lp_data.cc

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// Copyright 2010-2018 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
 
#include "ortools/lp_data/lp_data.h"
 
#include <algorithm>
#include <cmath>
#include <string>
#include <utility>
 
#include "absl/container/flat_hash_map.h"
#include "absl/strings/str_cat.h"
#include "absl/strings/str_format.h"
#include "ortools/base/logging.h"
#include "ortools/lp_data/lp_print_utils.h"
#include "ortools/lp_data/lp_utils.h"
#include "ortools/lp_data/matrix_utils.h"
#include "ortools/lp_data/permutation.h"
 
namespace operations_research {
namespace glop {
 
namespace {
 
// This should be the same as DCHECK(AreBoundsValid()), but the DCHECK() are
// split to give more meaningful information to the user in case of failure.
void DebugCheckBoundsValid(Fractional lower_bound, Fractional upper_bound) {
  DCHECK(!std::isnan(lower_bound));
  DCHECK(!std::isnan(upper_bound));
  DCHECK(!(lower_bound == kInfinity && upper_bound == kInfinity));
  DCHECK(!(lower_bound == -kInfinity && upper_bound == -kInfinity));
  DCHECK_LE(lower_bound, upper_bound);
  DCHECK(AreBoundsValid(lower_bound, upper_bound));
}
 
// Returns true if the bounds are the ones of a free or boxed row. Note that
// a fixed row is not counted as boxed.
bool AreBoundsFreeOrBoxed(Fractional lower_bound, Fractional upper_bound) {
  if (lower_bound == -kInfinity && upper_bound == kInfinity) return true;
  if (lower_bound != -kInfinity && upper_bound != kInfinity &&
      lower_bound != upper_bound) {
    return true;
  }
  return false;
}
 
template <class I, class T>
double Average(const absl::StrongVector<I, T>& v) {
  const size_t size = v.size();
  DCHECK_LT(0, size);
  double sum = 0.0;
  double n = 0.0;  // n is used in a calculation involving doubles.
  for (I i(0); i < size; ++i) {
    if (v[i] == 0.0) continue;
    ++n;
    sum += static_cast<double>(v[i].value());
  }
  return n == 0.0 ? 0.0 : sum / n;
}
 
template <class I, class T>
double StandardDeviation(const absl::StrongVector<I, T>& v) {
  const size_t size = v.size();
  double n = 0.0;  // n is used in a calculation involving doubles.
  double sigma_square = 0.0;
  double sigma = 0.0;
  for (I i(0); i < size; ++i) {
    double sample = static_cast<double>(v[i].value());
    if (sample == 0.0) continue;
    sigma_square += sample * sample;
    sigma += sample;
    ++n;
  }
  return n == 0.0 ? 0.0 : sqrt((sigma_square - sigma * sigma / n) / n);
}
 
// Returns 0 when the vector is empty.
template <class I, class T>
T GetMaxElement(const absl::StrongVector<I, T>& v) {
  const size_t size = v.size();
  if (size == 0) {
    return T(0);
  }
 
  T max_index = v[I(0)];
  for (I i(1); i < size; ++i) {
    if (max_index < v[i]) {
      max_index = v[i];
    }
  }
  return max_index;
}
 
}  // anonymous namespace
 
// --------------------------------------------------------
// LinearProgram
// --------------------------------------------------------
LinearProgram::LinearProgram()
    : matrix_(),
      transpose_matrix_(),
      constraint_lower_bounds_(),
      constraint_upper_bounds_(),
      constraint_names_(),
      objective_coefficients_(),
      variable_lower_bounds_(),
      variable_upper_bounds_(),
      variable_names_(),
      variable_types_(),
      integer_variables_list_(),
      variable_table_(),
      constraint_table_(),
      objective_offset_(0.0),
      objective_scaling_factor_(1.0),
      maximize_(false),
      columns_are_known_to_be_clean_(true),
      transpose_matrix_is_consistent_(true),
      integer_variables_list_is_consistent_(true),
      name_(),
      first_slack_variable_(kInvalidCol) {}
 
void LinearProgram::Clear() {
  matrix_.Clear();
  transpose_matrix_.Clear();
 
  constraint_lower_bounds_.clear();
  constraint_upper_bounds_.clear();
  constraint_names_.clear();
 
  objective_coefficients_.clear();
  variable_lower_bounds_.clear();
  variable_upper_bounds_.clear();
  variable_types_.clear();
  integer_variables_list_.clear();
  variable_names_.clear();
 
  constraint_table_.clear();
  variable_table_.clear();
 
  maximize_ = false;
  objective_offset_ = 0.0;
  objective_scaling_factor_ = 1.0;
  columns_are_known_to_be_clean_ = true;
  transpose_matrix_is_consistent_ = true;
  integer_variables_list_is_consistent_ = true;
  name_.clear();
  first_slack_variable_ = kInvalidCol;
}
 
ColIndex LinearProgram::CreateNewVariable() {
  DCHECK_EQ(kInvalidCol, first_slack_variable_)
      << "New variables can't be added to programs that already have slack "
         "variables. Consider calling LinearProgram::DeleteSlackVariables() "
         "before adding new variables to the problem.";
  objective_coefficients_.push_back(0.0);
  variable_lower_bounds_.push_back(0);
  variable_upper_bounds_.push_back(kInfinity);
  variable_types_.push_back(VariableType::CONTINUOUS);
  variable_names_.push_back("");
  transpose_matrix_is_consistent_ = false;
  return matrix_.AppendEmptyColumn();
}
 
ColIndex LinearProgram::CreateNewSlackVariable(bool is_integer_slack_variable,
                                               Fractional lower_bound,
                                               Fractional upper_bound,
                                               const std::string& name) {
  objective_coefficients_.push_back(0.0);
  variable_lower_bounds_.push_back(lower_bound);
  variable_upper_bounds_.push_back(upper_bound);
  variable_types_.push_back(is_integer_slack_variable
                                ? (VariableType::IMPLIED_INTEGER)
                                : (VariableType::CONTINUOUS));
  variable_names_.push_back(name);
  transpose_matrix_is_consistent_ = false;
  return matrix_.AppendEmptyColumn();
}
 
RowIndex LinearProgram::CreateNewConstraint() {
  DCHECK_EQ(kInvalidCol, first_slack_variable_)
      << "New constraints can't be added to programs that already have slack "
         "variables. Consider calling LinearProgram::DeleteSlackVariables() "
         "before adding new variables to the problem.";
  const RowIndex row(constraint_names_.size());
  matrix_.SetNumRows(row + 1);
  constraint_lower_bounds_.push_back(Fractional(0.0));
  constraint_upper_bounds_.push_back(Fractional(0.0));
  constraint_names_.push_back("");
  transpose_matrix_is_consistent_ = false;
  return row;
}
 
ColIndex LinearProgram::FindOrCreateVariable(const std::string& variable_id) {
  const absl::flat_hash_map<std::string, ColIndex>::iterator it =
      variable_table_.find(variable_id);
  if (it != variable_table_.end()) {
    return it->second;
  } else {
    const ColIndex col = CreateNewVariable();
    variable_names_[col] = variable_id;
    variable_table_[variable_id] = col;
    return col;
  }
}
 
