scheduling_cuts.cc

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// Copyright 2010-2021 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/scheduling_cuts.h"
 
#include <algorithm>
#include <cmath>
#include <cstdlib>
#include <functional>
#include <optional>
#include <string>
#include <tuple>
#include <utility>
#include <vector>
 
#include "absl/strings/str_cat.h"
#include "absl/types/span.h"
#include "ortools/base/logging.h"
#include "ortools/base/stl_util.h"
#include "ortools/base/strong_vector.h"
#include "ortools/sat/cuts.h"
#include "ortools/sat/diffn_util.h"
#include "ortools/sat/implied_bounds.h"
#include "ortools/sat/integer.h"
#include "ortools/sat/intervals.h"
#include "ortools/sat/linear_constraint.h"
#include "ortools/sat/linear_constraint_manager.h"
#include "ortools/sat/model.h"
#include "ortools/sat/sat_base.h"
#include "ortools/util/strong_integers.h"
 
namespace operations_research {
namespace sat {
 
namespace {
 
// Minimum amount of violation of the cut constraint by the solution. This
// is needed to avoid numerical issues and adding cuts with minor effect.
const double kMinCutViolation = 1e-4;
 
// Returns the lp value of a Literal.
double GetLiteralLpValue(
    const Literal lit,
    const absl::StrongVector<IntegerVariable, double>& lp_values,
    const IntegerEncoder* encoder) {
  const IntegerVariable direct_view = encoder->GetLiteralView(lit);
  if (direct_view != kNoIntegerVariable) {
    return lp_values[direct_view];
  }
  const IntegerVariable opposite_view = encoder->GetLiteralView(lit.Negated());
  DCHECK_NE(opposite_view, kNoIntegerVariable);
  return 1.0 - lp_values[opposite_view];
}
 
void AddIntegerVariableFromIntervals(SchedulingConstraintHelper* helper,
                                     Model* model,
                                     std::vector<IntegerVariable>* vars) {
  IntegerEncoder* encoder = model->GetOrCreate<IntegerEncoder>();
  for (int t = 0; t < helper->NumTasks(); ++t) {
    if (helper->Starts()[t].var != kNoIntegerVariable) {
      vars->push_back(helper->Starts()[t].var);
    }
    if (helper->Sizes()[t].var != kNoIntegerVariable) {
      vars->push_back(helper->Sizes()[t].var);
    }
    if (helper->Ends()[t].var != kNoIntegerVariable) {
      vars->push_back(helper->Ends()[t].var);
    }
    if (helper->IsOptional(t) && !helper->IsAbsent(t) &&
        !helper->IsPresent(t)) {
      const Literal l = helper->PresenceLiteral(t);
      if (encoder->GetLiteralView(l) == kNoIntegerVariable &&
          encoder->GetLiteralView(l.Negated()) == kNoIntegerVariable) {
        model->Add(NewIntegerVariableFromLiteral(l));
      }
      const IntegerVariable direct_view = encoder->GetLiteralView(l);
      if (direct_view != kNoIntegerVariable) {
        vars->push_back(direct_view);
      } else {
        vars->push_back(encoder->GetLiteralView(l.Negated()));
        DCHECK_NE(vars->back(), kNoIntegerVariable);
      }
    }
  }
}
 
}  // namespace
 
struct EnergyEvent {
  IntegerValue x_start_min;
  IntegerValue x_start_max;
  IntegerValue x_end_min;
  IntegerValue x_end_max;
  AffineExpression x_size;
  IntegerValue y_min = IntegerValue(0);  // Useful for no_overlap_2d.
  IntegerValue y_max = IntegerValue(0);  // Useful for no_overlap_2d.
  AffineExpression y_size;
 
  // Energy will be present only if the x_size * y_size could be linearized.
  std::optional<LinearExpression> energy;
  LiteralIndex presence_literal_index = kNoLiteralIndex;
 
  // Caches for MinOf(x_size), MinOf(y_size), and the LP values of the energy
  // and the presence literal.
  IntegerValue x_size_min;
  IntegerValue y_size_min;
  double literal_lp = 1.0;
  double energy_lp = 0.0;
 
  // Used to minimize the increase on the y axis for rectangles.
  double y_spread = 0.0;
 
  // The actual value of the presence literal of the interval(s) is checked
  // when the event is created. A value of kNoLiteralIndex indicates that either
  // the interval was not optional, or that its presence literal is true at
  // level zero.
  bool IsPresent() const { return presence_literal_index == kNoLiteralIndex; }
 
  // Computes the mandatory minimal overlap of the interval with the time window
  // [start, end].
  IntegerValue GetMinOverlap(IntegerValue start, IntegerValue end) const {
    return std::max(std::min({x_end_min - start, end - x_start_max, x_size_min,
                              end - start}),
                    IntegerValue(0));
  }
 
  std::string DebugString() const {
    return absl::StrCat(
        "EnergyEvent(x_start_min = ", x_start_min.value(),
        ", x_start_max = ", x_start_max.value(),
        ", x_end_min = ", x_end_min.value(),
        ", x_end_max = ", x_end_max.value(),
        ", x_size = ", x_size.DebugString(), ", y_min = ", y_min.value(),
        ", y_max = ", y_max.value(), ", y_size = ", y_size.DebugString(),
        ", energy = ", energy ? energy.value().DebugString() : "{}",
        ", presence_literal_index = ", presence_literal_index.value(), ")");
  }
};
 
