bop_ls.h

// 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.

// This file defines the needed classes to efficiently perform Local Search in
// Bop.
// Local Search is a technique used to locally improve an existing solution by
// flipping a limited number of variables. To be successful the produced
// solution has to satisfy all constraints of the problem and improve the
// objective cost.
//
// The class BopLocalSearchOptimizer is the only public interface for Local
// Search in Bop. For unit-testing purposes this file also contains the four
// internal classes AssignmentAndConstraintFeasibilityMaintainer,
// OneFlipConstraintRepairer, SatWrapper and LocalSearchAssignmentIterator.
// They are implementation details and should not be used outside of bop_ls.

#ifndef OR_TOOLS_BOP_BOP_LS_H_
#define OR_TOOLS_BOP_BOP_LS_H_

#include <array>

#include "absl/container/flat_hash_map.h"
#include "absl/container/flat_hash_set.h"
#include "absl/random/random.h"
#include "ortools/base/hash.h"
#include "ortools/base/random.h"
#include "ortools/bop/bop_base.h"
#include "ortools/bop/bop_solution.h"
#include "ortools/bop/bop_types.h"
#include "ortools/sat/boolean_problem.pb.h"
#include "ortools/sat/sat_solver.h"

namespace operations_research {
namespace bop {

// This class is used to ease the connection with the SAT solver.
//
// TODO(user): remove? the meat of the logic is used in just one place, so I am
// not sure having this extra layer improve the readability.
class SatWrapper {
 public:
  explicit SatWrapper(sat::SatSolver* sat_solver);

  // Returns the current state of the solver propagation trail.
  std::vector<sat::Literal> FullSatTrail() const;

  // Returns true if the problem is UNSAT.
  // Note that an UNSAT problem might not be marked as UNSAT at first because
  // the SAT solver is not able to prove it; After some decisions / learned
  // conflicts, the SAT solver might be able to prove UNSAT and so this will
  // return true.
  bool IsModelUnsat() const { return sat_solver_->IsModelUnsat(); }

  // Return the current solver VariablesAssignment.
  const sat::VariablesAssignment& SatAssignment() const {
    return sat_solver_->Assignment();
  }

  // Applies the decision that makes the given literal true and returns the
  // number of decisions to backtrack due to conflicts if any.
  // Two cases:
  //   - No conflicts: Returns 0 and fills the propagated_literals with the
  //     literals that have been propagated due to the decision including the
  //     the decision itself.
  //   - Conflicts: Returns the number of decisions to backtrack (the current
  //     decision included, i.e. returned value > 0) and fills the
  //     propagated_literals with the literals that the conflicts propagated.
  // Note that the decision variable should not be already assigned in SAT.
  int ApplyDecision(sat::Literal decision_literal,
                    std::vector<sat::Literal>* propagated_literals);

  // Backtracks the last decision if any.
  void BacktrackOneLevel();

  // Bactracks all the decisions.
  void BacktrackAll();

  // Extracts any new information learned during the search.
  void ExtractLearnedInfo(LearnedInfo* info);

  // Returns a deterministic number that should be correlated with the time
  // spent in the SAT wrapper. The order of magnitude should be close to the
  // time in seconds.
  double deterministic_time() const;

 private:
  sat::SatSolver* sat_solver_;
  DISALLOW_COPY_AND_ASSIGN(SatWrapper);
};

// Forward declaration.
class LocalSearchAssignmentIterator;

// This class defines a Local Search optimizer. The goal is to find a new
// solution with a better cost than the given solution by iterating on all
// assignments that can be reached in max_num_decisions decisions or less.
// The bop parameter max_number_of_explored_assignments_per_try_in_ls can be
// used to specify the number of new assignments to iterate on each time the
// method Optimize() is called. Limiting that parameter allows to reduce the
// time spent in the Optimize() method at once, and still explore all the
// reachable assignments (if Optimize() is called enough times).
// Note that due to propagation, the number of variables with a different value
// in the new solution can be greater than max_num_decisions.
class LocalSearchOptimizer : public BopOptimizerBase {
 public:
  LocalSearchOptimizer(const std::string& name, int max_num_decisions,
                       sat::SatSolver* sat_propagator);
  ~LocalSearchOptimizer() override;

 private:
  bool ShouldBeRun(const ProblemState& problem_state) const override;
  Status Optimize(const BopParameters& parameters,
                  const ProblemState& problem_state, LearnedInfo* learned_info,
                  TimeLimit* time_limit) override;

  int64 state_update_stamp_;