RowIndex LinearProgram::FindOrCreateConstraint(
    const std::string& constraint_id) {
  const absl::flat_hash_map<std::string, RowIndex>::iterator it =
      constraint_table_.find(constraint_id);
  if (it != constraint_table_.end()) {
    return it->second;
  } else {
    const RowIndex row = CreateNewConstraint();
    constraint_names_[row] = constraint_id;
    constraint_table_[constraint_id] = row;
    return row;
  }
}
 
void LinearProgram::SetVariableName(ColIndex col, absl::string_view name) {
  variable_names_[col] = std::string(name);
}
 
void LinearProgram::SetVariableType(ColIndex col, VariableType type) {
  const bool var_was_integer = IsVariableInteger(col);
  variable_types_[col] = type;
  const bool var_is_integer = IsVariableInteger(col);
  if (var_is_integer != var_was_integer) {
    integer_variables_list_is_consistent_ = false;
  }
}
 
void LinearProgram::SetConstraintName(RowIndex row, absl::string_view name) {
  constraint_names_[row] = std::string(name);
}
 
void LinearProgram::SetVariableBounds(ColIndex col, Fractional lower_bound,
                                      Fractional upper_bound) {
  if (dcheck_bounds_) DebugCheckBoundsValid(lower_bound, upper_bound);
  const bool var_was_binary = IsVariableBinary(col);
  variable_lower_bounds_[col] = lower_bound;
  variable_upper_bounds_[col] = upper_bound;
  const bool var_is_binary = IsVariableBinary(col);
  if (var_is_binary != var_was_binary) {
    integer_variables_list_is_consistent_ = false;
  }
}
 
void LinearProgram::UpdateAllIntegerVariableLists() const {
  if (integer_variables_list_is_consistent_) return;
  integer_variables_list_.clear();
  binary_variables_list_.clear();
  non_binary_variables_list_.clear();
  const ColIndex num_cols = num_variables();
  for (ColIndex col(0); col < num_cols; ++col) {
    if (IsVariableInteger(col)) {
      integer_variables_list_.push_back(col);
      if (IsVariableBinary(col)) {
        binary_variables_list_.push_back(col);
      } else {
        non_binary_variables_list_.push_back(col);
      }
    }
  }
  integer_variables_list_is_consistent_ = true;
}
 
const std::vector<ColIndex>& LinearProgram::IntegerVariablesList() const {
  UpdateAllIntegerVariableLists();
  return integer_variables_list_;
}
 
const std::vector<ColIndex>& LinearProgram::BinaryVariablesList() const {
  UpdateAllIntegerVariableLists();
  return binary_variables_list_;
}
 
const std::vector<ColIndex>& LinearProgram::NonBinaryVariablesList() const {
  UpdateAllIntegerVariableLists();
  return non_binary_variables_list_;
}
 
bool LinearProgram::IsVariableInteger(ColIndex col) const {
  return variable_types_[col] == VariableType::INTEGER ||
         variable_types_[col] == VariableType::IMPLIED_INTEGER;
}
 
bool LinearProgram::IsVariableBinary(ColIndex col) const {
  // TODO(user,user): bounds of binary variables (and of integer ones) should
  // be integer. Add a preprocessor for that.
  return IsVariableInteger(col) && (variable_lower_bounds_[col] < kEpsilon) &&
         (variable_lower_bounds_[col] > Fractional(-1)) &&
         (variable_upper_bounds_[col] > Fractional(1) - kEpsilon) &&
         (variable_upper_bounds_[col] < 2);
}
 
void LinearProgram::SetConstraintBounds(RowIndex row, Fractional lower_bound,
                                        Fractional upper_bound) {
  if (dcheck_bounds_) DebugCheckBoundsValid(lower_bound, upper_bound);
  ResizeRowsIfNeeded(row);
  constraint_lower_bounds_[row] = lower_bound;
  constraint_upper_bounds_[row] = upper_bound;
}
 
void LinearProgram::SetCoefficient(RowIndex row, ColIndex col,
                                   Fractional value) {
  DCHECK(IsFinite(value));
  ResizeRowsIfNeeded(row);
  columns_are_known_to_be_clean_ = false;
  transpose_matrix_is_consistent_ = false;
  matrix_.mutable_column(col)->SetCoefficient(row, value);
}
 
void LinearProgram::SetObjectiveCoefficient(ColIndex col, Fractional value) {
  DCHECK(IsFinite(value));
  objective_coefficients_[col] = value;
}
 
void LinearProgram::SetObjectiveOffset(Fractional objective_offset) {
  DCHECK(IsFinite(objective_offset));
  objective_offset_ = objective_offset;
}
 
void LinearProgram::SetObjectiveScalingFactor(
    Fractional objective_scaling_factor) {
  DCHECK(IsFinite(objective_scaling_factor));
  DCHECK_NE(0.0, objective_scaling_factor);
  objective_scaling_factor_ = objective_scaling_factor;
}
 
void LinearProgram::SetMaximizationProblem(bool maximize) {
  maximize_ = maximize;
}
 
void LinearProgram::CleanUp() {
  if (columns_are_known_to_be_clean_) return;
  matrix_.CleanUp();
  columns_are_known_to_be_clean_ = true;
  transpose_matrix_is_consistent_ = false;
}
 
bool LinearProgram::IsCleanedUp() const {
  if (columns_are_known_to_be_clean_) return true;
  columns_are_known_to_be_clean_ = matrix_.IsCleanedUp();
  return columns_are_known_to_be_clean_;
}
 
std::string LinearProgram::GetVariableName(ColIndex col) const {
  return col >= variable_names_.size() || variable_names_[col].empty()
             ? absl::StrFormat("c%d", col.value())
             : variable_names_[col];
}
 
std::string LinearProgram::GetConstraintName(RowIndex row) const {
  return row >= constraint_names_.size() || constraint_names_[row].empty()
             ? absl::StrFormat("r%d", row.value())
             : constraint_names_[row];
}
 
LinearProgram::VariableType LinearProgram::GetVariableType(ColIndex col) const {
  return variable_types_[col];
}
 
const SparseMatrix& LinearProgram::GetTransposeSparseMatrix() const {
  if (!transpose_matrix_is_consistent_) {
    transpose_matrix_.PopulateFromTranspose(matrix_);
    transpose_matrix_is_consistent_ = true;
  }
  DCHECK_EQ(transpose_matrix_.num_rows().value(), matrix_.num_cols().value());
  DCHECK_EQ(transpose_matrix_.num_cols().value(), matrix_.num_rows().value());
  return transpose_matrix_;
}
 
SparseMatrix* LinearProgram::GetMutableTransposeSparseMatrix() {
  if (!transpose_matrix_is_consistent_) {
    transpose_matrix_.PopulateFromTranspose(matrix_);
  }
  // This enables a client to start modifying the matrix and then abort and not
  // call UseTransposeMatrixAsReference(). Then, the other client of
  // GetTransposeSparseMatrix() will still see the correct matrix.
  transpose_matrix_is_consistent_ = false;
  return &transpose_matrix_;
}
 
void LinearProgram::UseTransposeMatrixAsReference() {
  DCHECK_EQ(transpose_matrix_.num_rows().value(), matrix_.num_cols().value());
  DCHECK_EQ(transpose_matrix_.num_cols().value(), matrix_.num_rows().value());
  matrix_.PopulateFromTranspose(transpose_matrix_);
  transpose_matrix_is_consistent_ = true;
}
 
void LinearProgram::ClearTransposeMatrix() {
  transpose_matrix_.Clear();
  transpose_matrix_is_consistent_ = false;
}
 
const SparseColumn& LinearProgram::GetSparseColumn(ColIndex col) const {
  return matrix_.column(col);
}
 
SparseColumn* LinearProgram::GetMutableSparseColumn(ColIndex col) {
  columns_are_known_to_be_clean_ = false;
  transpose_matrix_is_consistent_ = false;
  return matrix_.mutable_column(col);
}
 