// Note that we only support to cases:
// - capacity is non-zero and the y_min/y_max of the events must be zero.
// - capacity is zero, and the y_min/y_max can be set.
// This is DCHECKed in the code.
void GenerateEnergeticCuts(
    const std::string& cut_name,
    const absl::StrongVector<IntegerVariable, double>& lp_values,
    std::vector<EnergyEvent> events, const AffineExpression capacity,
    bool events_are_2d, Model* model, LinearConstraintManager* manager) {
  IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
  TopNCuts top_n_cuts(15);
 
  std::sort(events.begin(), events.end(),
            [](const EnergyEvent& a, const EnergyEvent& b) {
              return std::tie(a.x_start_min, a.y_spread, a.x_end_max) <
                     std::tie(b.x_start_min, b.y_spread, b.x_end_max);
            });
 
  const double capacity_lp = capacity.LpValue(lp_values);
 
  // The sum of all energies can be used to stop iterating early.
  double sum_of_all_energies = 0.0;
  for (const auto& e : events) {
    sum_of_all_energies += e.energy_lp;
  }
  CapacityProfile capacity_profile;
 
  IntegerValue processed_start = kMinIntegerValue;
  for (int i1 = 0; i1 + 1 < events.size(); ++i1) {
    // We want maximal cuts. For any x_start_min value, we only need to create
    // cuts starting from the first interval having this x_start_min value.
    //
    // Enabling this optimization reduces dramatically the number of generated
    // cuts in the rectangle_case case. So we only use it in the !events_are_2d
    // case.
    if (!events_are_2d) {
      if (events[i1].x_start_min == processed_start) {
        continue;
      } else {
        processed_start = events[i1].x_start_min;
      }
    }
 
    // For each start time, we will keep the most violated cut generated while
    // scanning the residual intervals.
    int max_violation_end_index = -1;
    double max_relative_violation = 1.0 + kMinCutViolation;
    IntegerValue max_violation_window_start(0);
    IntegerValue max_violation_window_end(0);
    IntegerValue max_violation_y_min(0);
    IntegerValue max_violation_y_max(0);
    IntegerValue max_violation_precise_area(0);
    bool max_violation_use_precise_area = false;
    std::vector<EnergyEvent> max_violation_lifted_events;
 
    // Accumulate intervals, areas, energies and check for potential cuts.
    double energy_lp = 0.0;
    IntegerValue window_min = kMaxIntegerValue;
    IntegerValue window_max = kMinIntegerValue;
    IntegerValue y_min = kMaxIntegerValue;
    IntegerValue y_max = kMinIntegerValue;
    capacity_profile.Clear();
 
    // We sort all tasks (x_start_min(task) >= x_start_min(start_index) by
    // increasing end max.
    std::vector<EnergyEvent> residual_events(events.begin() + i1, events.end());
    std::sort(residual_events.begin(), residual_events.end(),
              [](const EnergyEvent& a, const EnergyEvent& b) {
                return std::tie(a.x_end_max, a.y_spread) <
                       std::tie(b.x_end_max, b.y_spread);
              });
    // Let's process residual tasks and evaluate the violation of the cut at
    // each step. We follow the same structure as the cut creation code below.
    for (int i2 = 0; i2 < residual_events.size(); ++i2) {
      const EnergyEvent& e = residual_events[i2];
      energy_lp += e.energy_lp;
      window_min = std::min(window_min, e.x_start_min);
      window_max = std::max(window_max, e.x_end_max);
      y_min = std::min(y_min, e.y_min);
      y_max = std::max(y_max, e.y_max);
      if (events_are_2d) {
        capacity_profile.AddRectangle(e.x_start_min, e.x_end_max, e.y_min,
                                      e.y_max);
      }
 
      // Dominance rule. If the next interval also fits in
      // [window_min, window_max], the cut will be stronger with the
      // next interval/rectangle.
      if (i2 + 1 < residual_events.size() &&
          residual_events[i2 + 1].x_start_min >= window_min &&
          residual_events[i2 + 1].x_end_max <= window_max &&
          residual_events[i2 + 1].y_min >= y_min &&
          residual_events[i2 + 1].y_max <= y_max) {
        continue;
      }
 
      // Checks the current area vs the sum of all energies.
      DCHECK(capacity_lp == 0.0 || y_max == y_min);
      // we have two cases:
      //   - events_are_2d = false: the area is (window_max - window_min) *
      //   capacity.
      //     Its LP value is (window_max - window_min) * capacity_lp.
      //   - events_are_2d = true: The area is
      //   capacity_profile.GetBoundingArea().
      //     We can compare it to the bounding box area:
      //         (window_max - window_min) * (y_max - y_min).
      bool use_precise_area = false;
      IntegerValue precise_area(0);
      double area_lp = 0.0;
      if (events_are_2d) {
        const IntegerValue bbox_area =
            (window_max - window_min) * (y_max - y_min);
        precise_area = capacity_profile.GetBoundingArea();
        use_precise_area = precise_area < bbox_area;
        area_lp = ToDouble(precise_area);
      } else {
        area_lp = capacity_lp * ToDouble(window_max - window_min);
      }
 
      if (area_lp >= sum_of_all_energies) {
        break;
      }
 
      // Compute forced contributions from intervals not included in
      // [window_min..window_max].
      //
      // TODO(user): We could precompute possible intervals and store them
      // by x_start_max, x_end_min to reduce the complexity.
      //
      // Note: this is not useful in the no_overlap_2d case. Maybe because of
      // geometric nature of the overlap w.r.t. strict rectangles. i.e. a
      // rectangle  that can be lifted will most likely increase the y_range.
      std::vector<EnergyEvent> lifted_events;
      double lifted_contrib_lp = 0.0;
 
