ and\ }} y\) or \(x {\bf{\ or\ }} y\), where all variables are binary, we typically reformulate this as a set of inequalities. XOR is a bit more exotic, but I have never seen a usage for XNOR. Until now [1].
AND
First let's cover some more well-known boolean expression: AND and OR. The AND operator can be defined by a so-called truth-table [2]:
| \(x\) | \(y\) | \(x{\bf\ and\ }y\) |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
The construct \(z = x {\bf\ and \ }y\) can be written as three inequalities: \[\begin{align} &z \le x \\ & z \le y \\ & z \ge x+y-1\end{align}\] These constraints can be interpreted as: \[\begin{align} &z \le x &&:&&x=0\implies z=0 \\ & z \le y &&:&& y=0\implies z=0 \\ & z \ge x+y-1 &&:&& x=y=1 \implies z=1\end{align}\] Depending on the objective or the constraint, we may drop the \(\le\) or \(\ge\) constraints.
OR
The OR operator works like:
| \(x\) | \(y\) | \(x{\bf\ or\ }y\) |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
The inequalities that describe this are: \[\begin{align} & z \ge x \\ & z \ge y \\ & z \le x+y \end{align}\] These constraints implement the implications \[\begin{align} &x=1 \implies z=1 \\ & y=1 \implies z=1 \\ & x=y=0 \implies z=0\end{align}\].
XOR
Before working on XNOR, let's do XOR (exclusive or). We need:
| \(x\) | \(y\) | \(x{\bf\ xor\ }y\) |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
This is a bit more difficult. We need four inequalities: \[\begin{align} & z \le x+y \\& z\ge y-x \\ & z\ge x-y \\ & z \le 2-x-y\end{align}\] They implement the implications: \[\begin{align} & x=y=0 \implies z=0 \\ & x=0,y=1 \implies z=1 \\ & x=1,y=0 \implies z=1 \\ & x=y=1 \implies z=0 \end{align}\]
XNOR
XNOR is an XOR followed by a NOT: \[z = x {\bf\ xnor \ } y = {\bf not\ }(x {\bf\ xor \ } y)\] I.e.
| \(x\) | \(y\) | \(x{\bf\ xor\ }y\) | \(x{\bf\ xnor\ }y\) |
|---|---|---|---|
| 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
Obviously, we can write this as: \[\begin{align} & 1-z \le x+y \\& 1-z\ge y-x \\ & 1-z\ge x-y \\ & 1-z \le 2-x-y\end{align}\] Totally optionally we can re-arrange this as: \[\begin{align} & z \ge 1-x-y \\& z\le 1+x-y \\ & z\le 1-x+y \\ & z \ge x+y-1\end{align}\] Again, we can easily interpret these constraints as: \[\begin{align} & x=y=0 \implies z=1 \\ & x=0,y=1\implies z=0 \\ & x=1,y=0\implies z=0 \\ & x=y=1 \implies z=1\end{align}\]
Notes
Quadratic models can have these operations in disguise. E.g. \(x\cdot y\) is the same as \(x {\bf\ and\ }y\) for binary variables \(x,y\). \((x-y)^2 \) is the same as \(x {\bf\ xor\ }y\).
In many cases we can drop the \(\le\) or \(\ge\) constraints. Some examples are:
- An objective like \(\max \sum_{i,j} (x_i {\bf\ xor\ }x_j)\) allows us to write [3]: \[\begin{align}\max&\sum_{i,j} z_{i,j} \\ & z_{i,j} \le x_i+x_j \\ & z_{i,j} \le 2- x_i-x_j \\& x_i, z_{i,j} \in \{0,1\}\end{align}\]
- A constraint like \(\sum_{i,j} x_i\cdot x_j \le a\) can be written as: \[\begin{align}&\sum_{i,j} z_{i,j} \le a\\ & z_{i,j} \ge x_i+x_j-1 \\ & x_i,z_{i,j} \in \{0,1\}\end{align}\]