Fractional LinearProgram::GetObjectiveCoefficientForMinimizationVersion(
    ColIndex col) const {
  return maximize_ ? -objective_coefficients()[col]
                   : objective_coefficients()[col];
}
 
std::string LinearProgram::GetDimensionString() const {
  return absl::StrFormat(
      "%d rows, %d columns, %d entries", num_constraints().value(),
      num_variables().value(),
      // static_cast<int64> is needed because the Android port uses int32.
      static_cast<int64>(num_entries().value()));
}
 
std::string LinearProgram::GetObjectiveStatsString() const {
  int64 num_non_zeros = 0;
  Fractional min_value = +kInfinity;
  Fractional max_value = -kInfinity;
  for (ColIndex col(0); col < objective_coefficients_.size(); ++col) {
    const Fractional value = objective_coefficients_[col];
    if (value == 0) continue;
    min_value = std::min(min_value, value);
    max_value = std::max(max_value, value);
    ++num_non_zeros;
  }
  if (num_non_zeros == 0) {
    return "No objective term. This is a pure feasibility problem.";
  } else {
    return absl::StrFormat("%d non-zeros, range [%e, %e]", num_non_zeros,
                           min_value, max_value);
  }
}
 
bool LinearProgram::SolutionIsWithinVariableBounds(
    const DenseRow& solution, Fractional absolute_tolerance) const {
  DCHECK_EQ(solution.size(), num_variables());
  if (solution.size() != num_variables()) return false;
  const ColIndex num_cols = num_variables();
  for (ColIndex col = ColIndex(0); col < num_cols; ++col) {
    if (!IsFinite(solution[col])) return false;
    const Fractional lb_error = variable_lower_bounds()[col] - solution[col];
    const Fractional ub_error = solution[col] - variable_upper_bounds()[col];
    if (lb_error > absolute_tolerance || ub_error > absolute_tolerance) {
      return false;
    }
  }
  return true;
}
 
bool LinearProgram::SolutionIsLPFeasible(const DenseRow& solution,
                                         Fractional absolute_tolerance) const {
  if (!SolutionIsWithinVariableBounds(solution, absolute_tolerance)) {
    return false;
  }
  const SparseMatrix& transpose = GetTransposeSparseMatrix();
  const RowIndex num_rows = num_constraints();
  for (RowIndex row = RowIndex(0); row < num_rows; ++row) {
    const Fractional sum =
        ScalarProduct(solution, transpose.column(RowToColIndex(row)));
    if (!IsFinite(sum)) return false;
    const Fractional lb_error = constraint_lower_bounds()[row] - sum;
    const Fractional ub_error = sum - constraint_upper_bounds()[row];
    if (lb_error > absolute_tolerance || ub_error > absolute_tolerance) {
      return false;
    }
  }
  return true;
}
 
bool LinearProgram::SolutionIsInteger(const DenseRow& solution,
                                      Fractional absolute_tolerance) const {
  DCHECK_EQ(solution.size(), num_variables());
  if (solution.size() != num_variables()) return false;
  for (ColIndex col : IntegerVariablesList()) {
    if (!IsFinite(solution[col])) return false;
    const Fractional fractionality = fabs(solution[col] - round(solution[col]));
    if (fractionality > absolute_tolerance) return false;
  }
  return true;
}
 
bool LinearProgram::SolutionIsMIPFeasible(const DenseRow& solution,
                                          Fractional absolute_tolerance) const {
  return SolutionIsLPFeasible(solution, absolute_tolerance) &&
         SolutionIsInteger(solution, absolute_tolerance);
}
 
void LinearProgram::ComputeSlackVariableValues(DenseRow* solution) const {
  CHECK(solution != nullptr);
  const ColIndex num_cols = GetFirstSlackVariable();
  const SparseMatrix& transpose = GetTransposeSparseMatrix();
  const RowIndex num_rows = num_constraints();
  CHECK_EQ(solution->size(), num_variables());
  for (RowIndex row = RowIndex(0); row < num_rows; ++row) {
    const Fractional sum = PartialScalarProduct(
        *solution, transpose.column(RowToColIndex(row)), num_cols.value());
    const ColIndex slack_variable = GetSlackVariable(row);
    CHECK_NE(slack_variable, kInvalidCol);
    (*solution)[slack_variable] = -sum;
  }
}
 
Fractional LinearProgram::ApplyObjectiveScalingAndOffset(
    Fractional value) const {
  return objective_scaling_factor() * (value + objective_offset());
}
 
Fractional LinearProgram::RemoveObjectiveScalingAndOffset(
    Fractional value) const {
  return value / objective_scaling_factor() - objective_offset();
}
 
std::string LinearProgram::Dump() const {
  // Objective line.
  std::string output = maximize_ ? "max:" : "min:";
  if (objective_offset_ != 0.0) {
    output += Stringify(objective_offset_);
  }
  const ColIndex num_cols = num_variables();
  for (ColIndex col(0); col < num_cols; ++col) {
    const Fractional coeff = objective_coefficients()[col];
    if (coeff != 0.0) {
      output += StringifyMonomial(coeff, GetVariableName(col), false);
    }
  }
  output.append(";\n");
 
  // Constraints.
  const RowIndex num_rows = num_constraints();
  for (RowIndex row(0); row < num_rows; ++row) {
    const Fractional lower_bound = constraint_lower_bounds()[row];
    const Fractional upper_bound = constraint_upper_bounds()[row];
    output += GetConstraintName(row);
    output += ":";
    if (AreBoundsFreeOrBoxed(lower_bound, upper_bound)) {
      output += " ";
      output += Stringify(lower_bound);
      output += " <=";
    }
    for (ColIndex col(0); col < num_cols; ++col) {
      const Fractional coeff = matrix_.LookUpValue(row, col);
      output += StringifyMonomial(coeff, GetVariableName(col), false);
    }
    if (AreBoundsFreeOrBoxed(lower_bound, upper_bound)) {
      output += " <= ";
      output += Stringify(upper_bound);
    } else if (lower_bound == upper_bound) {
      output += " = ";
      output += Stringify(upper_bound);
    } else if (lower_bound != -kInfinity) {
      output += " >= ";
      output += Stringify(lower_bound);
    } else if (lower_bound != kInfinity) {
      output += " <= ";
      output += Stringify(upper_bound);
    }
    output += ";\n";
  }
 
  // Variables.
  for (ColIndex col(0); col < num_cols; ++col) {
    const Fractional lower_bound = variable_lower_bounds()[col];
    const Fractional upper_bound = variable_upper_bounds()[col];
    if (AreBoundsFreeOrBoxed(lower_bound, upper_bound)) {
      output += Stringify(lower_bound);
      output += " <= ";
    }
    output += GetVariableName(col);
    if (AreBoundsFreeOrBoxed(lower_bound, upper_bound)) {
      output += " <= ";
      output += Stringify(upper_bound);
    } else if (lower_bound == upper_bound) {
      output += " = ";
      output += Stringify(upper_bound);
    } else if (lower_bound != -kInfinity) {
      output += " >= ";
      output += Stringify(lower_bound);
    } else if (lower_bound != kInfinity) {
      output += " <= ";
      output += Stringify(upper_bound);
    }
    output += ";\n";
  }
 