      // Because of the 2D conflicts, lifted events are unlikely in the
      // no_overlap_2d case. Let's disable this expensive loop for the time
      // being.
      if (!events_are_2d) {
        const auto check_lifted_cuts = [&lifted_events, window_min, window_max,
                                        y_min, y_max, &lifted_contrib_lp,
                                        &lp_values](const EnergyEvent& e) {
          // It should not happen because of the 2 dominance rules above.
          if (e.x_start_min >= window_min && e.x_end_max <= window_max) {
            return;
          }
 
          // Do not add if it extends the y range.
          if (e.y_min < y_min || e.y_max > y_max) return;
 
          // Exit if the interval can be pushed left or right of the window.
          if (e.x_end_min <= window_min || e.x_start_max >= window_max) return;
 
          DCHECK_GT(window_max, window_min);
          DCHECK_GT(e.x_size_min, 0);
 
          const IntegerValue min_overlap =
              e.GetMinOverlap(window_min, window_max);
 
          if (min_overlap <= 0) return;
 
          lifted_events.push_back(e);
          double contrib_lp = 0.0;
          if (e.IsPresent()) {
            contrib_lp = e.y_size.LpValue(lp_values) * ToDouble(min_overlap);
          } else {
            contrib_lp = e.literal_lp * ToDouble(min_overlap * e.y_size_min);
          }
          // We know the energy is >= 0.0. Some epsilon in the simplex can make
          // contrib_lp a small negative number. We can correct this.
          lifted_contrib_lp += std::max(contrib_lp, 0.0);
        };
 
        for (int i3 = 0; i3 < i1; ++i3) {
          check_lifted_cuts(events[i3]);
        }
        for (int i3 = i2 + 1; i3 < residual_events.size(); ++i3) {
          check_lifted_cuts(residual_events[i3]);
        }
      }
 
      // Compute the violation of the potential cut.
      const double relative_violation =
          (energy_lp + lifted_contrib_lp) / area_lp;
      if (relative_violation > max_relative_violation) {
        max_violation_end_index = i2;
        max_relative_violation = relative_violation;
        max_violation_window_start = window_min;
        max_violation_window_end = window_max;
        std::swap(max_violation_lifted_events, lifted_events);
        max_violation_y_min = y_min;
        max_violation_y_max = y_max;
        max_violation_precise_area = precise_area;
        max_violation_use_precise_area = use_precise_area;
      }
    }
 
    if (max_violation_end_index == -1) continue;
 
    // A maximal violated cut has been found.
    bool cut_generated = true;
    bool has_opt_cuts = false;
    bool use_lifted_events = false;
    bool has_quadratic_cuts = false;
    bool use_energy = false;
 
    DCHECK(capacity_lp == 0.0 || max_violation_y_max == max_violation_y_min);
    // Build the cut.
    LinearConstraintBuilder cut(
        model, kMinIntegerValue,
        events_are_2d ? max_violation_precise_area : IntegerValue(0));
 
    // Adds the capacity in the disjunctive/cumulative case.
    if (!events_are_2d) {
      cut.AddTerm(capacity,
                  max_violation_window_start - max_violation_window_end);
    }
 
    for (int i2 = 0; i2 <= max_violation_end_index; ++i2) {
      const EnergyEvent& e = residual_events[i2];
      if (e.IsPresent()) {
        if (e.energy) {
          // We favor the energy info instead of the McCormick relaxation.
          cut.AddLinearExpression(e.energy.value());
          use_energy = true;
        } else {
          cut.AddQuadraticLowerBound(e.x_size, e.y_size, integer_trail);
          has_quadratic_cuts = true;
        }
      } else {
        has_opt_cuts = true;
        const IntegerValue min_energy =
            std::max(e.x_size_min * e.y_size_min,
                     e.energy ? e.energy.value().LevelZeroMin(integer_trail)
                              : IntegerValue(0));
        if (min_energy > e.x_size_min * e.y_size_min) {
          use_energy = true;
        }
        if (!cut.AddLiteralTerm(Literal(e.presence_literal_index),
                                min_energy)) {
          cut_generated = false;
          break;
        }
      }
    }
 
    for (const EnergyEvent& e : max_violation_lifted_events) {
      const IntegerValue min_overlap =
          e.GetMinOverlap(max_violation_window_start, max_violation_window_end);
      DCHECK_GT(min_overlap, 0);
      use_lifted_events = true;
 
