// -*- 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;

    }

  }

}