  // Integer variables.
  // TODO(user): if needed provide similar output for binary variables.
  const std::vector<ColIndex>& integer_variables = IntegerVariablesList();
  if (!integer_variables.empty()) {
    output += "int";
    for (ColIndex col : integer_variables) {
      output += " ";
      output += GetVariableName(col);
    }
    output += ";\n";
  }
 
  return output;
}
 
std::string LinearProgram::DumpSolution(const DenseRow& variable_values) const {
  DCHECK_EQ(variable_values.size(), num_variables());
  std::string output;
  for (ColIndex col(0); col < variable_values.size(); ++col) {
    if (!output.empty()) absl::StrAppend(&output, ", ");
    absl::StrAppend(&output, GetVariableName(col), " = ",
                    (variable_values[col]));
  }
  return output;
}
 
std::string LinearProgram::GetProblemStats() const {
  return ProblemStatFormatter(
      "%d,%d,%d,%d,%d,%d,%d,%d,%d,%d,%d,%d,%d,%d,"
      "%d,%d,%d,%d");
}
 
std::string LinearProgram::GetPrettyProblemStats() const {
  return ProblemStatFormatter(
      "Number of rows                               : %d\n"
      "Number of variables in file                  : %d\n"
      "Number of entries (non-zeros)                : %d\n"
      "Number of entries in the objective           : %d\n"
      "Number of entries in the right-hand side     : %d\n"
      "Number of <= constraints                     : %d\n"
      "Number of >= constraints                     : %d\n"
      "Number of = constraints                      : %d\n"
      "Number of range constraints                  : %d\n"
      "Number of non-negative variables             : %d\n"
      "Number of boxed variables                    : %d\n"
      "Number of free variables                     : %d\n"
      "Number of fixed variables                    : %d\n"
      "Number of other variables                    : %d\n"
      "Number of integer variables                  : %d\n"
      "Number of binary variables                   : %d\n"
      "Number of non-binary integer variables       : %d\n"
      "Number of continuous variables               : %d\n");
}
 
std::string LinearProgram::GetNonZeroStats() const {
  return NonZeroStatFormatter("%.2f%%,%d,%.2f,%.2f,%d,%.2f,%.2f");
}
 
std::string LinearProgram::GetPrettyNonZeroStats() const {
  return NonZeroStatFormatter(
      "Fill rate                                    : %.2f%%\n"
      "Entries in row (Max / average / std. dev.)   : %d / %.2f / %.2f\n"
      "Entries in column (Max / average / std. dev.): %d / %.2f / %.2f\n");
}
 
void LinearProgram::AddSlackVariablesWhereNecessary(
    bool detect_integer_constraints) {
  // Clean up the matrix. We're going to add entries, but we'll only be adding
  // them to new columns, and only one entry per column, which does not
  // invalidate the "cleanness" of the matrix.
  CleanUp();
 
  // Detect which constraints produce an integer slack variable. A constraint
  // has an integer slack variable, if it contains only integer variables with
  // integer coefficients. We do not check the bounds of the constraints,
  // because in such case, they will be tightened to integer values by the
  // preprocessors.
  //
  // We don't use the transpose, because it might not be valid and it would be
  // inefficient to update it and invalidate it again at the end of this
  // preprocessor.
  DenseBooleanColumn has_integer_slack_variable(num_constraints(),
                                                detect_integer_constraints);
  if (detect_integer_constraints) {
    for (ColIndex col(0); col < num_variables(); ++col) {
      const SparseColumn& column = matrix_.column(col);
      const bool is_integer_variable = IsVariableInteger(col);
      for (const SparseColumn::Entry& entry : column) {
        const RowIndex row = entry.row();
        has_integer_slack_variable[row] =
            has_integer_slack_variable[row] && is_integer_variable &&
            round(entry.coefficient()) == entry.coefficient();
      }
    }
  }
 
  // Extend the matrix of the problem with an identity matrix.
  const ColIndex original_num_variables = num_variables();
  for (RowIndex row(0); row < num_constraints(); ++row) {
    ColIndex slack_variable_index = GetSlackVariable(row);
    if (slack_variable_index != kInvalidCol &&
        slack_variable_index < original_num_variables) {
      // Slack variable is already present in this constraint.
      continue;
    }
    const ColIndex slack_col = CreateNewSlackVariable(
        has_integer_slack_variable[row], -constraint_upper_bounds_[row],
        -constraint_lower_bounds_[row], absl::StrCat("s", row.value()));
    SetCoefficient(row, slack_col, 1.0);
    SetConstraintBounds(row, 0.0, 0.0);
  }
 
  columns_are_known_to_be_clean_ = true;
  transpose_matrix_is_consistent_ = false;
  if (first_slack_variable_ == kInvalidCol) {
    first_slack_variable_ = original_num_variables;
  }
}
 
ColIndex LinearProgram::GetFirstSlackVariable() const {
  return first_slack_variable_;
}
 
ColIndex LinearProgram::GetSlackVariable(RowIndex row) const {
  DCHECK_GE(row, RowIndex(0));
  DCHECK_LT(row, num_constraints());
  if (first_slack_variable_ == kInvalidCol) {
    return kInvalidCol;
  }
  return first_slack_variable_ + RowToColIndex(row);
}
 
void LinearProgram::PopulateFromDual(const LinearProgram& dual,
                                     RowToColMapping* duplicated_rows) {
  const ColIndex dual_num_variables = dual.num_variables();
  const RowIndex dual_num_constraints = dual.num_constraints();
  Clear();
 
  // We always take the dual in its minimization form thanks to the
  // GetObjectiveCoefficientForMinimizationVersion() below, so this will always
  // be a maximization problem.
  SetMaximizationProblem(true);
 
  // Taking the dual does not change the offset nor the objective scaling
  // factor.
  SetObjectiveOffset(dual.objective_offset());
  SetObjectiveScalingFactor(dual.objective_scaling_factor());
 
  // Create the dual variables y, with bounds depending on the type
  // of constraints in the primal.
  for (RowIndex dual_row(0); dual_row < dual_num_constraints; ++dual_row) {
    CreateNewVariable();
    const ColIndex col = RowToColIndex(dual_row);
    const Fractional lower_bound = dual.constraint_lower_bounds()[dual_row];
    const Fractional upper_bound = dual.constraint_upper_bounds()[dual_row];
    if (lower_bound == upper_bound) {
      SetVariableBounds(col, -kInfinity, kInfinity);
      SetObjectiveCoefficient(col, lower_bound);
    } else if (upper_bound != kInfinity) {
      // Note that for a ranged constraint, the loop will be continued here.
      // This is wanted because we want to deal with the lower_bound afterwards.
      SetVariableBounds(col, -kInfinity, 0.0);
      SetObjectiveCoefficient(col, upper_bound);
    } else if (lower_bound != -kInfinity) {
      SetVariableBounds(col, 0.0, kInfinity);
      SetObjectiveCoefficient(col, lower_bound);
    } else {
      // This code does not support free rows in linear_program.
      LOG(DFATAL) << "PopulateFromDual() was called with a program "
                  << "containing free constraints.";
    }
  }
  // Create the dual slack variables v.
  for (ColIndex dual_col(0); dual_col < dual_num_variables; ++dual_col) {
    const Fractional lower_bound = dual.variable_lower_bounds()[dual_col];
    if (lower_bound != -kInfinity) {
      const ColIndex col = CreateNewVariable();
      SetObjectiveCoefficient(col, lower_bound);
      SetVariableBounds(col, 0.0, kInfinity);
      const RowIndex row = ColToRowIndex(dual_col);
      SetCoefficient(row, col, Fractional(1.0));
    }
  }
  // Create the dual surplus variables w.
  for (ColIndex dual_col(0); dual_col < dual_num_variables; ++dual_col) {
    const Fractional upper_bound = dual.variable_upper_bounds()[dual_col];
    if (upper_bound != kInfinity) {
      const ColIndex col = CreateNewVariable();
      SetObjectiveCoefficient(col, upper_bound);
      SetVariableBounds(col, -kInfinity, 0.0);
      const RowIndex row = ColToRowIndex(dual_col);
      SetCoefficient(row, col, Fractional(1.0));
    }
  }
  // Store the transpose of the matrix.
  for (ColIndex dual_col(0); dual_col < dual_num_variables; ++dual_col) {
    const RowIndex row = ColToRowIndex(dual_col);
    const Fractional row_bound =
        dual.GetObjectiveCoefficientForMinimizationVersion(dual_col);
    SetConstraintBounds(row, row_bound, row_bound);
    for (const SparseColumn::Entry e : dual.GetSparseColumn(dual_col)) {
      const RowIndex dual_row = e.row();
      const ColIndex col = RowToColIndex(dual_row);
      SetCoefficient(row, col, e.coefficient());
    }
  }
 