      if (e.IsPresent()) {
        cut.AddTerm(e.y_size, min_overlap);
      } else {
        has_opt_cuts = true;
        if (!cut.AddLiteralTerm(Literal(e.presence_literal_index),
                                min_overlap * e.y_size_min)) {
          cut_generated = false;
          break;
        }
      }
    }
 
    if (cut_generated) {
      std::string full_name = cut_name;
      if (has_opt_cuts) full_name.append("_opt");
      if (has_quadratic_cuts) full_name.append("_quad");
      if (use_lifted_events) full_name.append("_lifted");
      if (use_energy) full_name.append("_energy");
      if (max_violation_use_precise_area) full_name.append("_precise");
      top_n_cuts.AddCut(cut.Build(), full_name, lp_values);
    }
  }
  top_n_cuts.TransferToManager(lp_values, manager);
}
 
void AppendVariablesToCumulativeCut(
    const AffineExpression& capacity,
    const std::vector<AffineExpression>& demands, IntegerTrail* integer_trail,
    CutGenerator* result) {
  for (const AffineExpression& demand_expr : demands) {
    if (!integer_trail->IsFixed(demand_expr)) {
      result->vars.push_back(demand_expr.var);
    }
  }
  if (!integer_trail->IsFixed(capacity)) {
    result->vars.push_back(capacity.var);
  }
}
 
double ComputeEnergyLp(
    const EnergyEvent& e,
    const absl::StrongVector<IntegerVariable, double>& lp_values,
    IntegerTrail* integer_trail) {
  if (e.presence_literal_index == kNoLiteralIndex) {
    if (e.energy) {
      return e.energy.value().LpValue(lp_values);
    } else {  // demand and size are not fixed.
      // X * Y >= X * y_min + x_min * Y - x_min * y_min.
      return ToDouble(e.y_size_min) * e.x_size.LpValue(lp_values) +
             ToDouble(e.x_size_min) * e.y_size.LpValue(lp_values) -
             ToDouble(e.x_size_min * e.y_size_min);
    }
  } else {
    const IntegerValue min_energy =
        std::max(e.x_size_min * e.y_size_min,
                 e.energy ? e.energy.value().LevelZeroMin(integer_trail)
                          : IntegerValue(0));
    return e.literal_lp * ToDouble(min_energy);
  }
}
 
CutGenerator CreateCumulativeEnergyCutGenerator(
    const std::vector<IntervalVariable>& intervals,
    const AffineExpression& capacity,
    const std::vector<AffineExpression>& demands,
    const std::vector<LinearExpression>& energies, Model* model) {
  CutGenerator result;
 
  SchedulingConstraintHelper* helper =
      new SchedulingConstraintHelper(intervals, model);
  model->TakeOwnership(helper);
 
  Trail* trail = model->GetOrCreate<Trail>();
  IntegerEncoder* encoder = model->GetOrCreate<IntegerEncoder>();
  IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
  AppendVariablesToCumulativeCut(capacity, demands, integer_trail, &result);
  AddIntegerVariableFromIntervals(helper, model, &result.vars);
  for (const LinearExpression& energy : energies) {
    result.vars.insert(result.vars.end(), energy.vars.begin(),
                       energy.vars.end());
  }
  gtl::STLSortAndRemoveDuplicates(&result.vars);
 
  result.generate_cuts =
      [capacity, demands, energies, trail, integer_trail, helper, model,
       encoder](const absl::StrongVector<IntegerVariable, double>& lp_values,
                LinearConstraintManager* manager) {
        if (trail->CurrentDecisionLevel() > 0) return true;
 
        std::vector<EnergyEvent> events;
        for (int i = 0; i < helper->NumTasks(); ++i) {
          if (helper->IsAbsent(i)) continue;
          if (integer_trail->LevelZeroUpperBound(demands[i]) == 0 ||
              helper->SizeMin(i) == 0) {
            continue;
          }
 
          EnergyEvent e;
          e.x_start_min = helper->StartMin(i);
          e.x_start_max = helper->StartMax(i);
          e.x_end_min = helper->EndMin(i);
          e.x_end_max = helper->EndMax(i);
          e.x_size = helper->Sizes()[i];
          e.y_size = demands[i];
          if (ProductIsLinearized(energies[i])) {
            e.energy = energies[i];
          }
          e.x_size_min = helper->SizeMin(i);
          e.y_size_min = integer_trail->LevelZeroLowerBound(demands[i]);
          if (!helper->IsPresent(i)) {
            e.presence_literal_index = helper->PresenceLiteral(i).Index();
            e.literal_lp = GetLiteralLpValue(Literal(e.presence_literal_index),
                                             lp_values, encoder);
          }
          e.energy_lp = ComputeEnergyLp(e, lp_values, integer_trail);
          events.push_back(e);
        }
 
        GenerateEnergeticCuts("CumulativeEnergy", lp_values, events, capacity,
                              /*events_are_2d=*/false, model, manager);
        return true;
      };
 
  return result;
}
 
CutGenerator CreateNoOverlapEnergyCutGenerator(
    const std::vector<IntervalVariable>& intervals, Model* model) {
  CutGenerator result;
 
