Constructing the last-column backward diagonal constraints for the N-Queens Puzzle.

QueensILP_LastColBackwardDiag.java

/* Here we create the constraints for the "backward diagonals" where
 * we move along the last column.
 *   ie. x_i,n + the sum of x_i+m,n-m <= 1 for all i=1,...,n.
 */
double[][] columnBackwardConstraintsMatrix = new double[n][n*n];
for (int row = 0; row < n; row++) {
    for (int column = n*(row+1) - 1; column < n*n; column += (n-1)) {
        columnBackwardConstraintsMatrix[row][column] = 1;
    }
}

/* Note: Our iteration here always gives us an extra '1' at the end of
 * the top row, which we can simply remove here.
 */
columnBackwardConstraintsMatrix[0][n*n - 1] = 0;

for (double[] row : columnBackwardConstraintsMatrix) {
    lp.addConstraint(new LinearSmallerThanEqualsConstraint(row, 1, ""));
}