  // Take care of ranged constraints.
  duplicated_rows->assign(dual_num_constraints, kInvalidCol);
  for (RowIndex dual_row(0); dual_row < dual_num_constraints; ++dual_row) {
    const Fractional lower_bound = dual.constraint_lower_bounds()[dual_row];
    const Fractional upper_bound = dual.constraint_upper_bounds()[dual_row];
    if (AreBoundsFreeOrBoxed(lower_bound, upper_bound)) {
      DCHECK(upper_bound != kInfinity || lower_bound != -kInfinity);
 
      // upper_bound was done in a loop above, now do the lower_bound.
      const ColIndex col = CreateNewVariable();
      SetVariableBounds(col, 0.0, kInfinity);
      SetObjectiveCoefficient(col, lower_bound);
      matrix_.mutable_column(col)->PopulateFromSparseVector(
          matrix_.column(RowToColIndex(dual_row)));
      (*duplicated_rows)[dual_row] = col;
    }
  }
 
  // We know that the columns are ordered by rows.
  columns_are_known_to_be_clean_ = true;
  transpose_matrix_is_consistent_ = false;
}
 
void LinearProgram::PopulateFromLinearProgram(
    const LinearProgram& linear_program) {
  matrix_.PopulateFromSparseMatrix(linear_program.matrix_);
  if (linear_program.transpose_matrix_is_consistent_) {
    transpose_matrix_is_consistent_ = true;
    transpose_matrix_.PopulateFromSparseMatrix(
        linear_program.transpose_matrix_);
  } else {
    transpose_matrix_is_consistent_ = false;
    transpose_matrix_.Clear();
  }
 
  constraint_lower_bounds_ = linear_program.constraint_lower_bounds_;
  constraint_upper_bounds_ = linear_program.constraint_upper_bounds_;
  constraint_names_ = linear_program.constraint_names_;
  constraint_table_.clear();
 
  PopulateNameObjectiveAndVariablesFromLinearProgram(linear_program);
  first_slack_variable_ = linear_program.first_slack_variable_;
}
 
void LinearProgram::PopulateFromPermutedLinearProgram(
    const LinearProgram& lp, const RowPermutation& row_permutation,
    const ColumnPermutation& col_permutation) {
  DCHECK(lp.IsCleanedUp());
  DCHECK_EQ(row_permutation.size(), lp.num_constraints());
  DCHECK_EQ(col_permutation.size(), lp.num_variables());
  DCHECK_EQ(lp.GetFirstSlackVariable(), kInvalidCol);
  Clear();
 
  // Populate matrix coefficients.
  ColumnPermutation inverse_col_permutation;
  inverse_col_permutation.PopulateFromInverse(col_permutation);
  matrix_.PopulateFromPermutedMatrix(lp.matrix_, row_permutation,
                                     inverse_col_permutation);
  ClearTransposeMatrix();
 
  // Populate constraints.
  ApplyPermutation(row_permutation, lp.constraint_lower_bounds(),
                   &constraint_lower_bounds_);
  ApplyPermutation(row_permutation, lp.constraint_upper_bounds(),
                   &constraint_upper_bounds_);
 
  // Populate variables.
  ApplyPermutation(col_permutation, lp.objective_coefficients(),
                   &objective_coefficients_);
  ApplyPermutation(col_permutation, lp.variable_lower_bounds(),
                   &variable_lower_bounds_);
  ApplyPermutation(col_permutation, lp.variable_upper_bounds(),
                   &variable_upper_bounds_);
  ApplyPermutation(col_permutation, lp.variable_types(), &variable_types_);
  integer_variables_list_is_consistent_ = false;
 
  // There is no vector based accessor to names, because they may be created
  // on the fly.
  constraint_names_.resize(lp.num_constraints());
  for (RowIndex old_row(0); old_row < lp.num_constraints(); ++old_row) {
    const RowIndex new_row = row_permutation[old_row];
    constraint_names_[new_row] = lp.constraint_names_[old_row];
  }
  variable_names_.resize(lp.num_variables());
  for (ColIndex old_col(0); old_col < lp.num_variables(); ++old_col) {
    const ColIndex new_col = col_permutation[old_col];
    variable_names_[new_col] = lp.variable_names_[old_col];
  }
 
  // Populate singular fields.
  maximize_ = lp.maximize_;
  objective_offset_ = lp.objective_offset_;
  objective_scaling_factor_ = lp.objective_scaling_factor_;
  name_ = lp.name_;
}
 
void LinearProgram::PopulateFromLinearProgramVariables(
    const LinearProgram& linear_program) {
  matrix_.PopulateFromZero(RowIndex(0), linear_program.num_variables());
  first_slack_variable_ = kInvalidCol;
  transpose_matrix_is_consistent_ = false;
  transpose_matrix_.Clear();
 
  constraint_lower_bounds_.clear();
  constraint_upper_bounds_.clear();
  constraint_names_.clear();
  constraint_table_.clear();
 
  PopulateNameObjectiveAndVariablesFromLinearProgram(linear_program);
}
 
void LinearProgram::PopulateNameObjectiveAndVariablesFromLinearProgram(
    const LinearProgram& linear_program) {
  objective_coefficients_ = linear_program.objective_coefficients_;
  variable_lower_bounds_ = linear_program.variable_lower_bounds_;
  variable_upper_bounds_ = linear_program.variable_upper_bounds_;
  variable_names_ = linear_program.variable_names_;
  variable_types_ = linear_program.variable_types_;
  integer_variables_list_is_consistent_ =
      linear_program.integer_variables_list_is_consistent_;
  integer_variables_list_ = linear_program.integer_variables_list_;
  binary_variables_list_ = linear_program.binary_variables_list_;
  non_binary_variables_list_ = linear_program.non_binary_variables_list_;
  variable_table_.clear();
 
  maximize_ = linear_program.maximize_;
  objective_offset_ = linear_program.objective_offset_;
  objective_scaling_factor_ = linear_program.objective_scaling_factor_;
  columns_are_known_to_be_clean_ =
      linear_program.columns_are_known_to_be_clean_;
  name_ = linear_program.name_;
}
 
void LinearProgram::AddConstraints(
    const SparseMatrix& coefficients, const DenseColumn& left_hand_sides,
    const DenseColumn& right_hand_sides,
    const StrictITIVector<RowIndex, std::string>& names) {
  const RowIndex num_new_constraints = coefficients.num_rows();
  DCHECK_EQ(num_variables(), coefficients.num_cols());
  DCHECK_EQ(num_new_constraints, left_hand_sides.size());
  DCHECK_EQ(num_new_constraints, right_hand_sides.size());
  DCHECK_EQ(num_new_constraints, names.size());
 
  matrix_.AppendRowsFromSparseMatrix(coefficients);
  transpose_matrix_is_consistent_ = false;
  transpose_matrix_.Clear();
  columns_are_known_to_be_clean_ = false;
 
  // Copy constraint bounds and names from linear_program.
  constraint_lower_bounds_.insert(constraint_lower_bounds_.end(),
                                  left_hand_sides.begin(),
                                  left_hand_sides.end());
  constraint_upper_bounds_.insert(constraint_upper_bounds_.end(),
                                  right_hand_sides.begin(),
                                  right_hand_sides.end());
  constraint_names_.insert(constraint_names_.end(), names.begin(), names.end());
}
 
void LinearProgram::AddConstraintsWithSlackVariables(
    const SparseMatrix& coefficients, const DenseColumn& left_hand_sides,
    const DenseColumn& right_hand_sides,
    const StrictITIVector<RowIndex, std::string>& names,
    bool detect_integer_constraints_for_slack) {
  AddConstraints(coefficients, left_hand_sides, right_hand_sides, names);
  AddSlackVariablesWhereNecessary(detect_integer_constraints_for_slack);
}
 
bool LinearProgram::UpdateVariableBoundsToIntersection(
    const DenseRow& variable_lower_bounds,
    const DenseRow& variable_upper_bounds) {
  const ColIndex num_vars = num_variables();
  DCHECK_EQ(variable_lower_bounds.size(), num_vars);
  DCHECK_EQ(variable_upper_bounds.size(), num_vars);
 
