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%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sudoku.m %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%% building Aeq matrix for Sudoku Problem
n=4; % for 4 by 4 sudoku problem
%n=9; % for 9 by 9 sudoku problem
% x vector (n^3 by 1) contains the decision variables
% Aeq matrix of coefficients for equality constraints
% There are 4(n^2) + |G| constraints in total.
% (1) n^2 saying that each column can have only 1 of each integer 1:n
% (2) n^2 saying that each row can have only 1 of each integer 1:n
% (3) n^2 saying that each submatrix can have only 1 of each integer 1:n
% (4) n^2 saying that each element in the matrix must be covered
% (5) |G| constraints, one for each given element in the initial matrix
% NOTE: these givens are entered below in "Givens" matrix
Aeq=zeros(4*n^2, n^3);
%% First, fill in Aeq for the n^2 constraints requiring that each column
%% can have only 1 of each integer 1:n
for i=1:n^2
Aeq(i,(i-1)*n+1:i*n)=ones(1,n);
end
%% Second, fill in Aeq for the n^2 constraints requiring that each row
%% can have only 1 of each integer 1:n
for k=1:n
for i=1:n
for j=1:n
Aeq(n^2+i+(k-1)*n,1+(i-1)+(j-1)*n+(k-1)*n^2)=1;
end
end
end
%% Third, fill in Aeq for the n^2 constraints requiring that each submatrix
%% can have only 1 of each integer 1:n
for m=1:sqrt(n)
for l=1:sqrt(n)
for j=1:n
for k=1:sqrt(n)
Aeq(2*n^2+(m-1)*sqrt(n)*n+(l-1)*n+j,(j-1)*n^2+(l-1)*sqrt(n)+...
(m-1)*sqrt(n)*n+(k-1)*n+1:...
(j-1)*n^2+(l-1)*sqrt(n)+(m-1)*sqrt(n)*n+(k-1)*n+sqrt(n))=1;
end
end
end
end
%% Fourth, fill in Aeq for the n^2 constraints requiring that each element
%% of the matrix must contain an integer 1:n
for i=1:n
for j=1:n
for k=1:n
Aeq(3*n^2+(i-1)*n+j,(i-1)*n+j + (k-1)*n^2)=1;
end
end
end
%% Fifth, if any values of the Sudoku matrix are given, add these as
%% constraints
% Enter each given value in matrix as triplet (row, col, integer)
% Example for 4-by-4 sudoku
Givens=[1 4 4;
2 1 2;
2 3 1;
3 1 3;
3 4 2];
% turn these given elements into their appropriate position in the x vector
% of decision variables.
g=size(Givens,1);
for i=1:g
Aeq(4*n^2+i,Givens(i,1)+(Givens(i,2)-1)*n+(Givens(i,3)-1)*n^2)=1;
end
% beq=rhs vector (3n^2 by 1) for equality contraints
beq=ones(4*n^2+g,1);
% run matlab's "bintprog" command
options.MaxRLPIter=500000;
[x,fval,exitflag,output]=bintprog(zeros(n^3,1),[],[],Aeq,beq,[],options);
exitflag
output
% turn matlab's outputed solution vector x into sudoku terms
S=zeros(n,n);
for k=1:n
subx=find(x((k-1)*n^2+1:k*n^2));
for j=1:n
row=mod(subx(j),n);
if row==0
row=n;
end
S(row,j)=k;
end
end
% Sudoku Matrix S
S