One natural model (see [1]) is a pair of matrices of decision variables. The expression for the determinant is cumbersome but, at orders 4 and greater, there is some repetition which can be assigned to a single variable.

Exchanging a pair of rows or a pair of columns of a matrix negates the sign of the determinant. Hence, this problem does not quite have row and column symmetry. However, if we insist that the absolute value of the determinant is 1 then we can apply row and column symmetry breaking, for example by lexicographic and all-permutation ordering. If a solution has a determinant of -1 (and we are solving a variant where a positive determinant is required) we can simply exchange a pair of rows/columns in the solution.

The problem also has diagonal symmetry which can easily be broken by adding further symmetry-breaking constraints (again, see [1]).

  1. A. M. Frisch, C. Jefferson, I. Miguel, “Constraints for Breaking More Row and Column Symmetries,” Proceedings of the 9th International Conference on Principles and Practice of Constraint Programming, 2003.