linear_relaxation.cc

// 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/sat/linear_relaxation.h"

#include <vector>

#include "absl/container/flat_hash_set.h"
#include "ortools/base/iterator_adaptors.h"
#include "ortools/base/stl_util.h"
#include "ortools/sat/cp_model.pb.h"
#include "ortools/sat/cp_model_loader.h"
#include "ortools/sat/integer.h"
#include "ortools/sat/integer_expr.h"
#include "ortools/sat/linear_constraint.h"
#include "ortools/sat/linear_programming_constraint.h"
#include "ortools/sat/sat_base.h"

namespace operations_research {
namespace sat {

bool AppendFullEncodingRelaxation(IntegerVariable var, const Model& model,
                                  LinearRelaxation* relaxation) {
  const auto* encoder = model.Get<IntegerEncoder>();
  if (encoder == nullptr) return false;
  if (!encoder->VariableIsFullyEncoded(var)) return false;

  const auto& encoding = encoder->FullDomainEncoding(var);
  const IntegerValue var_min = model.Get<IntegerTrail>()->LowerBound(var);

  LinearConstraintBuilder at_least_one(&model, IntegerValue(1),
                                       kMaxIntegerValue);
  LinearConstraintBuilder encoding_ct(&model, var_min, var_min);
  encoding_ct.AddTerm(var, IntegerValue(1));

  // Create the constraint if all literal have a view.
  std::vector<Literal> at_most_one;

  for (const auto value_literal : encoding) {
    const Literal lit = value_literal.literal;
    const IntegerValue delta = value_literal.value - var_min;
    DCHECK_GE(delta, IntegerValue(0));
    at_most_one.push_back(lit);
    if (!at_least_one.AddLiteralTerm(lit, IntegerValue(1))) return false;
    if (delta != IntegerValue(0)) {
      if (!encoding_ct.AddLiteralTerm(lit, -delta)) return false;
    }
  }

  relaxation->linear_constraints.push_back(at_least_one.Build());
  relaxation->linear_constraints.push_back(encoding_ct.Build());
  relaxation->at_most_ones.push_back(at_most_one);
  return true;
}

namespace {

// TODO(user): Not super efficient.
std::pair<IntegerValue, IntegerValue> GetMinAndMaxNotEncoded(
    IntegerVariable var,
    const absl::flat_hash_set<IntegerValue>& encoded_values,
    const Model& model) {
  const auto* domains = model.Get<IntegerDomains>();
  if (domains == nullptr || var >= domains->size()) {
    return {kMaxIntegerValue, kMinIntegerValue};
  }

  // The domain can be large, but the list of values shouldn't, so this
  // runs in O(encoded_values.size());
  IntegerValue min = kMaxIntegerValue;
  for (const ClosedInterval interval : (*domains)[var]) {
    for (IntegerValue v(interval.start); v <= interval.end; ++v) {
      if (!gtl::ContainsKey(encoded_values, v)) {
        min = v;
        break;
      }
    }
    if (min != kMaxIntegerValue) break;
  }

  IntegerValue max = kMinIntegerValue;
  const auto& domain = (*domains)[var];
  for (int i = domain.NumIntervals() - 1; i >= 0; --i) {
    const ClosedInterval interval = domain[i];
    for (IntegerValue v(interval.end); v >= interval.start; --v) {
      if (!gtl::ContainsKey(encoded_values, v)) {
        max = v;
        break;
      }
    }
    if (max != kMinIntegerValue) break;
  }

  return {min, max};
}

}  // namespace

void AppendPartialEncodingRelaxation(IntegerVariable var, const Model& model,
                                     LinearRelaxation* relaxation) {
  const auto* encoder = model.Get<IntegerEncoder>();
  const auto* integer_trail = model.Get<IntegerTrail>();
  if (encoder == nullptr || integer_trail == nullptr) return;

  const std::vector<IntegerEncoder::ValueLiteralPair>& encoding =
      encoder->PartialDomainEncoding(var);
  if (encoding.empty()) return;

  std::vector<Literal> at_most_one_ct;
  absl::flat_hash_set<IntegerValue> encoded_values;
  for (const auto value_literal : encoding) {
    const Literal literal = value_literal.literal;

    // Note that we skip pairs that do not have an Integer view.
    if (encoder->GetLiteralView(literal) == kNoIntegerVariable &&
        encoder->GetLiteralView(literal.Negated()) == kNoIntegerVariable) {
      continue;
    }

    at_most_one_ct.push_back(literal);
    encoded_values.insert(value_literal.value);
  }
  if (encoded_values.empty()) return;

