integer_expr.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.

#ifndef OR_TOOLS_SAT_INTEGER_EXPR_H_
#define OR_TOOLS_SAT_INTEGER_EXPR_H_

#include <functional>
#include <vector>

#include "ortools/base/int_type.h"
#include "ortools/base/integral_types.h"
#include "ortools/base/logging.h"
#include "ortools/base/macros.h"
#include "ortools/base/mathutil.h"
#include "ortools/sat/integer.h"
#include "ortools/sat/linear_constraint.h"
#include "ortools/sat/model.h"
#include "ortools/sat/precedences.h"
#include "ortools/sat/sat_base.h"
#include "ortools/sat/sat_solver.h"
#include "ortools/util/rev.h"

namespace operations_research {
namespace sat {

// A really basic implementation of an upper-bounded sum of integer variables.
// The complexity is in O(num_variables) at each propagation.
//
// Note that we assume that there can be NO integer overflow. This must be
// checked at model validation time before this is even created.
//
// TODO(user): If one has many such constraint, it will be more efficient to
// propagate all of them at once rather than doing it one at the time.
//
// TODO(user): Explore tree structure to get a log(n) complexity.
//
// TODO(user): When the variables are Boolean, use directly the pseudo-Boolean
// constraint implementation. But we do need support for enforcement literals
// there.
class IntegerSumLE : public PropagatorInterface {
 public:
  // If refied_literal is kNoLiteralIndex then this is a normal constraint,
  // otherwise we enforce the implication refied_literal => constraint is true.
  // Note that we don't do the reverse implication here, it is usually done by
  // another IntegerSumLE constraint on the negated variables.
  IntegerSumLE(const std::vector<Literal>& enforcement_literals,
               const std::vector<IntegerVariable>& vars,
               const std::vector<IntegerValue>& coeffs,
               IntegerValue upper_bound, Model* model);

  // We propagate:
  // - If the sum of the individual lower-bound is > upper_bound, we fail.
  // - For all i, upper-bound of i
  //      <= upper_bound - Sum {individual lower-bound excluding i).
  bool Propagate() final;
  void RegisterWith(GenericLiteralWatcher* watcher);

  // Same as Propagate() but only consider current root level bounds. This is
  // mainly useful for the LP propagator since it can find relevant optimal
  // really late in the search tree.
  bool PropagateAtLevelZero();

 private:
  // Fills integer_reason_ with all the current lower_bounds. The real
  // explanation may require removing one of them, but as an optimization, we
  // always keep all the IntegerLiteral in integer_reason_, and swap them as
  // needed just before pushing something.
  void FillIntegerReason();

  const std::vector<Literal> enforcement_literals_;
  const IntegerValue upper_bound_;

  Trail* trail_;
  IntegerTrail* integer_trail_;
  TimeLimit* time_limit_;
  RevIntegerValueRepository* rev_integer_value_repository_;

  // Reversible sum of the lower bound of the fixed variables.
  bool is_registered_ = false;
  IntegerValue rev_lb_fixed_vars_;

  // Reversible number of fixed variables.
  int rev_num_fixed_vars_;

  // Those vectors are shuffled during search to ensure that the variables
  // (resp. coefficients) contained in the range [0, rev_num_fixed_vars_) of
  // vars_ (resp. coeffs_) are fixed (resp. belong to fixed variables).
  std::vector<IntegerVariable> vars_;
  std::vector<IntegerValue> coeffs_;
  std::vector<IntegerValue> max_variations_;

  std::vector<Literal> literal_reason_;

  // Parallel vectors.
  std::vector<IntegerLiteral> integer_reason_;
  std::vector<IntegerValue> reason_coeffs_;

  DISALLOW_COPY_AND_ASSIGN(IntegerSumLE);
};

// This assumes target = SUM_i coeffs[i] * vars[i], and detects that the target
// must be of the form (a*X + b).
//
// This propagator is quite specific and runs only at level zero. For now, this
// is mainly used for the objective variable. As we fix terms with high
// objective coefficient, it is possible the only terms left have a common
// divisor. This close app2-2.mps in less than a second instead of running
// forever to prove the optimal (in single thread).
class LevelZeroEquality : PropagatorInterface {
 public:
  LevelZeroEquality(IntegerVariable target,
                    const std::vector<IntegerVariable>& vars,
                    const std::vector<IntegerValue>& coeffs, Model* model);

  bool Propagate() final;

 private:
  const IntegerVariable target_;
  const std::vector<IntegerVariable> vars_;
  const std::vector<IntegerValue> coeffs_;