  // Maximum number of decisions the Local Search can take.
  // Note that there is no limit on the number of changed variables due to
  // propagation.
  const int max_num_decisions_;

  // A wrapper around the given sat_propagator.
  SatWrapper sat_wrapper_;

  // Iterator on all reachable assignments.
  // Note that this iterator is only reset when Synchronize() is called, i.e.
  // the iterator continues its iteration of the next assignments each time
  // Optimize() is called until everything is explored or a solution is found.
  std::unique_ptr<LocalSearchAssignmentIterator> assignment_iterator_;
};

//------------------------------------------------------------------------------
// Implementation details. The declarations of those utility classes are in
// the .h for testing reasons.
//------------------------------------------------------------------------------

// Maintains some information on a sparse set of integers in [0, n). More
// specifically this class:
// - Allows to dynamically add/remove element from the set.
// - Has a backtracking support.
// - Maintains the number of elements in the set.
// - Maintains a superset of the elements of the set that contains all the
//   modified elements.
template <typename IntType>
class BacktrackableIntegerSet {
 public:
  BacktrackableIntegerSet() {}

  // Prepares the class for integers in [0, n) and initializes the set to the
  // empty one. Note that this run in O(n). Once resized, it is better to call
  // BacktrackAll() instead of this to clear the set.
  void ClearAndResize(IntType n);

  // Changes the state of the given integer i to be either inside or outside the
  // set. Important: this should only be called with the opposite state of the
  // current one, otherwise size() will not be correct.
  void ChangeState(IntType i, bool should_be_inside);

  // Returns the current number of elements in the set.
  // Note that this is not its maximum size n.
  int size() const { return size_; }

  // Returns a superset of the current set of integers.
  const std::vector<IntType>& Superset() const { return stack_; }

  // BacktrackOneLevel() backtracks to the state the class was in when the
  // last AddBacktrackingLevel() was called. BacktrackAll() just restore the
  // class to its state just after the last ClearAndResize().
  void AddBacktrackingLevel();
  void BacktrackOneLevel();
  void BacktrackAll();

 private:
  int size_;

  // Contains the elements whose status has been changed at least once.
  std::vector<IntType> stack_;
  std::vector<bool> in_stack_;

  // Used for backtracking. Contains the size_ and the stack_.size() at the time
  // of each call to AddBacktrackingLevel() that is not yet backtracked over.
  std::vector<int> saved_sizes_;
  std::vector<int> saved_stack_sizes_;
};

// A simple and efficient class to hash a given set of integers in [0, n).
// It uses O(n) memory and produces a good hash (random linear function).
template <typename IntType>
class NonOrderedSetHasher {
 public:
  NonOrderedSetHasher() : random_("Random seed") {}

  // Initializes the NonOrderedSetHasher to hash sets of integer in [0, n).
  void Initialize(int size) {
    hashes_.resize(size);
    for (IntType i(0); i < size; ++i) {
      hashes_[i] = random_.Rand64();
    }
  }

  // Ignores the given set element in all subsequent hash computation. Note that
  // this will be reset by the next call to Initialize().
  void IgnoreElement(IntType e) { hashes_[e] = 0; }

  // Returns the hash of the given set. The hash is independent of the set
  // order, but there must be no duplicate element in the set. This uses a
  // simple random linear function which has really good hashing properties.
  uint64 Hash(const std::vector<IntType>& set) const {
    uint64 hash = 0;
    for (const IntType i : set) hash ^= hashes_[i];
    return hash;
  }

  // The hash of a set is simply the XOR of all its elements. This allows
  // to compute an hash incrementally or without the need of a vector<>.
  uint64 Hash(IntType e) const { return hashes_[e]; }

  // Returns true if Initialize() has been called with a non-zero size.
  bool IsInitialized() const { return !hashes_.empty(); }

 private:
  MTRandom random_;
  gtl::ITIVector<IntType, uint64> hashes_;
};