  // TODO(user): The PartialDomainEncoding() function automatically exclude
  // values that are no longer in the initial domain, so we could be a bit
  // tighter here. That said, this is supposed to be called just after the
  // presolve, so it shouldn't really matter.
  const auto pair = GetMinAndMaxNotEncoded(var, encoded_values, model);
  if (pair.first == kMaxIntegerValue) {
    // TODO(user): try to remove the duplication with
    // AppendFullEncodingRelaxation()? actually I am not sure we need the other
    // function since this one is just more general.
    LinearConstraintBuilder exactly_one_ct(&model, IntegerValue(1),
                                           IntegerValue(1));
    LinearConstraintBuilder encoding_ct(&model, IntegerValue(0),
                                        IntegerValue(0));
    encoding_ct.AddTerm(var, IntegerValue(1));
    for (const auto value_literal : encoding) {
      const Literal lit = value_literal.literal;
      CHECK(exactly_one_ct.AddLiteralTerm(lit, IntegerValue(1)));
      CHECK(
          encoding_ct.AddLiteralTerm(lit, IntegerValue(-value_literal.value)));
    }
    relaxation->linear_constraints.push_back(exactly_one_ct.Build());
    relaxation->linear_constraints.push_back(encoding_ct.Build());
    return;
  }

  // min + sum li * (xi - min) <= var.
  const IntegerValue d_min = pair.first;
  LinearConstraintBuilder lower_bound_ct(&model, d_min, kMaxIntegerValue);
  lower_bound_ct.AddTerm(var, IntegerValue(1));
  for (const auto value_literal : encoding) {
    CHECK(lower_bound_ct.AddLiteralTerm(value_literal.literal,
                                        d_min - value_literal.value));
  }

  // var <= max + sum li * (xi - max).
  const IntegerValue d_max = pair.second;
  LinearConstraintBuilder upper_bound_ct(&model, kMinIntegerValue, d_max);
  upper_bound_ct.AddTerm(var, IntegerValue(1));
  for (const auto value_literal : encoding) {
    CHECK(upper_bound_ct.AddLiteralTerm(value_literal.literal,
                                        d_max - value_literal.value));
  }

  // Note that empty/trivial constraints will be filtered later.
  relaxation->at_most_ones.push_back(at_most_one_ct);
  relaxation->linear_constraints.push_back(lower_bound_ct.Build());
  relaxation->linear_constraints.push_back(upper_bound_ct.Build());
}

void AppendPartialGreaterThanEncodingRelaxation(IntegerVariable var,
                                                const Model& model,
                                                LinearRelaxation* relaxation) {
  const auto* integer_trail = model.Get<IntegerTrail>();
  const auto* encoder = model.Get<IntegerEncoder>();
  if (integer_trail == nullptr || encoder == nullptr) return;

  const std::map<IntegerValue, Literal>& greater_than_encoding =
      encoder->PartialGreaterThanEncoding(var);
  if (greater_than_encoding.empty()) return;

  // Start by the var >= side.
  // And also add the implications between used literals.
  {
    IntegerValue prev_used_bound = integer_trail->LowerBound(var);
    LinearConstraintBuilder lb_constraint(&model, prev_used_bound,
                                          kMaxIntegerValue);
    lb_constraint.AddTerm(var, IntegerValue(1));
    LiteralIndex prev_literal_index = kNoLiteralIndex;
    for (const auto entry : greater_than_encoding) {
      if (entry.first <= prev_used_bound) continue;

      const LiteralIndex literal_index = entry.second.Index();
      const IntegerValue diff = prev_used_bound - entry.first;

      // Skip the entry if the literal doesn't have a view.
      if (!lb_constraint.AddLiteralTerm(entry.second, diff)) continue;
      if (prev_literal_index != kNoLiteralIndex) {
        // Add var <= prev_var, which is the same as var + not(prev_var) <= 1
        relaxation->at_most_ones.push_back(
            {Literal(literal_index), Literal(prev_literal_index).Negated()});
      }
      prev_used_bound = entry.first;
      prev_literal_index = literal_index;
    }
    relaxation->linear_constraints.push_back(lb_constraint.Build());
  }

  // Do the same for the var <= side by using NegationOfVar().
  // Note that we do not need to add the implications between literals again.
  {
    IntegerValue prev_used_bound = integer_trail->LowerBound(NegationOf(var));
    LinearConstraintBuilder lb_constraint(&model, prev_used_bound,
                                          kMaxIntegerValue);
    lb_constraint.AddTerm(var, IntegerValue(-1));
    for (const auto entry :
         encoder->PartialGreaterThanEncoding(NegationOf(var))) {
      if (entry.first <= prev_used_bound) continue;
      const IntegerValue diff = prev_used_bound - entry.first;