  IntegerValue gcd_ = IntegerValue(1);

  Trail* trail_;
  IntegerTrail* integer_trail_;
};

// A min (resp max) contraint of the form min == MIN(vars) can be decomposed
// into two inequalities:
//   1/ min <= MIN(vars), which is the same as for all v in vars, "min <= v".
//      This can be taken care of by the LowerOrEqual(min, v) constraint.
//   2/ min >= MIN(vars).
//
// And in turn, 2/ can be decomposed in:
//   a) lb(min) >= lb(MIN(vars)) = MIN(lb(var));
//   b) ub(min) >= ub(MIN(vars)) and we can't propagate anything here unless
//      there is just one possible variable 'v' that can be the min:
//         for all u != v, lb(u) > ub(min);
//      In this case, ub(min) >= ub(v).
//
// This constraint take care of a) and b). That is:
// - If the min of the lower bound of the vars increase, then the lower bound of
//   the min_var will be >= to it.
// - If there is only one candidate for the min, then if the ub(min) decrease,
//   the ub of the only candidate will be <= to it.
//
// Complexity: This is a basic implementation in O(num_vars) on each call to
// Propagate(), which will happen each time one or more variables in vars_
// changed.
//
// TODO(user): Implement a more efficient algorithm when the need arise.
class MinPropagator : public PropagatorInterface {
 public:
  MinPropagator(const std::vector<IntegerVariable>& vars,
                IntegerVariable min_var, IntegerTrail* integer_trail);

  bool Propagate() final;
  void RegisterWith(GenericLiteralWatcher* watcher);

 private:
  const std::vector<IntegerVariable> vars_;
  const IntegerVariable min_var_;
  IntegerTrail* integer_trail_;

  std::vector<IntegerLiteral> integer_reason_;

  DISALLOW_COPY_AND_ASSIGN(MinPropagator);
};

// Same as MinPropagator except this works on min = MIN(exprs) where exprs are
// linear expressions. It uses IntegerSumLE to propagate bounds on the exprs.
// Assumes Canonical expressions (all positive coefficients).
class LinMinPropagator : public PropagatorInterface {
 public:
  LinMinPropagator(const std::vector<LinearExpression>& exprs,
                   IntegerVariable min_var, Model* model);
  LinMinPropagator(const LinMinPropagator&) = delete;
  LinMinPropagator& operator=(const LinMinPropagator&) = delete;

  bool Propagate() final;
  void RegisterWith(GenericLiteralWatcher* watcher);

 private:
  // Lighter version of IntegerSumLE. This uses the current value of
  // integer_reason_ in addition to the reason for propagating the linear
  // constraint. The coeffs are assumed to be positive here.
  bool PropagateLinearUpperBound(const std::vector<IntegerVariable>& vars,
                                 const std::vector<IntegerValue>& coeffs,
                                 IntegerValue upper_bound);

  const std::vector<LinearExpression> exprs_;
  const IntegerVariable min_var_;
  std::vector<IntegerValue> expr_lbs_;
  Model* model_;
  IntegerTrail* integer_trail_;
  std::vector<IntegerLiteral> integer_reason_for_unique_candidate_;
  int rev_unique_candidate_ = 0;
};

// Propagates a * b = c. Basic version, we don't extract any special cases, and
// we only propagates the bounds.
//
// TODO(user): For now this only works on variables that are non-negative.
// TODO(user): Deal with overflow.
class PositiveProductPropagator : public PropagatorInterface {
 public:
  PositiveProductPropagator(IntegerVariable a, IntegerVariable b,
                            IntegerVariable p, IntegerTrail* integer_trail);

  bool Propagate() final;
  void RegisterWith(GenericLiteralWatcher* watcher);

 private:
  const IntegerVariable a_;
  const IntegerVariable b_;
  const IntegerVariable p_;
  IntegerTrail* integer_trail_;

  DISALLOW_COPY_AND_ASSIGN(PositiveProductPropagator);
};

