I came across a question about using an
xor condition in a MIP model. Here is a summary of the answer.
Expressing \(z=x\cdot y\) (or \(z=x\text{ and }y\) ) where \(x,y,z \in \{0,1\}\) as a set of linear constraints is wellknown:
and: 
\[\begin{align}
&z \le x \\
&z \le y \\
&z \ge x+y1
\end{align}\]

\(\>\>\>\Longleftrightarrow\>\>\> \)

\(z\)  \(x\)  \(y\) 

\(0\)  \(0\)  \(0\) 
\(0\)  \(0\)  \(1\) 
\(0\)  \(1\)  \(0\) 
\(1\)  \(1\)  \(1\) 

The relation \(z=x\text{ xor }y\) is slightly more complicated. The xor (exclusiveor) condition can also be written as \(z = x<>y\), i.e. \(z\) is 1 if \(x\) and \(y\) are different (and \(z\) is zero if they are the same). This we can write as the following set of linear inequalities:
xor: 
\[\begin{align}
& z \le x+y\\
& z \ge xy \\
& z \ge yx \\
& z \le 2 x  y
\end{align}\]

\(\>\>\>\Longleftrightarrow\>\>\> \)

\(z\)  \(x\)  \(y\) 

\(0\)  \(0\)  \(0\) 
\(1\)  \(0\)  \(1\) 
\(1\)  \(1\)  \(0\) 
\(0\)  \(1\)  \(1\) 

To be complete \(z=x\text{ or }y\) can be written as:
or: 
\[\begin{align}
& z \le x+y\\
& z \ge x \\
& z \ge y
\end{align}\]

\(\>\>\>\Longleftrightarrow\>\>\> \)

\(z\)  \(x\)  \(y\) 

\(0\)  \(0\)  \(0\) 
\(1\)  \(0\)  \(1\) 
\(1\)  \(1\)  \(0\) 
\(1\)  \(1\)  \(1\) 

In all of this we assumed \(x,y\) and \(z\) are binary variables.
Thanks! You help me a lot!
ReplyDeleteDear colleagues,
ReplyDeleterecently we understood how to represent any boolean function f(x) as system of linear inequalities in the following sense.
Let B^n={(x_1,x_2, ..., x_n): x_k={0,1}} be is a set of ndimensional unit cube's vertices, i.e. the set of all 01 arguments of some boolean function f:B^n > {0,1}. Then there exist a system of a number of linear inequalities of n+1 variables (x,y):
S(x,y)={a_i + l_i(x) + b_i*y >= 0: i={1:m}} such that for any x\in B^n, f(x)=y if and only if S(x,y) holds.
It is important that we do not need to assume that y is a discrete, 01, variable! For any 01 vector x, corresponding y value will be either 0, or 1.
And I can explicitly produce the system S(x,y) for any function f(x) given either by formula or in tabular form.
Can any body tell me any reference on this subject?
I can not believe that it a "fresh result" in math. programming ... But my rather intensive search over literature did not bring any similar assertion...
what about detecting equality or not (xor) between 2 consecutive integer variables? not only binary. thanks
ReplyDelete