// This class is used to incrementally maintain an assignment and the
// feasibility of the constraints of a given LinearBooleanProblem.
//
// The current assignment is initialized using a feasible reference solution,
// i.e. the reference solution satisfies all the constraints of the problem.
// The current assignment is updated using the Assign() method.
//
// Note that the current assignment is not a solution in the sense that it
// might not be feasible, ie. violates some constraints.
//
// The assignment can be accessed at any time using Assignment().
// The set of infeasible constraints can be accessed at any time using
// PossiblyInfeasibleConstraints().
//
// Note that this class is reversible, i.e. it is possible to backtrack to
// previously added backtracking levels.
// levels. Consider for instance variable a, b, c, and d.
//      Method called                 Assigned after method call
//   1- Assign({a, b})                         a b
//   2- AddBacktrackingLevel()                 a b |
//   3- Assign({c})                            a b | c
//   4- Assign({d})                            a b | c d
//   5- BacktrackOneLevel()                    a b
//   6- Assign({c})                            a b c
//   7- BacktrackOneLevel()
class AssignmentAndConstraintFeasibilityMaintainer {
 public:
  // Note that the constraint indices used in this class are not the same as
  // the one used in the given LinearBooleanProblem here.
  explicit AssignmentAndConstraintFeasibilityMaintainer(
      const sat::LinearBooleanProblem& problem);

  // When we construct the problem, we treat the objective as one constraint.
  // This is the index of this special "objective" constraint.
  static const ConstraintIndex kObjectiveConstraint;

  // Sets a new reference solution and reverts all internal structures to their
  // initial state. Note that the reference solution has to be feasible.
  void SetReferenceSolution(const BopSolution& reference_solution);

  // Behaves exactly like SetReferenceSolution() where the passed reference
  // is the current assignment held by this class. Note that the current
  // assignment must be feasible (i.e. IsFeasible() is true).
  void UseCurrentStateAsReference();

  // Assigns all literals. That updates the assignment, the constraint values,
  // and the infeasible constraints.
  // Note that the assignment of those literals can be reverted thanks to
  // AddBacktrackingLevel() and BacktrackOneLevel().
  // Note that a variable can't be assigned twice, even for the same literal.
  void Assign(const std::vector<sat::Literal>& literals);

  // Adds a new backtracking level to specify the state that will be restored
  // by BacktrackOneLevel().
  // See the example in the class comment.
  void AddBacktrackingLevel();

  // Backtracks internal structures to the previous level defined by
  // AddBacktrackingLevel(). As a consequence the state will be exactly as
  // before the previous call to AddBacktrackingLevel().
  // Note that backtracking the initial state has no effect.
  void BacktrackOneLevel();
  void BacktrackAll();

  // This returns the list of literal that appear in exactly all the current
  // infeasible constraints (ignoring the objective) and correspond to a flip in
  // a good direction for all the infeasible constraint. Performing this flip
  // may repair the problem without any propagations.
  //
  // Important: The returned reference is only valid until the next
  // PotentialOneFlipRepairs() call.
  const std::vector<sat::Literal>& PotentialOneFlipRepairs();

  // Returns true if there is no infeasible constraint in the current state.
  bool IsFeasible() const { return infeasible_constraint_set_.size() == 0; }

  // Returns the *exact* number of infeasible constraints.
  // Note that PossiblyInfeasibleConstraints() will potentially return a larger
  // number of constraints.
  int NumInfeasibleConstraints() const {
    return infeasible_constraint_set_.size();
  }

  // Returns a superset of all the infeasible constraints in the current state.
  const std::vector<ConstraintIndex>& PossiblyInfeasibleConstraints() const {
    return infeasible_constraint_set_.Superset();
  }

  // Returns the number of constraints of the problem, objective included,
  // i.e. the number of constraint in the problem + 1.
  size_t NumConstraints() const { return constraint_lower_bounds_.size(); }

  // Returns the value of the var in the assignment.
  // As the assignment is initialized with the reference solution, if the
  // variable has not been assigned through Assign(), the returned value is
  // the value of the variable in the reference solution.
  bool Assignment(VariableIndex var) const { return assignment_.Value(var); }

  // Returns the current assignment.
  const BopSolution& reference() const { return reference_; }