// Propagates a / b = c. Basic version, we don't extract any special cases, and
// we only propagates the bounds.
//
// TODO(user): For now this only works on variables that are non-negative.
// TODO(user): This only propagate the direction => c, do the reverse.
// TODO(user): Deal with overflow.
// TODO(user): Unit-test this like the ProductPropagator.
class DivisionPropagator : public PropagatorInterface {
 public:
  DivisionPropagator(IntegerVariable a, IntegerVariable b, IntegerVariable c,
                     IntegerTrail* integer_trail);

  bool Propagate() final;
  void RegisterWith(GenericLiteralWatcher* watcher);

 private:
  const IntegerVariable a_;
  const IntegerVariable b_;
  const IntegerVariable c_;
  IntegerTrail* integer_trail_;

  DISALLOW_COPY_AND_ASSIGN(DivisionPropagator);
};

// Propagates var_a / cst_b = var_c. Basic version, we don't extract any special
// cases, and we only propagates the bounds. cst_b must be > 0.
class FixedDivisionPropagator : public PropagatorInterface {
 public:
  FixedDivisionPropagator(IntegerVariable a, IntegerValue b, IntegerVariable c,
                          IntegerTrail* integer_trail);

  bool Propagate() final;
  void RegisterWith(GenericLiteralWatcher* watcher);

 private:
  const IntegerVariable a_;
  const IntegerValue b_;
  const IntegerVariable c_;
  IntegerTrail* integer_trail_;

  DISALLOW_COPY_AND_ASSIGN(FixedDivisionPropagator);
};

// Propagates x * x = s.
// TODO(user): Only works for x nonnegative.
class SquarePropagator : public PropagatorInterface {
 public:
  SquarePropagator(IntegerVariable x, IntegerVariable s,
                   IntegerTrail* integer_trail);

  bool Propagate() final;
  void RegisterWith(GenericLiteralWatcher* watcher);

 private:
  const IntegerVariable x_;
  const IntegerVariable s_;
  IntegerTrail* integer_trail_;

  DISALLOW_COPY_AND_ASSIGN(SquarePropagator);
};

// =============================================================================
// Model based functions.
// =============================================================================

// Weighted sum <= constant.
template <typename VectorInt>
inline std::function<void(Model*)> WeightedSumLowerOrEqual(
    const std::vector<IntegerVariable>& vars, const VectorInt& coefficients,
    int64 upper_bound) {
  // Special cases.
  CHECK_GE(vars.size(), 1);
  if (vars.size() == 1) {
    const int64 c = coefficients[0];
    CHECK_NE(c, 0);
    if (c > 0) {
      return LowerOrEqual(
          vars[0],
          FloorRatio(IntegerValue(upper_bound), IntegerValue(c)).value());
    } else {
      return GreaterOrEqual(
          vars[0],
          CeilRatio(IntegerValue(-upper_bound), IntegerValue(-c)).value());
    }
  }
  if (vars.size() == 2 && (coefficients[0] == 1 || coefficients[0] == -1) &&
      (coefficients[1] == 1 || coefficients[1] == -1)) {
    return Sum2LowerOrEqual(
        coefficients[0] == 1 ? vars[0] : NegationOf(vars[0]),
        coefficients[1] == 1 ? vars[1] : NegationOf(vars[1]), upper_bound);
  }
  if (vars.size() == 3 && (coefficients[0] == 1 || coefficients[0] == -1) &&
      (coefficients[1] == 1 || coefficients[1] == -1) &&
      (coefficients[2] == 1 || coefficients[2] == -1)) {
    return Sum3LowerOrEqual(
        coefficients[0] == 1 ? vars[0] : NegationOf(vars[0]),
        coefficients[1] == 1 ? vars[1] : NegationOf(vars[1]),
        coefficients[2] == 1 ? vars[2] : NegationOf(vars[2]), upper_bound);
  }

  return [=](Model* model) {
    // We split large constraints into a square root number of parts.
    // This is to avoid a bad complexity while propagating them since our
    // algorithm is not in O(num_changes).
    //
    // TODO(user): Alternatively, we could use a O(num_changes) propagation (a
    // bit tricky to implement), or a decomposition into a tree with more than
    // one level. Both requires experimentations.
    //
    // TODO(user): If the initial constraint was an equalilty we will create
    // the "intermediate" variable twice where we could have use the same for
    // both direction. Improve?
    const int num_vars = vars.size();
    if (num_vars > 100) {
      std::vector<IntegerVariable> bucket_sum_vars;