  DenseRow new_lower_bounds(num_vars, 0);
  DenseRow new_upper_bounds(num_vars, 0);
  for (ColIndex i(0); i < num_vars; ++i) {
    const Fractional new_lower_bound =
        std::max(variable_lower_bounds[i], variable_lower_bounds_[i]);
    const Fractional new_upper_bound =
        std::min(variable_upper_bounds[i], variable_upper_bounds_[i]);
    if (new_lower_bound > new_upper_bound) {
      return false;
    }
    new_lower_bounds[i] = new_lower_bound;
    new_upper_bounds[i] = new_upper_bound;
  }
  variable_lower_bounds_.swap(new_lower_bounds);
  variable_upper_bounds_.swap(new_upper_bounds);
  return true;
}
 
void LinearProgram::Swap(LinearProgram* linear_program) {
  matrix_.Swap(&linear_program->matrix_);
  transpose_matrix_.Swap(&linear_program->transpose_matrix_);
 
  constraint_lower_bounds_.swap(linear_program->constraint_lower_bounds_);
  constraint_upper_bounds_.swap(linear_program->constraint_upper_bounds_);
  constraint_names_.swap(linear_program->constraint_names_);
 
  objective_coefficients_.swap(linear_program->objective_coefficients_);
  variable_lower_bounds_.swap(linear_program->variable_lower_bounds_);
  variable_upper_bounds_.swap(linear_program->variable_upper_bounds_);
  variable_names_.swap(linear_program->variable_names_);
  variable_types_.swap(linear_program->variable_types_);
  integer_variables_list_.swap(linear_program->integer_variables_list_);
  binary_variables_list_.swap(linear_program->binary_variables_list_);
  non_binary_variables_list_.swap(linear_program->non_binary_variables_list_);
 
  variable_table_.swap(linear_program->variable_table_);
  constraint_table_.swap(linear_program->constraint_table_);
 
  std::swap(maximize_, linear_program->maximize_);
  std::swap(objective_offset_, linear_program->objective_offset_);
  std::swap(objective_scaling_factor_,
            linear_program->objective_scaling_factor_);
  std::swap(columns_are_known_to_be_clean_,
            linear_program->columns_are_known_to_be_clean_);
  std::swap(transpose_matrix_is_consistent_,
            linear_program->transpose_matrix_is_consistent_);
  std::swap(integer_variables_list_is_consistent_,
            linear_program->integer_variables_list_is_consistent_);
  name_.swap(linear_program->name_);
  std::swap(first_slack_variable_, linear_program->first_slack_variable_);
}
 
void LinearProgram::DeleteColumns(const DenseBooleanRow& columns_to_delete) {
  if (columns_to_delete.empty()) return;
  integer_variables_list_is_consistent_ = false;
  const ColIndex num_cols = num_variables();
  ColumnPermutation permutation(num_cols);
  ColIndex new_index(0);
  for (ColIndex col(0); col < num_cols; ++col) {
    permutation[col] = new_index;
    if (col >= columns_to_delete.size() || !columns_to_delete[col]) {
      objective_coefficients_[new_index] = objective_coefficients_[col];
      variable_lower_bounds_[new_index] = variable_lower_bounds_[col];
      variable_upper_bounds_[new_index] = variable_upper_bounds_[col];
      variable_names_[new_index] = variable_names_[col];
      variable_types_[new_index] = variable_types_[col];
      ++new_index;
    } else {
      permutation[col] = kInvalidCol;
    }
  }
 
  matrix_.DeleteColumns(columns_to_delete);
  objective_coefficients_.resize(new_index, 0.0);
  variable_lower_bounds_.resize(new_index, 0.0);
  variable_upper_bounds_.resize(new_index, 0.0);
  variable_types_.resize(new_index, VariableType::CONTINUOUS);
  variable_names_.resize(new_index, "");
 
  // Remove the id of the deleted columns and adjust the index of the other.
  absl::flat_hash_map<std::string, ColIndex>::iterator it =
      variable_table_.begin();
  while (it != variable_table_.end()) {
    const ColIndex col = it->second;
    if (col >= columns_to_delete.size() || !columns_to_delete[col]) {
      it->second = permutation[col];
      ++it;
    } else {
      // This safely deletes the entry and moves the iterator one step ahead.
      variable_table_.erase(it++);
    }
  }
 
  // Eventually update transpose_matrix_.
  if (transpose_matrix_is_consistent_) {
    transpose_matrix_.DeleteRows(
        ColToRowIndex(new_index),
        reinterpret_cast<const RowPermutation&>(permutation));
  }
}
 
void LinearProgram::DeleteSlackVariables() {
  DCHECK_NE(first_slack_variable_, kInvalidCol);
  DenseBooleanRow slack_variables(matrix_.num_cols(), false);
  // Restore the bounds on the constraints corresponding to the slack variables.
  for (ColIndex slack_variable = first_slack_variable_;
       slack_variable < matrix_.num_cols(); ++slack_variable) {
    const SparseColumn& column = matrix_.column(slack_variable);
    // Slack variables appear only in the constraints for which they were
    // created. We can find this constraint by looking at the (only) entry in
    // the columnm of the slack variable.
    DCHECK_EQ(column.num_entries(), 1);
    const RowIndex row = column.EntryRow(EntryIndex(0));
    DCHECK_EQ(constraint_lower_bounds_[row], 0.0);
    DCHECK_EQ(constraint_upper_bounds_[row], 0.0);
    SetConstraintBounds(row, -variable_upper_bounds_[slack_variable],
                        -variable_lower_bounds_[slack_variable]);
    slack_variables[slack_variable] = true;
  }
 
  DeleteColumns(slack_variables);
  first_slack_variable_ = kInvalidCol;
}
 
namespace {
 
// Note that we ignore zeros and infinities because they do not matter from a
// scaling perspective where this function is used.
template <typename FractionalRange>
void UpdateMinAndMaxMagnitude(const FractionalRange& range,
                              Fractional* min_magnitude,
                              Fractional* max_magnitude) {
  for (const Fractional value : range) {
    const Fractional magnitude = std::abs(value);
    if (magnitude == 0 || magnitude == kInfinity) continue;
    *min_magnitude = std::min(*min_magnitude, magnitude);
    *max_magnitude = std::max(*max_magnitude, magnitude);
  }
}
 
Fractional GetMedianScalingFactor(const DenseRow& range) {
  std::vector<Fractional> median;
  for (const Fractional value : range) {
    if (value == 0.0) continue;
    median.push_back(std::abs(value));
  }
  if (median.empty()) return 1.0;
  std::sort(median.begin(), median.end());
  return median[median.size() / 2];
}
 
Fractional GetMeanScalingFactor(const DenseRow& range) {
  Fractional mean = 0.0;
  int num_non_zeros = 0;
  for (const Fractional value : range) {
    if (value == 0.0) continue;
    ++num_non_zeros;
    mean += std::abs(value);
  }
  if (num_non_zeros == 0.0) return 1.0;
  return mean / static_cast<Fractional>(num_non_zeros);
}
 
Fractional ComputeDivisorSoThatRangeContainsOne(Fractional min_magnitude,
                                                Fractional max_magnitude) {
  if (min_magnitude > 1.0 && min_magnitude < kInfinity) {
    return min_magnitude;
  } else if (max_magnitude > 0.0 && max_magnitude < 1.0) {
    return max_magnitude;
  }
  return 1.0;
}
 