      // Skip the entry if the literal doesn't have a view.
      if (!lb_constraint.AddLiteralTerm(entry.second, diff)) continue;
      prev_used_bound = entry.first;
    }
    relaxation->linear_constraints.push_back(lb_constraint.Build());
  }
}

namespace {
// Adds enforcing_lit => target <= bounding_var to relaxation.
void AppendEnforcedUpperBound(const Literal enforcing_lit,
                              const IntegerVariable target,
                              const IntegerVariable bounding_var, Model* model,
                              LinearRelaxation* relaxation) {
  IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
  const IntegerValue max_target_value = integer_trail->UpperBound(target);
  const IntegerValue min_var_value = integer_trail->LowerBound(bounding_var);
  const IntegerValue max_term_value = max_target_value - min_var_value;
  LinearConstraintBuilder lc(model, kMinIntegerValue, max_term_value);
  lc.AddTerm(target, IntegerValue(1));
  lc.AddTerm(bounding_var, IntegerValue(-1));
  CHECK(lc.AddLiteralTerm(enforcing_lit, max_term_value));
  relaxation->linear_constraints.push_back(lc.Build());
}

// Adds {enforcing_lits} => rhs_domain_min <= expr <= rhs_domain_max.
// Requires expr offset to be 0.
void AppendEnforcedLinearExpression(
    const std::vector<Literal>& enforcing_literals,
    const LinearExpression& expr, const IntegerValue rhs_domain_min,
    const IntegerValue rhs_domain_max, const Model& model,
    LinearRelaxation* relaxation) {
  CHECK_EQ(expr.offset, IntegerValue(0));
  const LinearExpression canonical_expr = CanonicalizeExpr(expr);
  const IntegerTrail* integer_trail = model.Get<IntegerTrail>();
  const IntegerValue min_expr_value =
      LinExprLowerBound(canonical_expr, *integer_trail);

  if (rhs_domain_min > min_expr_value) {
    // And(ei) => terms >= rhs_domain_min
    // <=> Sum_i (~ei * (rhs_domain_min - min_expr_value)) + terms >=
    // rhs_domain_min
    LinearConstraintBuilder lc(&model, rhs_domain_min, kMaxIntegerValue);
    for (const Literal& literal : enforcing_literals) {
      CHECK(lc.AddLiteralTerm(literal.Negated(),
                              rhs_domain_min - min_expr_value));
    }
    for (int i = 0; i < canonical_expr.vars.size(); i++) {
      lc.AddTerm(canonical_expr.vars[i], canonical_expr.coeffs[i]);
    }
    relaxation->linear_constraints.push_back(lc.Build());
  }
  const IntegerValue max_expr_value =
      LinExprUpperBound(canonical_expr, *integer_trail);
  if (rhs_domain_max < max_expr_value) {
    // And(ei) => terms <= rhs_domain_max
    // <=> Sum_i (~ei * (rhs_domain_max - max_expr_value)) + terms <=
    // rhs_domain_max
    LinearConstraintBuilder lc(&model, kMinIntegerValue, rhs_domain_max);
    for (const Literal& literal : enforcing_literals) {
      CHECK(lc.AddLiteralTerm(literal.Negated(),
                              rhs_domain_max - max_expr_value));
    }
    for (int i = 0; i < canonical_expr.vars.size(); i++) {
      lc.AddTerm(canonical_expr.vars[i], canonical_expr.coeffs[i]);
    }
    relaxation->linear_constraints.push_back(lc.Build());
  }
}