      std::vector<IntegerVariable> local_vars;
      std::vector<IntegerValue> local_coeffs;

      int i = 0;
      const int num_buckets = static_cast<int>(std::round(std::sqrt(num_vars)));
      for (int b = 0; b < num_buckets; ++b) {
        local_vars.clear();
        local_coeffs.clear();
        int64 bucket_lb = 0;
        int64 bucket_ub = 0;
        const int limit = num_vars * (b + 1);
        for (; i * num_buckets < limit; ++i) {
          local_vars.push_back(vars[i]);
          local_coeffs.push_back(IntegerValue(coefficients[i]));
          const int64 term1 = model->Get(LowerBound(vars[i])) * coefficients[i];
          const int64 term2 = model->Get(UpperBound(vars[i])) * coefficients[i];
          bucket_lb += std::min(term1, term2);
          bucket_ub += std::max(term1, term2);
        }

        const IntegerVariable bucket_sum =
            model->Add(NewIntegerVariable(bucket_lb, bucket_ub));
        bucket_sum_vars.push_back(bucket_sum);
        local_vars.push_back(bucket_sum);
        local_coeffs.push_back(IntegerValue(-1));
        IntegerSumLE* constraint = new IntegerSumLE(
            {}, local_vars, local_coeffs, IntegerValue(0), model);
        constraint->RegisterWith(model->GetOrCreate<GenericLiteralWatcher>());
        model->TakeOwnership(constraint);
      }

      // Create the root-level sum.
      local_vars.clear();
      local_coeffs.clear();
      for (const IntegerVariable var : bucket_sum_vars) {
        local_vars.push_back(var);
        local_coeffs.push_back(IntegerValue(1));
      }
      IntegerSumLE* constraint = new IntegerSumLE(
          {}, local_vars, local_coeffs, IntegerValue(upper_bound), model);
      constraint->RegisterWith(model->GetOrCreate<GenericLiteralWatcher>());
      model->TakeOwnership(constraint);
      return;
    }

    IntegerSumLE* constraint = new IntegerSumLE(
        {}, vars,
        std::vector<IntegerValue>(coefficients.begin(), coefficients.end()),
        IntegerValue(upper_bound), model);
    constraint->RegisterWith(model->GetOrCreate<GenericLiteralWatcher>());
    model->TakeOwnership(constraint);
  };
}

// Weighted sum >= constant.
template <typename VectorInt>
inline std::function<void(Model*)> WeightedSumGreaterOrEqual(
    const std::vector<IntegerVariable>& vars, const VectorInt& coefficients,
    int64 lower_bound) {
  // We just negate everything and use an <= constraints.
  std::vector<int64> negated_coeffs(coefficients.begin(), coefficients.end());
  for (int64& ref : negated_coeffs) ref = -ref;
  return WeightedSumLowerOrEqual(vars, negated_coeffs, -lower_bound);
}

// Weighted sum == constant.
template <typename VectorInt>
inline std::function<void(Model*)> FixedWeightedSum(
    const std::vector<IntegerVariable>& vars, const VectorInt& coefficients,
    int64 value) {
  return [=](Model* model) {
    model->Add(WeightedSumGreaterOrEqual(vars, coefficients, value));
    model->Add(WeightedSumLowerOrEqual(vars, coefficients, value));
  };
}

// enforcement_literals => sum <= upper_bound
template <typename VectorInt>
inline std::function<void(Model*)> ConditionalWeightedSumLowerOrEqual(
    const std::vector<Literal>& enforcement_literals,
    const std::vector<IntegerVariable>& vars, const VectorInt& coefficients,
    int64 upper_bound) {
  // Special cases.
  CHECK_GE(vars.size(), 1);
  if (vars.size() == 1) {
    CHECK_NE(coefficients[0], 0);
    if (coefficients[0] > 0) {
      return Implication(
          enforcement_literals,
          IntegerLiteral::LowerOrEqual(
              vars[0], FloorRatio(IntegerValue(upper_bound),
                                  IntegerValue(coefficients[0]))));
    } else {
      return Implication(
          enforcement_literals,
          IntegerLiteral::GreaterOrEqual(
              vars[0], CeilRatio(IntegerValue(-upper_bound),
                                 IntegerValue(-coefficients[0]))));
    }
  }

