Is there a better way of defining a constraint on positive integer variables such that no two variables are the same and are uniquely assigned a value

Question

So suppose I have integer variables $x_1,x_2,\dots,x_N$ and I enforce that the integer variables are bounded i.e $1 \leq x_i \leq N$

I was interested in posing a constraint so that in the collection $\{ x_i\}$ I would see all values $1,\dots,N$. This is to say that each variable $x_i$ is assigned uniquely a value in the interval $1,\dots,N$.

One way to do this constraint I thought was to just do:

$$\prod_{i=1}^{N} x_i = N! = N(N-1)(N-2)\cdots(2)(1)$$

I believe this fits the bill, but is there a better way of how to pose this constraint? I would suspect integer solvers would have difficulty with this constraint - perhaps there is a better formulation? I could not figure out a straightforward linear representation, and would be interested in seeing if one exists. I know from linear programming I could do $|x_i-x_j| =|e_{i,j}|\geq 1, i\neq j$, but I think this eventually will lead to a similar constraint $e^+e^-$=0.

Answer 1

Introduce binary variables $y_{ij}\in \{0,1\}$ that take value $1$ if and only if $x_i$ is assigned to value $j\in \{1,...,N\}$, and use the following constraints: \begin{align} x_i &= \sum_{j=1}^N j \cdot y_{ij} \quad &\forall i \in \{1,...,N\}\\ \sum_{j=1}^N y_{ij} &= 1 \quad &\forall i \in \{1,...,N\}\\ \sum_{i=1}^N y_{ij} &= 1 \quad &\forall j \in \{1,...,N\} \\ \end{align}

Answer 2

This is called an alldifferent constraint and is supported directly by constraint programming solvers. @Kuifje gave a traditional linear formulation (+1) that uses binary variables, but it is worth pointing out that there is another linear formulation that uses only the original variables. See Theorem 19 here: https://www.andrew.cmu.edu/user/vanhoeve/papers/alldiff.pdf