Consider the NQueens problem: How can N queens be placed on an NxN chessboard so that no two of them attack each other? [ Solve it on Google Colab! ]
 Constraint
alldiff
enforces a set of integer variables to take distinct values.
s.t. OneJobPerMachine:
alldiff {j in JOBS} MachineForJob[j];

alldiff
can be used conditionally:
s.t. VisitOnce {j in GUESTS}:
IsHost[j] = 0 ==> alldiff {t in TIMES} Visit[j,t];
 Older MIP drivers need a manual reformulation of
alldiff
with binary variables:
s.t. OneMachinePerJob {j in JOBS}:
sum {k in MACHINES} Assign[j,k] = 1;
s.t. OneJobPerMachine {k in MACHINES}:
sum {j in JOBS} Assign[j,k] = 1;
Back to the NQueens problem
New MP Librarybased, as well as Constraint Programming drivers, accept alldiff
directly:
param n integer > 0; # N queens
var Row {1..n} integer >= 1 <= n;
s.t. row_attacks: alldiff ({j in 1..n} Row[j]);
s.t. diag_attacks: alldiff ({j in 1..n} Row[j]+j);
s.t. rdiag_attacks: alldiff ({j in 1..n} Row[j]j);
A reformulated model for older MIP drivers:
param n integer > 0;
var X{1..n, 1..n} binary;
# X[i,j] is one if there is a queen at (i,j); else zero
s.t. column_attacks {j in 1..n}:
sum {i in 1..n} X[i,j] = 1;
s.t. row_attacks {i in 1..n}:
sum {j in 1..n} X[i,j] = 1;
s.t. diagonal1_attacks {k in 2..2*n}:
sum {i in 1..n, j in 1..n: i+j=k} X[i,j] <= 1;
s.t. diagonal2_attacks {k in (n1)..(n1)}:
sum {i in 1..n, j in 1..n: ij=k} X[i,j] <= 1;
Running both models with HiGHS:
ampl: include nqueens.mod;
ampl: let n := 8;
ampl: option solver highs;
ampl: solve;
HiGHS 1.4.0: optimal solution
0 simplex iterations
1 branching nodes
Objective = find a feasible point.
Another problem can be modeled with alldiff
: Sudoku. A GUIbased Colab Notebook.
MP Documentation on the alldiff
operator can be found at https://amplmp.readthedocs.io/.