}  // namespace

// Add a linear relaxation of the CP constraint to the set of linear
// constraints. The highest linearization_level is, the more types of constraint
// we encode. This method should be called only for linearization_level > 0.
//
// Note: IntProd is linearized dynamically using the cut generators.
//
// TODO(user): In full generality, we could encode all the constraint as an LP.
// TODO(user,user): Add unit tests for this method.
void TryToLinearizeConstraint(const CpModelProto& model_proto,
                              const ConstraintProto& ct, Model* model,
                              int linearization_level,
                              LinearRelaxation* relaxation) {
  CHECK_EQ(model->GetOrCreate<SatSolver>()->CurrentDecisionLevel(), 0);
  DCHECK_GT(linearization_level, 0);
  auto* mapping = model->GetOrCreate<CpModelMapping>();
  if (ct.constraint_case() == ConstraintProto::ConstraintCase::kBoolOr) {
    if (linearization_level < 2) return;
    LinearConstraintBuilder lc(model, IntegerValue(1), kMaxIntegerValue);
    for (const int enforcement_ref : ct.enforcement_literal()) {
      CHECK(lc.AddLiteralTerm(mapping->Literal(NegatedRef(enforcement_ref)),
                              IntegerValue(1)));
    }
    for (const int ref : ct.bool_or().literals()) {
      CHECK(lc.AddLiteralTerm(mapping->Literal(ref), IntegerValue(1)));
    }
    relaxation->linear_constraints.push_back(lc.Build());
  } else if (ct.constraint_case() ==
             ConstraintProto::ConstraintCase::kBoolAnd) {
    // TODO(user): These constraints can be many, and if they are not regrouped
    // in big at most ones, then they should probably only added lazily as cuts.
    // Regroup this with future clique-cut separation logic.
    if (linearization_level < 2) return;
    if (!HasEnforcementLiteral(ct)) return;
    if (ct.enforcement_literal().size() == 1) {
      const Literal enforcement = mapping->Literal(ct.enforcement_literal(0));
      for (const int ref : ct.bool_and().literals()) {
        relaxation->at_most_ones.push_back(
            {enforcement, mapping->Literal(ref).Negated()});
      }
      return;
    }

    // Andi(e_i) => Andj(x_j)
    // <=> num_rhs_terms <= Sum_j(x_j) + num_rhs_terms * Sum_i(~e_i)
    int num_literals = ct.bool_and().literals_size();
    LinearConstraintBuilder lc(model, IntegerValue(num_literals),
                               kMaxIntegerValue);
    for (const int ref : ct.bool_and().literals()) {
      CHECK(lc.AddLiteralTerm(mapping->Literal(ref), IntegerValue(1)));
    }
    for (const int enforcement_ref : ct.enforcement_literal()) {
      CHECK(lc.AddLiteralTerm(mapping->Literal(NegatedRef(enforcement_ref)),
                              IntegerValue(num_literals)));
    }
    relaxation->linear_constraints.push_back(lc.Build());
  } else if (ct.constraint_case() ==
             ConstraintProto::ConstraintCase::kAtMostOne) {
    if (HasEnforcementLiteral(ct)) return;
    std::vector<Literal> at_most_one;
    for (const int ref : ct.at_most_one().literals()) {
      at_most_one.push_back(mapping->Literal(ref));
    }
    relaxation->at_most_ones.push_back(at_most_one);
  } else if (ct.constraint_case() == ConstraintProto::ConstraintCase::kIntMax) {
    if (HasEnforcementLiteral(ct)) return;
    const IntegerVariable target = mapping->Integer(ct.int_max().target());
    const std::vector<IntegerVariable> vars =
        mapping->Integers(ct.int_max().vars());
    AppendMaxRelaxation(target, vars, linearization_level, model, relaxation);

  } else if (ct.constraint_case() == ConstraintProto::ConstraintCase::kIntMin) {
    if (HasEnforcementLiteral(ct)) return;
    const IntegerVariable negative_target =
        NegationOf(mapping->Integer(ct.int_min().target()));
    const std::vector<IntegerVariable> negative_vars =
        NegationOf(mapping->Integers(ct.int_min().vars()));
    AppendMaxRelaxation(negative_target, negative_vars, linearization_level,
                        model, relaxation);
  } else if (ct.constraint_case() == ConstraintProto::ConstraintCase::kLinear) {
    AppendLinearConstraintRelaxation(ct, linearization_level, *model,
                                     relaxation);
  } else if (ct.constraint_case() ==
             ConstraintProto::ConstraintCase::kCircuit) {
    if (HasEnforcementLiteral(ct)) return;
    const int num_arcs = ct.circuit().literals_size();
    CHECK_EQ(num_arcs, ct.circuit().tails_size());
    CHECK_EQ(num_arcs, ct.circuit().heads_size());

    // Each node must have exactly one incoming and one outgoing arc (note that
    // it can be the unique self-arc of this node too).
    std::map<int, std::vector<Literal>> incoming_arc_constraints;
    std::map<int, std::vector<Literal>> outgoing_arc_constraints;
    for (int i = 0; i < num_arcs; i++) {
      const Literal arc = mapping->Literal(ct.circuit().literals(i));
      const int tail = ct.circuit().tails(i);
      const int head = ct.circuit().heads(i);