}  // namespace
 
Fractional LinearProgram::ScaleObjective(
    GlopParameters::CostScalingAlgorithm method) {
  Fractional min_magnitude = kInfinity;
  Fractional max_magnitude = 0.0;
  UpdateMinAndMaxMagnitude(objective_coefficients(), &min_magnitude,
                           &max_magnitude);
  Fractional cost_scaling_factor = 1.0;
  switch (method) {
    case GlopParameters::NO_COST_SCALING:
      break;
    case GlopParameters::CONTAIN_ONE_COST_SCALING:
      cost_scaling_factor =
          ComputeDivisorSoThatRangeContainsOne(min_magnitude, max_magnitude);
      break;
    case GlopParameters::MEAN_COST_SCALING:
      cost_scaling_factor = GetMeanScalingFactor(objective_coefficients());
      break;
    case GlopParameters::MEDIAN_COST_SCALING:
      cost_scaling_factor = GetMedianScalingFactor(objective_coefficients());
      break;
  }
  if (cost_scaling_factor != 1.0) {
    for (ColIndex col(0); col < num_variables(); ++col) {
      if (objective_coefficients()[col] == 0.0) continue;
      SetObjectiveCoefficient(
          col, objective_coefficients()[col] / cost_scaling_factor);
    }
    SetObjectiveScalingFactor(objective_scaling_factor() * cost_scaling_factor);
    SetObjectiveOffset(objective_offset() / cost_scaling_factor);
  }
  VLOG(1) << "Objective magnitude range is [" << min_magnitude << ", "
          << max_magnitude << "] (dividing by " << cost_scaling_factor << ").";
  return cost_scaling_factor;
}
 
Fractional LinearProgram::ScaleBounds() {
  Fractional min_magnitude = kInfinity;
  Fractional max_magnitude = 0.0;
  UpdateMinAndMaxMagnitude(variable_lower_bounds(), &min_magnitude,
                           &max_magnitude);
  UpdateMinAndMaxMagnitude(variable_upper_bounds(), &min_magnitude,
                           &max_magnitude);
  UpdateMinAndMaxMagnitude(constraint_lower_bounds(), &min_magnitude,
                           &max_magnitude);
  UpdateMinAndMaxMagnitude(constraint_upper_bounds(), &min_magnitude,
                           &max_magnitude);
  const Fractional bound_scaling_factor =
      ComputeDivisorSoThatRangeContainsOne(min_magnitude, max_magnitude);
  if (bound_scaling_factor != 1.0) {
    SetObjectiveScalingFactor(objective_scaling_factor() *
                              bound_scaling_factor);
    SetObjectiveOffset(objective_offset() / bound_scaling_factor);
    for (ColIndex col(0); col < num_variables(); ++col) {
      SetVariableBounds(col,
                        variable_lower_bounds()[col] / bound_scaling_factor,
                        variable_upper_bounds()[col] / bound_scaling_factor);
    }
    for (RowIndex row(0); row < num_constraints(); ++row) {
      SetConstraintBounds(
          row, constraint_lower_bounds()[row] / bound_scaling_factor,
          constraint_upper_bounds()[row] / bound_scaling_factor);
    }
  }
 
  VLOG(1) << "Bounds magnitude range is [" << min_magnitude << ", "
          << max_magnitude << "] (dividing bounds by " << bound_scaling_factor
          << ").";
  return bound_scaling_factor;
}
 
void LinearProgram::DeleteRows(const DenseBooleanColumn& rows_to_delete) {
  if (rows_to_delete.empty()) return;
 
  // Deal with row-indexed data and construct the row mapping that will need to
  // be applied to every column entry.
  const RowIndex num_rows = num_constraints();
  RowPermutation permutation(num_rows);
  RowIndex new_index(0);
  for (RowIndex row(0); row < num_rows; ++row) {
    if (row >= rows_to_delete.size() || !rows_to_delete[row]) {
      constraint_lower_bounds_[new_index] = constraint_lower_bounds_[row];
      constraint_upper_bounds_[new_index] = constraint_upper_bounds_[row];
      constraint_names_[new_index].swap(constraint_names_[row]);
      permutation[row] = new_index;
      ++new_index;
    } else {
      permutation[row] = kInvalidRow;
    }
  }
  constraint_lower_bounds_.resize(new_index, 0.0);
  constraint_upper_bounds_.resize(new_index, 0.0);
  constraint_names_.resize(new_index, "");
 
  // Remove the rows from the matrix.
  matrix_.DeleteRows(new_index, permutation);
 
  // Remove the id of the deleted rows and adjust the index of the other.
  absl::flat_hash_map<std::string, RowIndex>::iterator it =
      constraint_table_.begin();
  while (it != constraint_table_.end()) {
    const RowIndex row = it->second;
    if (permutation[row] != kInvalidRow) {
      it->second = permutation[row];
      ++it;
    } else {
      // This safely deletes the entry and moves the iterator one step ahead.
      constraint_table_.erase(it++);
    }
  }
 
  // Eventually update transpose_matrix_.
  if (transpose_matrix_is_consistent_) {
    transpose_matrix_.DeleteColumns(
        reinterpret_cast<const DenseBooleanRow&>(rows_to_delete));
  }
}
 
bool LinearProgram::IsValid() const {
  if (!IsFinite(objective_offset_)) return false;
  if (!IsFinite(objective_scaling_factor_)) return false;
  if (objective_scaling_factor_ == 0.0) return false;
  const ColIndex num_cols = num_variables();
  for (ColIndex col(0); col < num_cols; ++col) {
    if (!AreBoundsValid(variable_lower_bounds()[col],
                        variable_upper_bounds()[col])) {
      return false;
    }
    if (!IsFinite(objective_coefficients()[col])) {
      return false;
    }
    for (const SparseColumn::Entry e : GetSparseColumn(col)) {
      if (!IsFinite(e.coefficient())) return false;
    }
  }
  if (constraint_upper_bounds_.size() != constraint_lower_bounds_.size()) {
    return false;
  }
  for (RowIndex row(0); row < constraint_lower_bounds_.size(); ++row) {
    if (!AreBoundsValid(constraint_lower_bounds()[row],
                        constraint_upper_bounds()[row])) {
      return false;
    }
  }
  return true;
}
 
std::string LinearProgram::ProblemStatFormatter(
    const absl::string_view format) const {
  int num_objective_non_zeros = 0;
  int num_non_negative_variables = 0;
  int num_boxed_variables = 0;
  int num_free_variables = 0;
  int num_fixed_variables = 0;
  int num_other_variables = 0;
  const ColIndex num_cols = num_variables();
  for (ColIndex col(0); col < num_cols; ++col) {
    if (objective_coefficients()[col] != 0.0) {
      ++num_objective_non_zeros;
    }
 
    const Fractional lower_bound = variable_lower_bounds()[col];
    const Fractional upper_bound = variable_upper_bounds()[col];
    const bool lower_bounded = (lower_bound != -kInfinity);
    const bool upper_bounded = (upper_bound != kInfinity);
 
    if (!lower_bounded && !upper_bounded) {
      ++num_free_variables;
    } else if (lower_bound == 0.0 && !upper_bounded) {
      ++num_non_negative_variables;
    } else if (!upper_bounded || !lower_bounded) {
      ++num_other_variables;
    } else if (lower_bound == upper_bound) {
      ++num_fixed_variables;
    } else {
      ++num_boxed_variables;
    }
  }
 
