+ remove_artificial_variables();
+
+ // Solve the phase 2 problem. We already know it to be feasible.
+ while(pivot()) ;
+
+ solved = true;
+
+ return true;
+}
+
+void Layout::LinearProgram::prepare_columns()
+{
+ /* See if any variables are already basic. A basic variable must only have
+ a nonzero coefficient on one row, and its product with the constant column
+ must not be negative. Only one variable can be basic for any given row. */
+ vector<float> basic_coeff(n_rows, 0.0f);
+ const vector<float> &constants = columns.back().values;
+ for(vector<Column>::iterator i=columns.begin(); i!=columns.end(); ++i)
+ {
+ if(i->values.size()>=2 && i->values.back()!=0.0f && (constants.size()<i->values.size() || i->values.back()*constants[i->values.size()-1]>=0.0f) && basic_coeff[i->values.size()-1]==0.0f)
+ {
+ bool basic = true;
+ for(unsigned j=1; (basic && j+1<i->values.size()); ++j)
+ basic = (i->values[j]==0.0f);
+ if(basic)
+ {
+ i->basic = i->values.size()-1;
+ basic_coeff[i->basic] = -i->values.front()/i->values.back();
+ i->values.clear();
+ }
+ }
+ }
+
+ // Price out the newly-created basic variables.
+ for(vector<Column>::iterator i=columns.begin(); i!=columns.end(); ++i)
+ if(!i->values.empty())
+ {
+ for(unsigned j=0; j<i->values.size(); ++j)
+ i->values.front() += basic_coeff[j]*i->values[j];
+ }
+}
+
+void Layout::LinearProgram::add_artificial_variables()
+{
+ vector<unsigned> artificial_rows(n_rows-1);
+ for(unsigned i=0; i<artificial_rows.size(); ++i)
+ artificial_rows[i] = i+1;
+
+ for(vector<Column>::iterator i=columns.begin(); i!=columns.end(); ++i)
+ if(i->basic)
+ artificial_rows[i->basic-1] = 0;
+ artificial_rows.erase(std::remove(artificial_rows.begin(), artificial_rows.end(), 0), artificial_rows.end());
+
+ /* Force all non-basic columns fully into existence and relocate objective
+ row to bottom in preparation of phase 1. A new objective row is calculated
+ by pricing out the constraint rows. */
+ for(vector<Column>::iterator i=columns.begin(); i!=columns.end(); ++i)
+ if(!i->basic)
+ {
+ float objective = 0.0f;
+ if(!i->values.empty())
+ {
+ objective = i->values.front();
+ i->values.front() = 0.0f;
+ for(vector<unsigned>::iterator j=artificial_rows.begin(); j!=artificial_rows.end(); ++j)
+ if(*j<i->values.size())
+ i->values.front() += i->values[*j];
+ }
+ i->values.resize(n_rows+1, 0.0f);
+ i->values.back() = objective;
+ }
+
+ if(artificial_rows.empty())
+ return;
+
+ /* Create artificial variables for phase 1. This ensures that each row has
+ a basic variable associated with it. The original objective row already
+ contains the implicit objective variable, which is basic. */
+ columns.resize(n_columns+artificial_rows.size());
+ columns.back() = columns[n_columns-1];
+ columns[n_columns-1].values.clear();
+ for(unsigned i=0; i<artificial_rows.size(); ++i)
+ columns[n_columns+i-1].basic = artificial_rows[i];
+}
+
+void Layout::LinearProgram::remove_artificial_variables()
+{
+ /* See if any artificial variables are still basic. This could be because
+ the program is degenerate. To avoid trouble later on, use pivots to make
+ some of the original variables basic instead.
+
+ I don't fully understand why this is needed, but it appears to work. */
+ for(unsigned i=n_columns-1; i+1<columns.size(); ++i)
+ if(columns[i].basic && columns.back().values[columns[i].basic]==0.0f)
+ {
+ for(unsigned j=0; j+1<n_columns; ++j)
+ if(!columns[j].basic && columns[j].values[columns[i].basic]!=0.0f)
+ {
+ make_basic_column(j, columns[i].basic);
+ break;
+ }
+ }
+