      // Make sure this literal has a view.
      model->Add(NewIntegerVariableFromLiteral(arc));
      outgoing_arc_constraints[tail].push_back(arc);
      incoming_arc_constraints[head].push_back(arc);
    }
    for (const auto* node_map :
         {&outgoing_arc_constraints, &incoming_arc_constraints}) {
      for (const auto& entry : *node_map) {
        const std::vector<Literal>& exactly_one = entry.second;
        if (exactly_one.size() > 1) {
          LinearConstraintBuilder at_least_one_lc(model, IntegerValue(1),
                                                  kMaxIntegerValue);
          for (const Literal l : exactly_one) {
            CHECK(at_least_one_lc.AddLiteralTerm(l, IntegerValue(1)));
          }

          // We separate the two constraints.
          relaxation->at_most_ones.push_back(exactly_one);
          relaxation->linear_constraints.push_back(at_least_one_lc.Build());
        }
      }
    }
  } else if (ct.constraint_case() ==
             ConstraintProto::ConstraintCase::kElement) {
    const IntegerVariable index = mapping->Integer(ct.element().index());
    const IntegerVariable target = mapping->Integer(ct.element().target());
    const std::vector<IntegerVariable> vars =
        mapping->Integers(ct.element().vars());

    // We only relax the case where all the vars are constant.
    // target = sum (index == i) * fixed_vars[i].
    LinearConstraintBuilder constraint(model, IntegerValue(0), IntegerValue(0));
    constraint.AddTerm(target, IntegerValue(-1));
    IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
    for (const auto literal_value : model->Add(FullyEncodeVariable((index)))) {
      const IntegerVariable var = vars[literal_value.value.value()];
      if (!model->Get(IsFixed(var))) return;

      // Make sure this literal has a view.
      model->Add(NewIntegerVariableFromLiteral(literal_value.literal));
      CHECK(constraint.AddLiteralTerm(literal_value.literal,
                                      integer_trail->LowerBound(var)));
    }

    relaxation->linear_constraints.push_back(constraint.Build());
  } else if (ct.constraint_case() ==
             ConstraintProto::ConstraintCase::kInterval) {
    if (linearization_level < 3) return;
    if (HasEnforcementLiteral(ct)) return;
    const IntegerVariable start = mapping->Integer(ct.interval().start());
    const IntegerVariable size = mapping->Integer(ct.interval().size());
    const IntegerVariable end = mapping->Integer(ct.interval().end());
    LinearConstraintBuilder lc(model, IntegerValue(0), IntegerValue(0));
    lc.AddTerm(start, IntegerValue(1));
    lc.AddTerm(size, IntegerValue(1));
    lc.AddTerm(end, IntegerValue(-1));
    relaxation->linear_constraints.push_back(lc.Build());
  } else if (ct.constraint_case() ==
             ConstraintProto::ConstraintCase::kNoOverlap) {
    AppendNoOverlapRelaxation(model_proto, ct, linearization_level, model,
                              relaxation);
  }
}

// TODO(user,user): Support optional interval in the relaxation.
void AppendNoOverlapRelaxation(const CpModelProto& model_proto,
                               const ConstraintProto& ct,
                               int linearization_level, Model* model,
                               LinearRelaxation* relaxation) {
  CHECK(ct.has_no_overlap());
  if (linearization_level < 3) return;
  if (HasEnforcementLiteral(ct)) return;
  if (ct.no_overlap().intervals_size() < 2) return;
  auto* mapping = model->GetOrCreate<CpModelMapping>();
  const int64 num_intervals = ct.no_overlap().intervals_size();
  IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
  IntegerEncoder* encoder = model->GetOrCreate<IntegerEncoder>();
  for (int index1 = 0; index1 < num_intervals; ++index1) {
    const int interval_index1 = ct.no_overlap().intervals(index1);
    if (HasEnforcementLiteral(model_proto.constraints(interval_index1)))
      continue;
    const IntervalConstraintProto interval1 =
        model_proto.constraints(interval_index1).interval();
    const IntegerVariable start1 = mapping->Integer(interval1.start());
    const IntegerVariable end1 = mapping->Integer(interval1.end());
    for (int index2 = index1 + 1; index2 < num_intervals; ++index2) {
      const int interval_index2 = ct.no_overlap().intervals(index2);
      if (HasEnforcementLiteral(model_proto.constraints(interval_index2))) {
        continue;
      }
      const IntervalConstraintProto interval2 =
          model_proto.constraints(interval_index2).interval();
      const IntegerVariable start2 = mapping->Integer(interval2.start());
      const IntegerVariable end2 = mapping->Integer(interval2.end());
      // Encode only the interesting pairs.
      if (integer_trail->UpperBound(end1) <=
              integer_trail->LowerBound(start2) ||
          integer_trail->UpperBound(end2) <=
              integer_trail->LowerBound(start1)) {
        continue;
      }