  // Returns the lower bound of the constraint.
  int64 ConstraintLowerBound(ConstraintIndex constraint) const {
    return constraint_lower_bounds_[constraint];
  }

  // Returns the upper bound of the constraint.
  int64 ConstraintUpperBound(ConstraintIndex constraint) const {
    return constraint_upper_bounds_[constraint];
  }

  // Returns the value of the constraint. The value is computed using the
  // variable values in the assignment. Note that a constraint is feasible iff
  // its value is between its two bounds (inclusive).
  int64 ConstraintValue(ConstraintIndex constraint) const {
    return constraint_values_[constraint];
  }

  // Returns true if the given constraint is currently feasible.
  bool ConstraintIsFeasible(ConstraintIndex constraint) const {
    const int64 value = ConstraintValue(constraint);
    return value >= ConstraintLowerBound(constraint) &&
           value <= ConstraintUpperBound(constraint);
  }

  std::string DebugString() const;

 private:
  // This is lazily called by PotentialOneFlipRepairs() once.
  void InitializeConstraintSetHasher();

  // This is used by PotentialOneFlipRepairs(). It encodes a ConstraintIndex
  // together with a "repair" direction depending on the bound that make a
  // constraint infeasible. An "up" direction means that the constraint activity
  // is lower than the lower bound and we need to make the activity move up to
  // fix the infeasibility.
  DEFINE_INT_TYPE(ConstraintIndexWithDirection, int32);
  ConstraintIndexWithDirection FromConstraintIndex(ConstraintIndex index,
                                                   bool up) const {
    return ConstraintIndexWithDirection(2 * index.value() + (up ? 1 : 0));
  }

  // Over constrains the objective cost by the given delta. This should only be
  // called on a feasible reference solution and a fully backtracked state.
  void MakeObjectiveConstraintInfeasible(int delta);

  // Local structure to represent the sparse matrix by variable used for fast
  // update of the contraint values.
  struct ConstraintEntry {
    ConstraintEntry(ConstraintIndex c, int64 w) : constraint(c), weight(w) {}
    ConstraintIndex constraint;
    int64 weight;
  };

  gtl::ITIVector<VariableIndex, gtl::ITIVector<EntryIndex, ConstraintEntry>>
      by_variable_matrix_;
  gtl::ITIVector<ConstraintIndex, int64> constraint_lower_bounds_;
  gtl::ITIVector<ConstraintIndex, int64> constraint_upper_bounds_;

  BopSolution assignment_;
  BopSolution reference_;

  gtl::ITIVector<ConstraintIndex, int64> constraint_values_;
  BacktrackableIntegerSet<ConstraintIndex> infeasible_constraint_set_;

  // This contains the list of variable flipped in assignment_.
  // flipped_var_trail_backtrack_levels_[i-1] is the index in flipped_var_trail_
  // of the first variable flipped after the i-th AddBacktrackingLevel() call.
  std::vector<int> flipped_var_trail_backtrack_levels_;
  std::vector<VariableIndex> flipped_var_trail_;

  // Members used by PotentialOneFlipRepairs().
  std::vector<sat::Literal> tmp_potential_repairs_;
  NonOrderedSetHasher<ConstraintIndexWithDirection> constraint_set_hasher_;
  absl::flat_hash_map<uint64, std::vector<sat::Literal>>
      hash_to_potential_repairs_;

  DISALLOW_COPY_AND_ASSIGN(AssignmentAndConstraintFeasibilityMaintainer);
};

// This class is an utility class used to select which infeasible constraint to
// repair and identify one variable to flip to actually repair the constraint.
// A constraint 'lb <= sum_i(w_i * x_i) <= ub', with 'lb' the lower bound,
// 'ub' the upper bound, 'w_i' the weight of the i-th term and 'x_i' the
// boolean variable appearing in the i-th term, is infeasible for a given
// assignment iff its value 'sum_i(w_i * x_i)' is outside of the bounds.
// Repairing-a-constraint-in-one-flip means making the constraint feasible by
// just flipping the value of one unassigned variable of the current assignment
// from the AssignmentAndConstraintFeasibilityMaintainer.
// For performance reasons, the pairs weight / variable (w_i, x_i) are stored
// in a sparse manner as a vector of terms (w_i, x_i). In the following the
// TermIndex term_index refers to the position of the term in the vector.
class OneFlipConstraintRepairer {
 public:
  // Note that the constraint indices used in this class follow the same
  // convention as the one used in the
  // AssignmentAndConstraintFeasibilityMaintainer.
  //
  // TODO(user): maybe merge the two classes? maintaining this implicit indices
  // convention between the two classes sounds like a bad idea.
  OneFlipConstraintRepairer(
      const sat::LinearBooleanProblem& problem,
      const AssignmentAndConstraintFeasibilityMaintainer& maintainer,
      const sat::VariablesAssignment& sat_assignment);