  Trail* trail = model->GetOrCreate<Trail>();
  IntegerEncoder* encoder = model->GetOrCreate<IntegerEncoder>();
  SchedulingConstraintHelper* helper =
      new SchedulingConstraintHelper(intervals, model);
  model->TakeOwnership(helper);
 
  AddIntegerVariableFromIntervals(helper, model, &result.vars);
  gtl::STLSortAndRemoveDuplicates(&result.vars);
 
  // We need to convert AffineExpression to LinearExpression for the energy.
  std::vector<LinearExpression> sizes;
  sizes.reserve(intervals.size());
  for (int i = 0; i < intervals.size(); ++i) {
    LinearConstraintBuilder builder(model);
    builder.AddTerm(helper->Sizes()[i], IntegerValue(1));
    sizes.push_back(builder.BuildExpression());
  }
 
  result.generate_cuts =
      [sizes, trail, helper, model, encoder](
          const absl::StrongVector<IntegerVariable, double>& lp_values,
          LinearConstraintManager* manager) {
        if (trail->CurrentDecisionLevel() > 0) return true;
 
        std::vector<EnergyEvent> events;
        for (int i = 0; i < helper->NumTasks(); ++i) {
          if (helper->IsAbsent(i)) continue;
          if (helper->SizeMin(i) == 0) {
            continue;
          }
 
          EnergyEvent e;
          e.x_start_min = helper->StartMin(i);
          e.x_start_max = helper->StartMax(i);
          e.x_end_min = helper->EndMin(i);
          e.x_end_max = helper->EndMax(i);
          e.x_size = helper->Sizes()[i];
          e.y_size = IntegerValue(1);
          e.energy = sizes[i];
          e.x_size_min = helper->SizeMin(i);
          e.y_size_min = IntegerValue(1);
          if (helper->IsPresent(i)) {
            e.energy_lp = e.energy->LpValue(lp_values);
          } else {
            e.presence_literal_index = helper->PresenceLiteral(i).Index();
            e.literal_lp = GetLiteralLpValue(Literal(e.presence_literal_index),
                                             lp_values, encoder);
            e.energy_lp = e.literal_lp * ToDouble(e.x_size_min);
          }
          events.push_back(e);
        }
 
        GenerateEnergeticCuts("NoOverlapEnergy", lp_values, events,
                              IntegerValue(1), /*events_are_2d=*/false, model,
                              manager);
        return true;
      };
  return result;
}
 
void GenerateNoOverlap2dEnergyCut(
    const std::vector<LinearExpression>& energies, absl::Span<int> rectangles,
    const std::string& cut_name,
    const absl::StrongVector<IntegerVariable, double>& lp_values, Model* model,
    IntegerTrail* integer_trail, IntegerEncoder* encoder,
    LinearConstraintManager* manager, SchedulingConstraintHelper* x_helper,
    SchedulingConstraintHelper* y_helper) {
  std::vector<EnergyEvent> events;
  for (const int rect : rectangles) {
    if (y_helper->SizeMax(rect) == 0 || x_helper->SizeMax(rect) == 0) {
      continue;
    }
 
    EnergyEvent e;
    e.x_start_min = x_helper->StartMin(rect);
    e.x_start_max = x_helper->StartMax(rect);
    e.x_end_min = x_helper->EndMin(rect);
    e.x_end_max = x_helper->EndMax(rect);
    e.x_size = x_helper->Sizes()[rect];
    e.y_min = y_helper->StartMin(rect);
    e.y_max = y_helper->EndMax(rect);
    e.y_size = y_helper->Sizes()[rect];
    if (ProductIsLinearized(energies[rect])) {
      e.energy = energies[rect];
    }
    e.presence_literal_index =
        x_helper->IsPresent(rect)
            ? (y_helper->IsPresent(rect)
                   ? kNoLiteralIndex
                   : y_helper->PresenceLiteral(rect).Index())
            : x_helper->PresenceLiteral(rect).Index();
    if (e.presence_literal_index != kNoLiteralIndex) {
      e.literal_lp = GetLiteralLpValue(Literal(e.presence_literal_index),
                                       lp_values, encoder);
    }
    e.x_size_min = x_helper->SizeMin(rect);
    e.y_size_min = y_helper->SizeMin(rect);
    e.energy_lp = ComputeEnergyLp(e, lp_values, integer_trail);
    events.push_back(e);
  }
 
  if (events.empty()) return;
 
  // Compute y_spread.
  double average_d = 0.0;
  for (const auto& e : events) {
    average_d += ToDouble(e.y_min + e.y_max);
  }
  const double average = average_d / 2.0 / static_cast<double>(events.size());
  for (auto& e : events) {
    e.y_spread = std::abs(ToDouble(e.y_max) - average) +
                 std::abs(average - ToDouble(e.y_min));
  }
  GenerateEnergeticCuts(cut_name, lp_values, events, IntegerValue(0),
                        /*events_are_2d=*/true, model, manager);
}
 