  if (vars.size() == 2 && (coefficients[0] == 1 || coefficients[0] == -1) &&
      (coefficients[1] == 1 || coefficients[1] == -1)) {
    return ConditionalSum2LowerOrEqual(
        coefficients[0] == 1 ? vars[0] : NegationOf(vars[0]),
        coefficients[1] == 1 ? vars[1] : NegationOf(vars[1]), upper_bound,
        enforcement_literals);
  }
  if (vars.size() == 3 && (coefficients[0] == 1 || coefficients[0] == -1) &&
      (coefficients[1] == 1 || coefficients[1] == -1) &&
      (coefficients[2] == 1 || coefficients[2] == -1)) {
    return ConditionalSum3LowerOrEqual(
        coefficients[0] == 1 ? vars[0] : NegationOf(vars[0]),
        coefficients[1] == 1 ? vars[1] : NegationOf(vars[1]),
        coefficients[2] == 1 ? vars[2] : NegationOf(vars[2]), upper_bound,
        enforcement_literals);
  }

  return [=](Model* model) {
    // If value == min(expression), then we can avoid creating the sum.
    IntegerValue expression_min(0);
    auto* integer_trail = model->GetOrCreate<IntegerTrail>();
    for (int i = 0; i < vars.size(); ++i) {
      expression_min +=
          coefficients[i] * (coefficients[i] >= 0
                                 ? integer_trail->LowerBound(vars[i])
                                 : integer_trail->UpperBound(vars[i]));
    }
    if (expression_min == upper_bound) {
      // Tricky: as we create integer literal, we might propagate stuff and
      // the bounds might change, so if the expression_min increase with the
      // bound we use, then the literal must be false.
      IntegerValue non_cached_min;
      for (int i = 0; i < vars.size(); ++i) {
        if (coefficients[i] > 0) {
          const IntegerValue lb = integer_trail->LowerBound(vars[i]);
          non_cached_min += coefficients[i] * lb;
          model->Add(Implication(enforcement_literals,
                                 IntegerLiteral::LowerOrEqual(vars[i], lb)));
        } else if (coefficients[i] < 0) {
          const IntegerValue ub = integer_trail->UpperBound(vars[i]);
          non_cached_min += coefficients[i] * ub;
          model->Add(Implication(enforcement_literals,
                                 IntegerLiteral::GreaterOrEqual(vars[i], ub)));
        }
      }
      if (non_cached_min > expression_min) {
        std::vector<Literal> clause;
        for (const Literal l : enforcement_literals) {
          clause.push_back(l.Negated());
        }
        model->Add(ClauseConstraint(clause));
      }
    } else {
      IntegerSumLE* constraint = new IntegerSumLE(
          enforcement_literals, vars,
          std::vector<IntegerValue>(coefficients.begin(), coefficients.end()),
          IntegerValue(upper_bound), model);
      constraint->RegisterWith(model->GetOrCreate<GenericLiteralWatcher>());
      model->TakeOwnership(constraint);
    }
  };
}

// enforcement_literals => sum >= lower_bound
template <typename VectorInt>
inline std::function<void(Model*)> ConditionalWeightedSumGreaterOrEqual(
    const std::vector<Literal>& enforcement_literals,
    const std::vector<IntegerVariable>& vars, const VectorInt& coefficients,
    int64 lower_bound) {
  // We just negate everything and use an <= constraint.
  std::vector<int64> negated_coeffs(coefficients.begin(), coefficients.end());
  for (int64& ref : negated_coeffs) ref = -ref;
  return ConditionalWeightedSumLowerOrEqual(enforcement_literals, vars,
                                            negated_coeffs, -lower_bound);
}

// Weighted sum <= constant reified.
template <typename VectorInt>
inline std::function<void(Model*)> WeightedSumLowerOrEqualReif(
    Literal is_le, const std::vector<IntegerVariable>& vars,
    const VectorInt& coefficients, int64 upper_bound) {
  return [=](Model* model) {
    model->Add(ConditionalWeightedSumLowerOrEqual({is_le}, vars, coefficients,
                                                  upper_bound));
    model->Add(ConditionalWeightedSumGreaterOrEqual(
        {is_le.Negated()}, vars, coefficients, upper_bound + 1));
  };
}