  static const ConstraintIndex kInvalidConstraint;
  static const TermIndex kInitTerm;
  static const TermIndex kInvalidTerm;

  // Returns the index of a constraint to repair. This will always return the
  // index of a constraint that can be repaired in one flip if there is one.
  // Note however that if there is only one possible candidate, it will be
  // returned without checking that it can indeed be repaired in one flip.
  // This is because the later check can be expensive, and is not needed in our
  // context.
  ConstraintIndex ConstraintToRepair() const;

  // Returns the index of the next term which repairs the constraint when the
  // value of its variable is flipped. This method explores terms with an
  // index strictly greater than start_term_index and then terms with an index
  // smaller than or equal to init_term_index if any.
  // Returns kInvalidTerm when no reparing terms are found.
  //
  // Note that if init_term_index == start_term_index, then all the terms will
  // be explored. Both TermIndex arguments can take values in [-1, constraint
  // size).
  TermIndex NextRepairingTerm(ConstraintIndex ct_index,
                              TermIndex init_term_index,
                              TermIndex start_term_index) const;

  // Returns true if the constraint is infeasible and if flipping the variable
  // at the given index will repair it.
  bool RepairIsValid(ConstraintIndex ct_index, TermIndex term_index) const;

  // Returns the literal formed by the variable at the given constraint term and
  // assigned to the opposite value of this variable in the current assignment.
  sat::Literal GetFlip(ConstraintIndex ct_index, TermIndex term_index) const;

  // Local structure to represent the sparse matrix by constraint used for fast
  // lookups.
  struct ConstraintTerm {
    ConstraintTerm(VariableIndex v, int64 w) : var(v), weight(w) {}
    VariableIndex var;
    int64 weight;
  };

 private:
  // Sorts the terms of each constraints in the by_constraint_matrix_ to iterate
  // on most promising variables first.
  void SortTermsOfEachConstraints(int num_variables);

  gtl::ITIVector<ConstraintIndex, gtl::ITIVector<TermIndex, ConstraintTerm>>
      by_constraint_matrix_;
  const AssignmentAndConstraintFeasibilityMaintainer& maintainer_;
  const sat::VariablesAssignment& sat_assignment_;

  DISALLOW_COPY_AND_ASSIGN(OneFlipConstraintRepairer);
};

// This class is used to iterate on all assignments that can be obtained by
// deliberately flipping 'n' variables from the reference solution, 'n' being
// smaller than or equal to max_num_decisions.
// Note that one deliberate variable flip may lead to many other flips due to
// constraint propagation, those additional flips are not counted in 'n'.
class LocalSearchAssignmentIterator {
 public:
  LocalSearchAssignmentIterator(const ProblemState& problem_state,
                                int max_num_decisions,
                                int max_num_broken_constraints,
                                SatWrapper* sat_wrapper);
  ~LocalSearchAssignmentIterator();

  // Parameters of the LS algorithm.
  void UseTranspositionTable(bool v) { use_transposition_table_ = v; }
  void UsePotentialOneFlipRepairs(bool v) {
    use_potential_one_flip_repairs_ = v;
  }

  // Synchronizes the iterator with the problem state, e.g. set fixed variables,
  // set the reference solution. Call this only when a new solution has been
  // found. This will restart the LS.
  void Synchronize(const ProblemState& problem_state);

  // Synchronize the SatWrapper with our current search state. This needs to be
  // called before calls to NextAssignment() if the underlying SatWrapper was
  // used by someone else than this class.
  void SynchronizeSatWrapper();

  // Move to the next assignment. Returns false when the search is finished.
  bool NextAssignment();