CutGenerator CreateNoOverlap2dEnergyCutGenerator(
    const std::vector<IntervalVariable>& x_intervals,
    const std::vector<IntervalVariable>& y_intervals, Model* model) {
  CutGenerator result;
  IntervalsRepository* intervals_repository =
      model->GetOrCreate<IntervalsRepository>();
  const int num_rectangles = x_intervals.size();
 
  std::vector<LinearExpression> energies;
  std::vector<AffineExpression> x_sizes;
  std::vector<AffineExpression> y_sizes;
  for (int i = 0; i < num_rectangles; ++i) {
    x_sizes.push_back(intervals_repository->Size(x_intervals[i]));
    y_sizes.push_back(intervals_repository->Size(y_intervals[i]));
  }
  LinearizeInnerProduct(x_sizes, y_sizes, model, &energies);
 
  SchedulingConstraintHelper* x_helper =
      new SchedulingConstraintHelper(x_intervals, model);
  model->TakeOwnership(x_helper);
 
  SchedulingConstraintHelper* y_helper =
      new SchedulingConstraintHelper(y_intervals, model);
  model->TakeOwnership(y_helper);
 
  AddIntegerVariableFromIntervals(x_helper, model, &result.vars);
  AddIntegerVariableFromIntervals(y_helper, model, &result.vars);
  gtl::STLSortAndRemoveDuplicates(&result.vars);
 
  Trail* trail = model->GetOrCreate<Trail>();
  IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
  IntegerEncoder* encoder = model->GetOrCreate<IntegerEncoder>();
 
  result.generate_cuts =
      [integer_trail, trail, encoder, x_helper, y_helper, model, energies](
          const absl::StrongVector<IntegerVariable, double>& lp_values,
          LinearConstraintManager* manager) {
        if (trail->CurrentDecisionLevel() > 0) return true;
 
        if (!x_helper->SynchronizeAndSetTimeDirection(true)) return false;
        if (!y_helper->SynchronizeAndSetTimeDirection(true)) return false;
 
        const int num_rectangles = x_helper->NumTasks();
        std::vector<int> active_rectangles;
        std::vector<Rectangle> cached_rectangles(num_rectangles);
        for (int rect = 0; rect < num_rectangles; ++rect) {
          if (y_helper->IsAbsent(rect) || y_helper->IsAbsent(rect)) continue;
          // We do not consider rectangles controlled by 2 different unassigned
          // enforcement literals.
          if (!x_helper->IsPresent(rect) && !y_helper->IsPresent(rect) &&
              x_helper->PresenceLiteral(rect) !=
                  y_helper->PresenceLiteral(rect)) {
            continue;
          }
 
          // TODO(user): It might be possible/better to use some shifted value
          // here, but for now this code is not in the hot spot, so better be
          // defensive and only do connected components on really disjoint
          // rectangles.
          Rectangle& rectangle = cached_rectangles[rect];
          rectangle.x_min = x_helper->StartMin(rect);
          rectangle.x_max = x_helper->EndMax(rect);
          rectangle.y_min = y_helper->StartMin(rect);
          rectangle.y_max = y_helper->EndMax(rect);
 
          active_rectangles.push_back(rect);
        }
 
        if (active_rectangles.size() <= 1) return true;
 
        std::vector<absl::Span<int>> components =
            GetOverlappingRectangleComponents(
                cached_rectangles, absl::MakeSpan(active_rectangles));
 
        // Forward pass. No need to do a backward pass.
        for (absl::Span<int> rectangles : components) {
          if (rectangles.size() <= 1) continue;
 
          GenerateNoOverlap2dEnergyCut(
              energies, rectangles, "NoOverlap2dXEnergy", lp_values, model,
              integer_trail, encoder, manager, x_helper, y_helper);
          GenerateNoOverlap2dEnergyCut(
              energies, rectangles, "NoOverlap2dYEnergy", lp_values, model,
              integer_trail, encoder, manager, y_helper, x_helper);
        }
 
        return true;
      };
  return result;
}
 
CutGenerator CreateCumulativeTimeTableCutGenerator(
    const std::vector<IntervalVariable>& intervals,
    const AffineExpression& capacity,
    const std::vector<AffineExpression>& demands, Model* model) {
  CutGenerator result;
 
  SchedulingConstraintHelper* helper =
      new SchedulingConstraintHelper(intervals, model);
  model->TakeOwnership(helper);
 
  IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
  AppendVariablesToCumulativeCut(capacity, demands, integer_trail, &result);
 
  AddIntegerVariableFromIntervals(helper, model, &result.vars);
  gtl::STLSortAndRemoveDuplicates(&result.vars);
 
  struct TimeTableEvent {
    int interval_index;
    IntegerValue time;
    bool positive;
    AffineExpression demand;
  };
 
  Trail* trail = model->GetOrCreate<Trail>();
 
  result.generate_cuts =
      [helper, capacity, demands, trail, integer_trail, model](
          const absl::StrongVector<IntegerVariable, double>& lp_values,
          LinearConstraintManager* manager) {
        if (trail->CurrentDecisionLevel() > 0) return true;
 
        std::vector<TimeTableEvent> events;
        // Iterate through the intervals. If start_max < end_min, the demand
        // is mandatory.
        for (int i = 0; i < helper->NumTasks(); ++i) {
          if (helper->IsAbsent(i)) continue;
 
          const IntegerValue start_max = helper->StartMax(i);
          const IntegerValue end_min = helper->EndMin(i);
 
          if (start_max >= end_min) continue;
 
