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If $x,y \in [1,n]$, how to write $x \neq y$ in integer-linear constraint?

Possible Answer:

$x−y\geq 1−(1−t)\times n$ and $y−x \geq 1−t\times n$ where $t \in \{0,1\}$

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    $\begingroup$ Do you refer to integer linear programming, or linear programming? $\endgroup$
    – D.W.
    Commented Apr 22 at 20:08
  • $\begingroup$ It is not a linear constraint, say $-1 \neq 1$ and $1\neq -1$ but $-1+1 = 1-1$. $=$ and $\geq$ are linear. $\endgroup$
    – Ethan
    Commented Apr 23 at 20:44

1 Answer 1

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With linear programming: you can't.

With integer linear programming, here is one solution: introduce zero-or-one variable $t$, and the inequalities

$$x \le y - 1 + (1-t)n, \qquad x \ge y+1 - tn.$$

Intuition: if $t$ is 1, then we must have $x<y$; if $t$ is 0, then we must have $x>y$.

I have assumed that $x,y$ are integer variables and $n$ is a known constant.

My thanks to @Subhankar Ghosal for an improvement to my initial answer, which I have incorporated here.

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  • $\begingroup$ Can we have the following? $x - y \geq 1 - (1-t)n$ and $y - x \geq 1 - tn$ $\endgroup$ Commented Apr 25 at 12:59
  • $\begingroup$ @SubhankarGhosal, oh good point, that is a better solution. I have revised my answer accordingly. $\endgroup$
    – D.W.
    Commented Apr 25 at 16:56

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