// Weighted sum >= constant reified.
template <typename VectorInt>
inline std::function<void(Model*)> WeightedSumGreaterOrEqualReif(
    Literal is_ge, const std::vector<IntegerVariable>& vars,
    const VectorInt& coefficients, int64 lower_bound) {
  return [=](Model* model) {
    model->Add(ConditionalWeightedSumGreaterOrEqual({is_ge}, vars, coefficients,
                                                    lower_bound));
    model->Add(ConditionalWeightedSumLowerOrEqual(
        {is_ge.Negated()}, vars, coefficients, lower_bound - 1));
  };
}

// LinearConstraint version.
inline void LoadLinearConstraint(const LinearConstraint& cst, Model* model) {
  if (cst.vars.empty()) {
    if (cst.lb <= 0 && cst.ub >= 0) return;
    model->GetOrCreate<SatSolver>()->NotifyThatModelIsUnsat();
    return;
  }

  // TODO(user): Remove the conversion!
  std::vector<int64> converted_coeffs;
  for (const IntegerValue v : cst.coeffs) converted_coeffs.push_back(v.value());
  if (cst.ub < kMaxIntegerValue) {
    model->Add(
        WeightedSumLowerOrEqual(cst.vars, converted_coeffs, cst.ub.value()));
  }
  if (cst.lb > kMinIntegerValue) {
    model->Add(
        WeightedSumGreaterOrEqual(cst.vars, converted_coeffs, cst.lb.value()));
  }
}

inline void LoadConditionalLinearConstraint(
    const absl::Span<const Literal> enforcement_literals,
    const LinearConstraint& cst, Model* model) {
  if (enforcement_literals.empty()) {
    return LoadLinearConstraint(cst, model);
  }
  if (cst.vars.empty()) {
    if (cst.lb <= 0 && cst.ub >= 0) return;
    return model->Add(ClauseConstraint(enforcement_literals));
  }

  // TODO(user): Remove the conversion!
  std::vector<Literal> converted_literals(enforcement_literals.begin(),
                                          enforcement_literals.end());
  std::vector<int64> converted_coeffs;
  for (const IntegerValue v : cst.coeffs) converted_coeffs.push_back(v.value());

  if (cst.ub < kMaxIntegerValue) {
    model->Add(ConditionalWeightedSumLowerOrEqual(
        converted_literals, cst.vars, converted_coeffs, cst.ub.value()));
  }
  if (cst.lb > kMinIntegerValue) {
    model->Add(ConditionalWeightedSumGreaterOrEqual(
        converted_literals, cst.vars, converted_coeffs, cst.lb.value()));
  }
}

// Weighted sum == constant reified.
// TODO(user): Simplify if the constant is at the edge of the possible values.
template <typename VectorInt>
inline std::function<void(Model*)> FixedWeightedSumReif(
    Literal is_eq, const std::vector<IntegerVariable>& vars,
    const VectorInt& coefficients, int64 value) {
  return [=](Model* model) {
    // We creates two extra Boolean variables in this case. The alternative is
    // to code a custom propagator for the direction equality => reified.
    const Literal is_le = Literal(model->Add(NewBooleanVariable()), true);
    const Literal is_ge = Literal(model->Add(NewBooleanVariable()), true);
    model->Add(ReifiedBoolAnd({is_le, is_ge}, is_eq));
    model->Add(WeightedSumLowerOrEqualReif(is_le, vars, coefficients, value));
    model->Add(WeightedSumGreaterOrEqualReif(is_ge, vars, coefficients, value));
  };
}

// Weighted sum != constant.
// TODO(user): Simplify if the constant is at the edge of the possible values.
template <typename VectorInt>
inline std::function<void(Model*)> WeightedSumNotEqual(
    const std::vector<IntegerVariable>& vars, const VectorInt& coefficients,
    int64 value) {
  return [=](Model* model) {
    // Exactly one of these alternative must be true.
    const Literal is_lt = Literal(model->Add(NewBooleanVariable()), true);
    const Literal is_gt = is_lt.Negated();
    model->Add(ConditionalWeightedSumLowerOrEqual(is_lt, vars, coefficients,
                                                  value - 1));
    model->Add(ConditionalWeightedSumGreaterOrEqual(is_gt, vars, coefficients,
                                                    value + 1));
  };
}