      const bool interval_1_can_precede_2 =
          integer_trail->LowerBound(end1) <= integer_trail->UpperBound(start2);
      const bool interval_2_can_precede_1 =
          integer_trail->LowerBound(end2) <= integer_trail->UpperBound(start1);

      if (interval_1_can_precede_2 && interval_2_can_precede_1) {
        const IntegerVariable interval1_precedes_interval2 =
            model->Add(NewIntegerVariable(0, 1));
        const Literal interval1_precedes_interval2_lit =
            encoder->GetOrCreateLiteralAssociatedToEquality(
                interval1_precedes_interval2, IntegerValue(1));
        // interval1_precedes_interval2 => interval1.end <= interval2.start
        // ~interval1_precedes_interval2 => interval2.end <= interval1.start
        AppendEnforcedUpperBound(interval1_precedes_interval2_lit, end1, start2,
                                 model, relaxation);
        AppendEnforcedUpperBound(interval1_precedes_interval2_lit.Negated(),
                                 end2, start1, model, relaxation);
      } else if (interval_1_can_precede_2) {
        // interval1.end <= interval2.start
        LinearConstraintBuilder lc(model, kMinIntegerValue, IntegerValue(0));
        lc.AddTerm(end1, IntegerValue(1));
        lc.AddTerm(start2, IntegerValue(-1));
        relaxation->linear_constraints.push_back(lc.Build());
      } else if (interval_2_can_precede_1) {
        // interval2.end <= interval1.start
        LinearConstraintBuilder lc(model, kMinIntegerValue, IntegerValue(0));
        lc.AddTerm(end2, IntegerValue(1));
        lc.AddTerm(start1, IntegerValue(-1));
        relaxation->linear_constraints.push_back(lc.Build());
      }
    }
  }
}

void AppendMaxRelaxation(IntegerVariable target,
                         const std::vector<IntegerVariable>& vars,
                         int linearization_level, Model* model,
                         LinearRelaxation* relaxation) {
  // Case X = max(X_1, X_2, ..., X_N)
  // Part 1: Encode X >= max(X_1, X_2, ..., X_N)
  for (const IntegerVariable var : vars) {
    // This deal with the corner case X = max(X, Y, Z, ..) !
    // Note that this can be presolved into X >= Y, X >= Z, ...
    if (target == var) continue;
    LinearConstraintBuilder lc(model, kMinIntegerValue, IntegerValue(0));
    lc.AddTerm(var, IntegerValue(1));
    lc.AddTerm(target, IntegerValue(-1));
    relaxation->linear_constraints.push_back(lc.Build());
  }

  // Part 2: Encode upper bound on X.
  if (linearization_level < 2) return;
  GenericLiteralWatcher* watcher = model->GetOrCreate<GenericLiteralWatcher>();
  // For size = 2, we do this with 1 less variable.
  IntegerEncoder* encoder = model->GetOrCreate<IntegerEncoder>();
  if (vars.size() == 2) {
    IntegerVariable y = model->Add(NewIntegerVariable(0, 1));
    const Literal y_lit =
        encoder->GetOrCreateLiteralAssociatedToEquality(y, IntegerValue(1));
    AppendEnforcedUpperBound(y_lit, target, vars[0], model, relaxation);

    // TODO(user,user): It makes more sense to use ConditionalLowerOrEqual()
    // here, but that degrades perf on the road*.fzn problem. Understand why.
    IntegerSumLE* upper_bound1 = new IntegerSumLE(
        {y_lit}, {target, vars[0]}, {IntegerValue(1), IntegerValue(-1)},
        IntegerValue(0), model);
    upper_bound1->RegisterWith(watcher);
    model->TakeOwnership(upper_bound1);
    AppendEnforcedUpperBound(y_lit.Negated(), target, vars[1], model,
                             relaxation);
    IntegerSumLE* upper_bound2 = new IntegerSumLE(
        {y_lit.Negated()}, {target, vars[1]},
        {IntegerValue(1), IntegerValue(-1)}, IntegerValue(0), model);
    upper_bound2->RegisterWith(watcher);
    model->TakeOwnership(upper_bound2);
    return;
  }
  // For each X_i, we encode y_i => X <= X_i. And at least one of the y_i is
  // true. Note that the correct y_i will be chosen because of the first part in
  // linearlization (X >= X_i).
  // TODO(user): Only lower bound is needed, experiment.
  LinearConstraintBuilder lc_exactly_one(model, IntegerValue(1),
                                         IntegerValue(1));
  std::vector<Literal> exactly_one_literals;
  exactly_one_literals.reserve(vars.size());
  for (const IntegerVariable var : vars) {
    if (target == var) continue;
    // y => X <= X_i.
    // <=> max_term_value * y + X - X_i <= max_term_value.
    // where max_tern_value is X_ub - X_i_lb.
    IntegerVariable y = model->Add(NewIntegerVariable(0, 1));
    const Literal y_lit =
        encoder->GetOrCreateLiteralAssociatedToEquality(y, IntegerValue(1));