  int num_range_constraints = 0;
  int num_less_than_constraints = 0;
  int num_greater_than_constraints = 0;
  int num_equal_constraints = 0;
  int num_rhs_non_zeros = 0;
  const RowIndex num_rows = num_constraints();
  for (RowIndex row(0); row < num_rows; ++row) {
    const Fractional lower_bound = constraint_lower_bounds()[row];
    const Fractional upper_bound = constraint_upper_bounds()[row];
    if (AreBoundsFreeOrBoxed(lower_bound, upper_bound)) {
      // TODO(user): we currently count a free row as a range constraint.
      // Add a new category?
      ++num_range_constraints;
      continue;
    }
    if (lower_bound == upper_bound) {
      ++num_equal_constraints;
      if (lower_bound != 0) {
        ++num_rhs_non_zeros;
      }
      continue;
    }
    if (lower_bound == -kInfinity) {
      ++num_less_than_constraints;
      if (upper_bound != 0) {
        ++num_rhs_non_zeros;
      }
      continue;
    }
    if (upper_bound == kInfinity) {
      ++num_greater_than_constraints;
      if (lower_bound != 0) {
        ++num_rhs_non_zeros;
      }
      continue;
    }
    LOG(DFATAL) << "There is a bug since all possible cases for the row bounds "
                   "should have been accounted for. row="
                << row;
  }
 
  const int num_integer_variables = IntegerVariablesList().size();
  const int num_binary_variables = BinaryVariablesList().size();
  const int num_non_binary_variables = NonBinaryVariablesList().size();
  const int num_continuous_variables =
      ColToIntIndex(num_variables()) - num_integer_variables;
  auto format_runtime =
      absl::ParsedFormat<'d', 'd', 'd', 'd', 'd', 'd', 'd', 'd', 'd', 'd', 'd',
                         'd', 'd', 'd', 'd', 'd', 'd', 'd'>::New(format);
  CHECK(format_runtime);
  return absl::StrFormat(
      *format_runtime, RowToIntIndex(num_constraints()),
      ColToIntIndex(num_variables()), matrix_.num_entries().value(),
      num_objective_non_zeros, num_rhs_non_zeros, num_less_than_constraints,
      num_greater_than_constraints, num_equal_constraints,
      num_range_constraints, num_non_negative_variables, num_boxed_variables,
      num_free_variables, num_fixed_variables, num_other_variables,
      num_integer_variables, num_binary_variables, num_non_binary_variables,
      num_continuous_variables);
}
 
std::string LinearProgram::NonZeroStatFormatter(
    const absl::string_view format) const {
  StrictITIVector<RowIndex, EntryIndex> num_entries_in_row(num_constraints(),
                                                           EntryIndex(0));
  StrictITIVector<ColIndex, EntryIndex> num_entries_in_column(num_variables(),
                                                              EntryIndex(0));
  EntryIndex num_entries(0);
  const ColIndex num_cols = num_variables();
  for (ColIndex col(0); col < num_cols; ++col) {
    const SparseColumn& sparse_column = GetSparseColumn(col);
    num_entries += sparse_column.num_entries();
    num_entries_in_column[col] = sparse_column.num_entries();
    for (const SparseColumn::Entry e : sparse_column) {
      ++num_entries_in_row[e.row()];
    }
  }
 
  // To avoid division by 0 if there are no columns or no rows, we set
  // height and width to be at least one.
  const int64 height = std::max(RowToIntIndex(num_constraints()), 1);
  const int64 width = std::max(ColToIntIndex(num_variables()), 1);
  const double fill_rate = 100.0 * static_cast<double>(num_entries.value()) /
                           static_cast<double>(height * width);
 
  auto format_runtime =
      absl::ParsedFormat<'f', 'd', 'f', 'f', 'd', 'f', 'f'>::New(format);
  return absl::StrFormat(
      *format_runtime, fill_rate, GetMaxElement(num_entries_in_row).value(),
      Average(num_entries_in_row), StandardDeviation(num_entries_in_row),
      GetMaxElement(num_entries_in_column).value(),
      Average(num_entries_in_column), StandardDeviation(num_entries_in_column));
}
 
void LinearProgram::ResizeRowsIfNeeded(RowIndex row) {
  DCHECK_GE(row, 0);
  if (row >= num_constraints()) {
    transpose_matrix_is_consistent_ = false;
    matrix_.SetNumRows(row + 1);
    constraint_lower_bounds_.resize(row + 1, Fractional(0.0));
    constraint_upper_bounds_.resize(row + 1, Fractional(0.0));
    constraint_names_.resize(row + 1, "");
  }
}
 
bool LinearProgram::IsInEquationForm() const {
  for (RowIndex constraint(0); constraint < num_constraints(); ++constraint) {
    if (constraint_lower_bounds_[constraint] != 0.0 ||
        constraint_upper_bounds_[constraint] != 0.0) {
      return false;
    }
  }
  const ColIndex num_slack_variables =
      num_variables() - GetFirstSlackVariable();
  return num_constraints().value() == num_slack_variables.value() &&
         IsRightMostSquareMatrixIdentity(matrix_);
}
 
bool LinearProgram::BoundsOfIntegerVariablesAreInteger(
    Fractional tolerance) const {
  for (const ColIndex col : IntegerVariablesList()) {
    if ((IsFinite(variable_lower_bounds_[col]) &&
         !IsIntegerWithinTolerance(variable_lower_bounds_[col], tolerance)) ||
        (IsFinite(variable_upper_bounds_[col]) &&
         !IsIntegerWithinTolerance(variable_upper_bounds_[col], tolerance))) {
      VLOG(1) << "Bounds of variable " << col.value() << " are non-integer ("
              << variable_lower_bounds_[col] << ", "
              << variable_upper_bounds_[col] << ").";
      return false;
    }
  }
  return true;
}
 
bool LinearProgram::BoundsOfIntegerConstraintsAreInteger(
    Fractional tolerance) const {
  // Using transpose for this is faster (complexity = O(number of non zeros in
  // matrix)) than directly iterating through entries (complexity = O(number of
  // constraints * number of variables)).
  const SparseMatrix& transpose = GetTransposeSparseMatrix();
  for (RowIndex row = RowIndex(0); row < num_constraints(); ++row) {
    bool integer_constraint = true;
    for (const SparseColumn::Entry var : transpose.column(RowToColIndex(row))) {
      if (!IsVariableInteger(RowToColIndex(var.row()))) {
        integer_constraint = false;
        break;
      }
      if (!IsIntegerWithinTolerance(var.coefficient(), tolerance)) {
        integer_constraint = false;
        break;
      }
    }
    if (integer_constraint) {
      if ((IsFinite(constraint_lower_bounds_[row]) &&
           !IsIntegerWithinTolerance(constraint_lower_bounds_[row],
                                     tolerance)) ||
          (IsFinite(constraint_upper_bounds_[row]) &&
           !IsIntegerWithinTolerance(constraint_upper_bounds_[row],
                                     tolerance))) {
        VLOG(1) << "Bounds of constraint " << row.value()
                << " are non-integer (" << constraint_lower_bounds_[row] << ", "
                << constraint_upper_bounds_[row] << ").";
        return false;
      }
    }
  }
  return true;
}
 
// --------------------------------------------------------
// ProblemSolution
// --------------------------------------------------------
std::string ProblemSolution::DebugString() const {
  std::string s = "Problem status: " + GetProblemStatusString(status);
  for (ColIndex col(0); col < primal_values.size(); ++col) {
    absl::StrAppendFormat(&s, "\n  Var #%d: %s %g", col.value(),
                          GetVariableStatusString(variable_statuses[col]),
                          primal_values[col]);
  }
  s += "\n------------------------------";
  for (RowIndex row(0); row < dual_values.size(); ++row) {
    absl::StrAppendFormat(&s, "\n  Constraint #%d: %s %g", row.value(),
                          GetConstraintStatusString(constraint_statuses[row]),
                          dual_values[row]);
  }
  return s;
}
 
}  // namespace glop
}  // namespace operations_research