  // Returns the last feasible assignment.
  const BopSolution& LastReferenceAssignment() const {
    return maintainer_.reference();
  }

  // Returns true if the current assignment has a better solution than the one
  // passed to the last Synchronize() call.
  bool BetterSolutionHasBeenFound() const {
    return better_solution_has_been_found_;
  }

  // Returns a deterministic number that should be correlated with the time
  // spent in the iterator. The order of magnitude should be close to the time
  // in seconds.
  double deterministic_time() const;

  std::string DebugString() const;

 private:
  // This is called when a better solution has been found to restore the search
  // to the new "root" node.
  void UseCurrentStateAsReference();

  // See transposition_table_ below.
  static constexpr size_t kStoredMaxDecisions = 4;

  // Internal structure used to represent a node of the search tree during local
  // search.
  struct SearchNode {
    SearchNode()
        : constraint(OneFlipConstraintRepairer::kInvalidConstraint),
          term_index(OneFlipConstraintRepairer::kInvalidTerm) {}
    SearchNode(ConstraintIndex c, TermIndex t) : constraint(c), term_index(t) {}
    ConstraintIndex constraint;
    TermIndex term_index;
  };

  // Applies the decision. Automatically backtracks when SAT detects conflicts.
  void ApplyDecision(sat::Literal literal);

  // Adds one more decision to repair infeasible constraints.
  // Returns true in case of success.
  bool GoDeeper();

  // Backtracks and moves to the next decision in the search tree.
  void Backtrack();

  // Looks if the current decisions (in search_nodes_) plus the new one (given
  // by l) lead to a position already present in transposition_table_.
  bool NewStateIsInTranspositionTable(sat::Literal l);

  // Inserts the current set of decisions in transposition_table_.
  void InsertInTranspositionTable();

  // Initializes the given array with the current decisions in search_nodes_ and
  // by filling the other positions with 0.
  void InitializeTranspositionTableKey(
      std::array<int32, kStoredMaxDecisions>* a);

  // Looks for the next repairing term in the given constraints while skipping
  // the position already present in transposition_table_. A given TermIndex of
  // -1 means that this is the first time we explore this constraint.
  bool EnqueueNextRepairingTermIfAny(ConstraintIndex ct_to_repair,
                                     TermIndex index);

  const int max_num_decisions_;
  const int max_num_broken_constraints_;
  bool better_solution_has_been_found_;
  AssignmentAndConstraintFeasibilityMaintainer maintainer_;
  SatWrapper* const sat_wrapper_;
  OneFlipConstraintRepairer repairer_;
  std::vector<SearchNode> search_nodes_;
  gtl::ITIVector<ConstraintIndex, TermIndex> initial_term_index_;

  // Temporary vector used by ApplyDecision().
  std::vector<sat::Literal> tmp_propagated_literals_;

  // For each set of explored decisions, we store it in this table so that we
  // don't explore decisions (a, b) and later (b, a) for instance. The decisions
  // are converted to int32, sorted and padded with 0 before beeing inserted
  // here.
  //
  // TODO(user): We may still miss some equivalent states because it is possible
  // that completely differents decisions lead to exactly the same state.
  // However this is more time consuming to detect because we must apply the
  // last decision first before trying to compare the states.
  //
  // TODO(user): Currently, we only store kStoredMaxDecisions or less decisions.
  // Ideally, this should be related to the maximum number of decision in the
  // LS, but that requires templating the whole LS optimizer.
  bool use_transposition_table_;
  absl::flat_hash_set<std::array<int32, kStoredMaxDecisions>>
      transposition_table_;

  bool use_potential_one_flip_repairs_;

  // The number of explored nodes.
  int64 num_nodes_;

  // The number of skipped nodes thanks to the transposition table.
  int64 num_skipped_nodes_;

  // The overall number of better solution found. And the ones found by the
  // use_potential_one_flip_repairs_ heuristic.
  int64 num_improvements_;
  int64 num_improvements_by_one_flip_repairs_;
  int64 num_inspected_one_flip_repairs_;

  DISALLOW_COPY_AND_ASSIGN(LocalSearchAssignmentIterator);
};

}  // namespace bop
}  // namespace operations_research
#endif  // OR_TOOLS_BOP_BOP_LS_H_