// Model-based function to create an IntegerVariable that corresponds to the
// given weighted sum of other IntegerVariables.
//
// Note that this is templated so that it can seamlessly accept vector<int> or
// vector<int64>.
//
// TODO(user): invert the coefficients/vars arguments.
template <typename VectorInt>
inline std::function<IntegerVariable(Model*)> NewWeightedSum(
    const VectorInt& coefficients, const std::vector<IntegerVariable>& vars) {
  return [=](Model* model) {
    std::vector<IntegerVariable> new_vars = vars;
    // To avoid overflow in the FixedWeightedSum() constraint, we need to
    // compute the basic bounds on the sum.
    //
    // TODO(user): deal with overflow here too!
    int64 sum_lb(0);
    int64 sum_ub(0);
    for (int i = 0; i < new_vars.size(); ++i) {
      if (coefficients[i] > 0) {
        sum_lb += coefficients[i] * model->Get(LowerBound(new_vars[i]));
        sum_ub += coefficients[i] * model->Get(UpperBound(new_vars[i]));
      } else {
        sum_lb += coefficients[i] * model->Get(UpperBound(new_vars[i]));
        sum_ub += coefficients[i] * model->Get(LowerBound(new_vars[i]));
      }
    }

    const IntegerVariable sum = model->Add(NewIntegerVariable(sum_lb, sum_ub));
    new_vars.push_back(sum);
    std::vector<int64> new_coeffs(coefficients.begin(), coefficients.end());
    new_coeffs.push_back(-1);
    model->Add(FixedWeightedSum(new_vars, new_coeffs, 0));
    return sum;
  };
}

// Expresses the fact that an existing integer variable is equal to the minimum
// of other integer variables.
inline std::function<void(Model*)> IsEqualToMinOf(
    IntegerVariable min_var, const std::vector<IntegerVariable>& vars) {
  return [=](Model* model) {
    for (const IntegerVariable& var : vars) {
      model->Add(LowerOrEqual(min_var, var));
    }

    MinPropagator* constraint =
        new MinPropagator(vars, min_var, model->GetOrCreate<IntegerTrail>());
    constraint->RegisterWith(model->GetOrCreate<GenericLiteralWatcher>());
    model->TakeOwnership(constraint);
  };
}

// Expresses the fact that an existing integer variable is equal to the minimum
// of linear expressions. Assumes Canonical expressions (all positive
// coefficients).
inline std::function<void(Model*)> IsEqualToMinOf(
    const LinearExpression& min_expr,
    const std::vector<LinearExpression>& exprs) {
  return [=](Model* model) {
    IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();

    IntegerVariable min_var;
    if (min_expr.vars.size() == 1 &&
        std::abs(min_expr.coeffs[0].value()) == 1) {
      if (min_expr.coeffs[0].value() == 1) {
        min_var = min_expr.vars[0];
      } else {
        min_var = NegationOf(min_expr.vars[0]);
      }
    } else {
      // Create a new variable if the expression is not just a single variable.
      IntegerValue min_lb = LinExprLowerBound(min_expr, *integer_trail);
      IntegerValue min_ub = LinExprUpperBound(min_expr, *integer_trail);
      min_var = integer_trail->AddIntegerVariable(min_lb, min_ub);

      // min_var = min_expr
      std::vector<IntegerVariable> min_sum_vars = min_expr.vars;
      std::vector<int64> min_sum_coeffs;
      for (IntegerValue coeff : min_expr.coeffs) {
        min_sum_coeffs.push_back(coeff.value());
      }
      min_sum_vars.push_back(min_var);
      min_sum_coeffs.push_back(-1);

      model->Add(FixedWeightedSum(min_sum_vars, min_sum_coeffs,
                                  -min_expr.offset.value()));
    }
    for (const LinearExpression& expr : exprs) {
      // min_var <= expr
      std::vector<IntegerVariable> vars = expr.vars;
      std::vector<int64> coeffs;
      for (IntegerValue coeff : expr.coeffs) {
        coeffs.push_back(coeff.value());
      }
      vars.push_back(min_var);
      coeffs.push_back(-1);
      model->Add(WeightedSumGreaterOrEqual(vars, coeffs, -expr.offset.value()));
    }

    LinMinPropagator* constraint = new LinMinPropagator(exprs, min_var, model);
    constraint->RegisterWith(model->GetOrCreate<GenericLiteralWatcher>());
    model->TakeOwnership(constraint);
  };
}