    AppendEnforcedUpperBound(y_lit, target, var, model, relaxation);
    IntegerSumLE* upper_bound_constraint = new IntegerSumLE(
        {y_lit}, {target, var}, {IntegerValue(1), IntegerValue(-1)},
        IntegerValue(0), model);
    upper_bound_constraint->RegisterWith(watcher);
    model->TakeOwnership(upper_bound_constraint);
    exactly_one_literals.push_back(y_lit);

    CHECK(lc_exactly_one.AddLiteralTerm(y_lit, IntegerValue(1)));
  }
  model->Add(ExactlyOneConstraint(exactly_one_literals));
  relaxation->linear_constraints.push_back(lc_exactly_one.Build());
}

std::vector<IntegerVariable> AppendLinMaxRelaxation(
    IntegerVariable target, const std::vector<LinearExpression>& exprs,
    Model* model, LinearRelaxation* relaxation) {
  // We want to linearize X = max(exprs[1], exprs[2], ..., exprs[d]).
  // Part 1: Encode X >= max(exprs[1], exprs[2], ..., exprs[d])
  for (const LinearExpression& expr : exprs) {
    LinearConstraintBuilder lc(model, kMinIntegerValue, -expr.offset);
    for (int i = 0; i < expr.vars.size(); ++i) {
      lc.AddTerm(expr.vars[i], expr.coeffs[i]);
    }
    lc.AddTerm(target, IntegerValue(-1));
    relaxation->linear_constraints.push_back(lc.Build());
  }

  // Part 2: Encode upper bound on X.

  // Add linking constraint to the CP solver
  // sum zi = 1 and for all i, zi => max = expr_i.
  const int num_exprs = exprs.size();
  IntegerEncoder* encoder = model->GetOrCreate<IntegerEncoder>();
  GenericLiteralWatcher* watcher = model->GetOrCreate<GenericLiteralWatcher>();

  // TODO(user): For the case where num_exprs = 2, Create only one z var.
  std::vector<IntegerVariable> z_vars;
  std::vector<Literal> z_lits;
  z_vars.reserve(num_exprs);
  z_lits.reserve(num_exprs);
  LinearConstraintBuilder lc_exactly_one(model, IntegerValue(1),
                                         IntegerValue(1));
  std::vector<Literal> exactly_one_literals;
  for (int i = 0; i < num_exprs; ++i) {
    IntegerVariable z = model->Add(NewIntegerVariable(0, 1));
    z_vars.push_back(z);
    const Literal z_lit =
        encoder->GetOrCreateLiteralAssociatedToEquality(z, IntegerValue(1));
    z_lits.push_back(z_lit);
    LinearExpression local_expr;
    local_expr.vars = NegationOf(exprs[i].vars);
    local_expr.vars.push_back(target);
    local_expr.coeffs = exprs[i].coeffs;
    local_expr.coeffs.push_back(IntegerValue(1));
    IntegerSumLE* upper_bound = new IntegerSumLE(
        {z_lit}, local_expr.vars, local_expr.coeffs, exprs[i].offset, model);
    upper_bound->RegisterWith(watcher);
    model->TakeOwnership(upper_bound);

    CHECK(lc_exactly_one.AddLiteralTerm(z_lit, IntegerValue(1)));
  }
  model->Add(ExactlyOneConstraint(z_lits));

  // For the relaxation, we use different constraints with a stronger linear
  // relaxation as explained in the .h
  // TODO(user): Consider passing the x_vars to this method instead of
  // computing it here.
  std::vector<IntegerVariable> x_vars;
  for (int i = 0; i < num_exprs; ++i) {
    x_vars.insert(x_vars.end(), exprs[i].vars.begin(), exprs[i].vars.end());
  }
  gtl::STLSortAndRemoveDuplicates(&x_vars);
  // All expressions should only contain positive variables.
  DCHECK(std::all_of(x_vars.begin(), x_vars.end(), [](IntegerVariable var) {
    return VariableIsPositive(var);
  }));

  std::vector<std::vector<IntegerValue>> sum_of_max_corner_diff(
      num_exprs, std::vector<IntegerValue>(num_exprs, IntegerValue(0)));

  IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
  for (int i = 0; i < num_exprs; ++i) {
    for (int j = 0; j < num_exprs; ++j) {
      if (i == j) continue;
      for (const IntegerVariable x_var : x_vars) {
        const IntegerValue lb = integer_trail->LevelZeroLowerBound(x_var);
        const IntegerValue ub = integer_trail->LevelZeroUpperBound(x_var);
        const IntegerValue diff =
            GetCoefficient(x_var, exprs[j]) - GetCoefficient(x_var, exprs[i]);
        sum_of_max_corner_diff[i][j] += std::max(diff * lb, diff * ub);
      }
    }
  }
  for (int i = 0; i < num_exprs; ++i) {
    LinearConstraintBuilder lc(model, kMinIntegerValue, IntegerValue(0));
    lc.AddTerm(target, IntegerValue(1));
    for (int j = 0; j < exprs[i].vars.size(); ++j) {
      lc.AddTerm(exprs[i].vars[j], -exprs[i].coeffs[j]);
    }
    for (int j = 0; j < num_exprs; ++j) {
      CHECK(lc.AddLiteralTerm(z_lits[j],
                              -exprs[j].offset - sum_of_max_corner_diff[i][j]));
    }
    relaxation->linear_constraints.push_back(lc.Build());
  }

  relaxation->linear_constraints.push_back(lc_exactly_one.Build());
  return z_vars;
}

void AppendLinearConstraintRelaxation(const ConstraintProto& constraint_proto,
                                      const int linearization_level,
                                      const Model& model,
                                      LinearRelaxation* relaxation) {
  auto* mapping = model.Get<CpModelMapping>();

  // Note that we ignore the holes in the domain.
  //
  // TODO(user): In LoadLinearConstraint() we already created intermediate
  // Booleans for each disjoint interval, we should reuse them here if
  // possible.
  //
  // TODO(user): process the "at most one" part of a == 1 separately?
  const IntegerValue rhs_domain_min =
      IntegerValue(constraint_proto.linear().domain(0));
  const IntegerValue rhs_domain_max =
      IntegerValue(constraint_proto.linear().domain(
          constraint_proto.linear().domain_size() - 1));
  if (rhs_domain_min == kint64min && rhs_domain_max == kint64max) return;

  if (!HasEnforcementLiteral(constraint_proto)) {
    LinearConstraintBuilder lc(&model, rhs_domain_min, rhs_domain_max);
    for (int i = 0; i < constraint_proto.linear().vars_size(); i++) {
      const int ref = constraint_proto.linear().vars(i);
      const int64 coeff = constraint_proto.linear().coeffs(i);
      lc.AddTerm(mapping->Integer(ref), IntegerValue(coeff));
    }
    relaxation->linear_constraints.push_back(lc.Build());
    return;
  }

  // Reified version.
  if (linearization_level < 2) return;

  // We linearize fully reified constraints of size 1 all together for a given
  // variable. But we need to process half-reified ones.
  if (!mapping->IsHalfEncodingConstraint(&constraint_proto) &&
      constraint_proto.linear().vars_size() <= 1) {
    return;
  }

  std::vector<Literal> enforcing_literals;
  enforcing_literals.reserve(constraint_proto.enforcement_literal_size());
  for (const int enforcement_ref : constraint_proto.enforcement_literal()) {
    enforcing_literals.push_back(mapping->Literal(enforcement_ref));
  }
  LinearExpression expr;
  expr.vars.reserve(constraint_proto.linear().vars_size());
  expr.coeffs.reserve(constraint_proto.linear().vars_size());
  for (int i = 0; i < constraint_proto.linear().vars_size(); i++) {
    int ref = constraint_proto.linear().vars(i);
    IntegerValue coeff(constraint_proto.linear().coeffs(i));
    if (!RefIsPositive(ref)) {
      ref = PositiveRef(ref);
      coeff = -coeff;
    }
    const IntegerVariable int_var = mapping->Integer(ref);
    expr.vars.push_back(int_var);
    expr.coeffs.push_back(coeff);
  }
  AppendEnforcedLinearExpression(enforcing_literals, expr, rhs_domain_min,
                                 rhs_domain_max, model, relaxation);
}

}  // namespace sat
}  // namespace operations_research