// -*- c++ -*-

 #include <iostream>

 #include <fstream>

 #include <lemon/time_measure.h>

 #include <lp_solver_base.h>

 using std::cout;

 using std::endl;

 using namespace lemon;

 /*

   for n=3,4 , the program is very fast

   for n=5, with glpk, the run takes hours

  */

 int main(int, char **) {

   const int n=4;

   const double row_sum=(1.0+n*n)*n/2;

   Timer ts;

   ts.reset();

   typedef LPGLPK LPSolver;

   typedef LPSolver::Col Col;

   LPSolver lp;

   typedef std::map<std::pair<int, int>, Col> Coords;

   Coords x;

   // we create a new variable for each entry 

   // of the magic square

   for (int i=1; i<=n; ++i) {

     for (int j=1; j<=n; ++j) { 

       Col col=lp.addCol();

       x[std::make_pair(i,j)]=col;

       lp.setColLowerBound(col, 1.0);

       lp.setColUpperBound(col, double(n*n));

     }

   }

   LPSolver::Expression expr3, expr4;

   for (int i=1; i<=n; ++i) {

     LPSolver::Expression expr1, expr2;

     for (int j=1; j<=n; ++j) {

       expr1+=x[std::make_pair(i, j)];

       expr2+=x[std::make_pair(j, i)];

     }

     // sum of rows and columns

     lp.addRow(expr1==row_sum);

     lp.addRow(expr2==row_sum);

     expr3+=x[std::make_pair(i, i)];

     expr4+=x[std::make_pair(i, (n+1)-i)];

   }

   // sum of the diagonal entries

   lp.addRow(expr3==row_sum);

   lp.addRow(expr4==row_sum);

   lp.solveSimplex();

   cout << "elapsed time: " << ts << endl;

   for (int i=1; i<=n; ++i) {

     for (int j=1; j<=n; ++j) { 

       cout << "x("<<i<<","<<j<<")="<<lp.getPrimal(x[std::make_pair(i,j)]) 

 	   << endl;

     }

   }

   // we make new binary variables for each pair of 

   // entries of the square to achieve that 

   // the values of different entries are different

   lp.setMIP();

   for (Coords::const_iterator it=x.begin(); it!=x.end(); ++it) {

     Coords::const_iterator jt=it; ++jt;

     for(; jt!=x.end(); ++jt) {

       Col col1=(*it).second;

       Col col2=(*jt).second;

       Col col=lp.addCol();

       lp.setColLowerBound(col, 0.0);

       lp.setColUpperBound(col, 1.0);

       lp.addRow(double(-n*n+1.0)<=1.0*col2-1.0*col1-double(n*n)*col<=-1.0);

       lp.setColInt(col);

     }

   }

   cout << "elapsed time: " << ts << endl;

   lp.solveSimplex();

   // let's solve the integer problem

   lp.solveBandB();

   cout << "elapsed time: " << ts << endl;

   for (int i=1; i<=n; ++i) {

     for (int j=1; j<=n; ++j) { 

       cout << "x("<<i<<","<<j<<")="<<lp.getMIPPrimal(x[std::make_pair(i,j)]) 

 	   << endl;

     }

   }

 }