// Expresses the fact that an existing integer variable is equal to the maximum
// of other integer variables.
inline std::function<void(Model*)> IsEqualToMaxOf(
    IntegerVariable max_var, const std::vector<IntegerVariable>& vars) {
  return [=](Model* model) {
    std::vector<IntegerVariable> negated_vars;
    for (const IntegerVariable& var : vars) {
      negated_vars.push_back(NegationOf(var));
      model->Add(GreaterOrEqual(max_var, var));
    }

    MinPropagator* constraint = new MinPropagator(
        negated_vars, NegationOf(max_var), model->GetOrCreate<IntegerTrail>());
    constraint->RegisterWith(model->GetOrCreate<GenericLiteralWatcher>());
    model->TakeOwnership(constraint);
  };
}

// Expresses the fact that an existing integer variable is equal to one of
// the given values, each selected by a given literal.
std::function<void(Model*)> IsOneOf(IntegerVariable var,
                                    const std::vector<Literal>& selectors,
                                    const std::vector<IntegerValue>& values);

template <class T>
void RegisterAndTransferOwnership(Model* model, T* ct) {
  ct->RegisterWith(model->GetOrCreate<GenericLiteralWatcher>());
  model->TakeOwnership(ct);
}
// Adds the constraint: a * b = p.
inline std::function<void(Model*)> ProductConstraint(IntegerVariable a,
                                                     IntegerVariable b,
                                                     IntegerVariable p) {
  return [=](Model* model) {
    IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
    if (a == b) {
      if (model->Get(LowerBound(a)) >= 0) {
        RegisterAndTransferOwnership(model,
                                     new SquarePropagator(a, p, integer_trail));
      } else if (model->Get(UpperBound(a)) <= 0) {
        RegisterAndTransferOwnership(
            model, new SquarePropagator(NegationOf(a), p, integer_trail));
      } else {
        LOG(FATAL) << "Not supported";
      }
    } else if (model->Get(LowerBound(a)) >= 0 &&
               model->Get(LowerBound(b)) >= 0) {
      RegisterAndTransferOwnership(
          model, new PositiveProductPropagator(a, b, p, integer_trail));
    } else if (model->Get(LowerBound(a)) >= 0 &&
               model->Get(UpperBound(b)) <= 0) {
      RegisterAndTransferOwnership(
          model, new PositiveProductPropagator(a, NegationOf(b), NegationOf(p),
                                               integer_trail));
    } else if (model->Get(UpperBound(a)) <= 0 &&
               model->Get(LowerBound(b)) >= 0) {
      RegisterAndTransferOwnership(
          model, new PositiveProductPropagator(NegationOf(a), b, NegationOf(p),
                                               integer_trail));
    } else if (model->Get(UpperBound(a)) <= 0 &&
               model->Get(UpperBound(b)) <= 0) {
      RegisterAndTransferOwnership(
          model, new PositiveProductPropagator(NegationOf(a), NegationOf(b), p,
                                               integer_trail));
    } else {
      LOG(FATAL) << "Not supported";
    }
  };
}

// Adds the constraint: a / b = c.
inline std::function<void(Model*)> DivisionConstraint(IntegerVariable a,
                                                      IntegerVariable b,
                                                      IntegerVariable c) {
  return [=](Model* model) {
    IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
    DivisionPropagator* constraint =
        new DivisionPropagator(a, b, c, integer_trail);
    constraint->RegisterWith(model->GetOrCreate<GenericLiteralWatcher>());
    model->TakeOwnership(constraint);
  };
}

// Adds the constraint: a / b = c where b is a constant.
inline std::function<void(Model*)> FixedDivisionConstraint(IntegerVariable a,
                                                           IntegerValue b,
                                                           IntegerVariable c) {
  return [=](Model* model) {
    IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
    FixedDivisionPropagator* constraint =
        b > 0
            ? new FixedDivisionPropagator(a, b, c, integer_trail)
            : new FixedDivisionPropagator(NegationOf(a), -b, c, integer_trail);
    constraint->RegisterWith(model->GetOrCreate<GenericLiteralWatcher>());
    model->TakeOwnership(constraint);
  };
}

}  // namespace sat
}  // namespace operations_research

#endif  // OR_TOOLS_SAT_INTEGER_EXPR_H_