          TimeTableEvent e1;
          e1.interval_index = i;
          e1.time = start_max;
          e1.demand = demands[i];
          e1.positive = true;
 
          TimeTableEvent e2 = e1;
          e2.time = end_min;
          e2.positive = false;
          events.push_back(e1);
          events.push_back(e2);
        }
 
        // Sort events by time.
        // It is also important that all positive event with the same time as
        // negative events appear after for the correctness of the algo below.
        std::sort(events.begin(), events.end(),
                  [](const TimeTableEvent& i, const TimeTableEvent& j) {
                    if (i.time == j.time) {
                      if (i.positive == j.positive) {
                        return i.interval_index < j.interval_index;
                      }
                      return !i.positive;
                    }
                    return i.time < j.time;
                  });
 
        std::vector<TimeTableEvent> cut_events;
        bool added_positive_event = false;
        for (const TimeTableEvent& e : events) {
          if (e.positive) {
            added_positive_event = true;
            cut_events.push_back(e);
            continue;
          }
          if (added_positive_event && cut_events.size() > 1) {
            // Create cut.
            bool cut_generated = true;
            LinearConstraintBuilder cut(model, kMinIntegerValue,
                                        IntegerValue(0));
            cut.AddTerm(capacity, IntegerValue(-1));
            for (const TimeTableEvent& cut_event : cut_events) {
              if (helper->IsPresent(cut_event.interval_index)) {
                cut.AddTerm(cut_event.demand, IntegerValue(1));
              } else {
                cut_generated &= cut.AddLiteralTerm(
                    helper->PresenceLiteral(cut_event.interval_index),
                    integer_trail->LowerBound(cut_event.demand));
                if (!cut_generated) break;
              }
            }
            if (cut_generated) {
              // Violation of the cut is checked by AddCut so we don't check
              // it here.
              manager->AddCut(cut.Build(), "CumulativeTimeTable", lp_values);
            }
          }
          // Remove the event.
          int new_size = 0;
          for (int i = 0; i < cut_events.size(); ++i) {
            if (cut_events[i].interval_index == e.interval_index) {
              continue;
            }
            cut_events[new_size] = cut_events[i];
            new_size++;
          }
          cut_events.resize(new_size);
          added_positive_event = false;
        }
        return true;
      };
  return result;
}
 
// Cached Information about one interval.
struct PrecedenceEvent {
  IntegerValue start_min;
  IntegerValue start_max;
  AffineExpression start;
  IntegerValue end_min;
  IntegerValue end_max;
  AffineExpression end;
  IntegerValue demand_min;
};
 
void GeneratePrecedenceCuts(
    const std::string& cut_name,
    const absl::StrongVector<IntegerVariable, double>& lp_values,
    std::vector<PrecedenceEvent> events, IntegerValue capacity_max,
    Model* model, LinearConstraintManager* manager) {
  const int num_events = events.size();
  if (num_events <= 1) return;
 
  std::sort(events.begin(), events.end(),
            [](const PrecedenceEvent& e1, const PrecedenceEvent& e2) {
              return e1.start_min < e2.start_min ||
                     (e1.start_min == e2.start_min && e1.end_max < e2.end_max);
            });
 
  const double tolerance = 1e-4;
 
  for (int i = 0; i + 1 < num_events; ++i) {
    const PrecedenceEvent& e1 = events[i];
    for (int j = i + 1; j < num_events; ++j) {
      const PrecedenceEvent& e2 = events[j];
      if (e2.start_min >= e1.end_max) break;  // Break out of the index2 loop.
 
      // Encode only the interesting pairs.
      if (e1.demand_min + e2.demand_min <= capacity_max) continue;
 
      const bool interval_1_can_precede_2 = e1.end_min <= e2.start_max;
      const bool interval_2_can_precede_1 = e2.end_min <= e1.start_max;
 
      if (interval_1_can_precede_2 && !interval_2_can_precede_1 &&
          e1.end.LpValue(lp_values) >=
              e2.start.LpValue(lp_values) + tolerance) {
        // interval1.end <= interval2.start
        LinearConstraintBuilder cut(model, kMinIntegerValue, IntegerValue(0));
        cut.AddTerm(e1.end, IntegerValue(1));
        cut.AddTerm(e2.start, IntegerValue(-1));
      } else if (interval_2_can_precede_1 && !interval_1_can_precede_2 &&
                 e2.end.LpValue(lp_values) >=
                     e1.start.LpValue(lp_values) + tolerance) {
        // interval2.end <= interval1.start
        LinearConstraintBuilder cut(model, kMinIntegerValue, IntegerValue(0));
        cut.AddTerm(e2.end, IntegerValue(1));
        cut.AddTerm(e1.start, IntegerValue(-1));
        manager->AddCut(cut.Build(), cut_name, lp_values);
      }
    }
  }
}
 
CutGenerator CreateCumulativePrecedenceCutGenerator(
    const std::vector<IntervalVariable>& intervals,
    const AffineExpression& capacity,
    const std::vector<AffineExpression>& demands, Model* model) {
  CutGenerator result;
 
  SchedulingConstraintHelper* helper =
      new SchedulingConstraintHelper(intervals, model);
  model->TakeOwnership(helper);
 
  IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
  AppendVariablesToCumulativeCut(capacity, demands, integer_trail, &result);
 
  AddIntegerVariableFromIntervals(helper, model, &result.vars);
  gtl::STLSortAndRemoveDuplicates(&result.vars);
 
  Trail* trail = model->GetOrCreate<Trail>();
 
  result.generate_cuts =
      [trail, integer_trail, helper, demands, capacity, model](
          const absl::StrongVector<IntegerVariable, double>& lp_values,
          LinearConstraintManager* manager) {
        if (trail->CurrentDecisionLevel() > 0) return true;
 
        const IntegerValue capacity_max = integer_trail->UpperBound(capacity);
        std::vector<PrecedenceEvent> events;
        for (int t = 0; t < helper->NumTasks(); ++t) {
          if (!helper->IsPresent(t)) continue;
          PrecedenceEvent event;
          event.start_min = helper->StartMin(t);
          event.start_max = helper->StartMax(t);
          event.start = helper->Starts()[t];
          event.end_min = helper->EndMin(t);
          event.end_max = helper->EndMax(t);
          event.end = helper->Ends()[t];
          event.demand_min = integer_trail->LowerBound(demands[t]);
          events.push_back(event);
        }
        GeneratePrecedenceCuts("CumulativePrecedence", lp_values,
                               std::move(events), capacity_max, model, manager);
        return true;
      };
  return result;
}
 
CutGenerator CreateNoOverlapPrecedenceCutGenerator(
    const std::vector<IntervalVariable>& intervals, Model* model) {
  CutGenerator result;
 
  SchedulingConstraintHelper* helper =
      new SchedulingConstraintHelper(intervals, model);
  model->TakeOwnership(helper);
 
  AddIntegerVariableFromIntervals(helper, model, &result.vars);
  gtl::STLSortAndRemoveDuplicates(&result.vars);
 
  Trail* trail = model->GetOrCreate<Trail>();
 
  result.generate_cuts =
      [trail, helper, model](
          const absl::StrongVector<IntegerVariable, double>& lp_values,
          LinearConstraintManager* manager) {
        if (trail->CurrentDecisionLevel() > 0) return true;
 
        std::vector<PrecedenceEvent> events;
        for (int t = 0; t < helper->NumTasks(); ++t) {
          if (!helper->IsPresent(t)) continue;
          PrecedenceEvent event;
          event.start_min = helper->StartMin(t);
          event.start_max = helper->StartMax(t);
          event.start = helper->Starts()[t];
          event.end_min = helper->EndMin(t);
          event.end_max = helper->EndMax(t);
          event.end = helper->Ends()[t];
          event.demand_min = IntegerValue(1);
          events.push_back(event);
        }
        GeneratePrecedenceCuts("NoOverlapPrecedence", lp_values,
                               std::move(events), IntegerValue(1), model,
                               manager);
        return true;
      };
 
  return result;
}
 
// Stores the event for a rectangle along the two axis x and y.
//   For a no_overlap constraint, y is always of size 1 between 0 and 1.
//   For a cumulative constraint, y is the demand that must be between 0 and
//       capacity_max.
//   For a no_overlap_2d constraint, y the other dimension of the rect.
struct CtEvent {
  // The start min of the x interval.
  IntegerValue x_start_min;
 
  // The size min of the x interval.
  IntegerValue x_size_min;
 
  // The end of the x interval.
  AffineExpression x_end;
 
  // The lp value of the end of the x interval.
  double x_lp_end;
 
  // The start min of the y interval.
  IntegerValue y_start_min;
 
  // The end max of the y interval.
  IntegerValue y_end_max;
 
  // The min energy of the task (this is always larger or equal to x_size_min *
  // y_size_min).
  IntegerValue energy_min;
 
  // Indicates if the events used the optional energy information from the
  // model.
  bool use_energy = false;
 
  // Indicates if the cut is lifted, that is if it includes tasks that are not
  // strictly contained in the current time window.
  bool lifted = false;
 
  std::string DebugString() const {
    return absl::StrCat("CtEvent(x_end = ", x_end.DebugString(),
                        ", x_start_min = ", x_start_min.value(),
                        ", x_size_min = ", x_size_min.value(),
                        ", x_lp_end = ", x_lp_end,
                        ", y_start_min = ", y_start_min.value(),
                        ", y_end_max = ", y_end_max.value(),
                        ", energy_min = ", energy_min.value(),
                        ", use_energy = ", use_energy, ", lifted = ", lifted);
  }
};
 
// We generate the cut from the Smith's rule from:
// M. Queyranne, Structure of a simple scheduling polyhedron,
// Mathematical Programming 58 (1993), 263–285
//
// The original cut is:
//    sum(end_min_i * duration_min_i) >=
//        (sum(duration_min_i^2) + sum(duration_min_i)^2) / 2
// We strenghten this cuts by noticing that if all tasks starts after S,
// then replacing end_min_i by (end_min_i - S) is still valid.
//
// A second difference is that we look at a set of intervals starting
// after a given start_min, sorted by relative (end_lp - start_min).
void GenerateCompletionTimeCuts(
    const std::string& cut_name,
    const absl::StrongVector<IntegerVariable, double>& lp_values,
    std::vector<CtEvent> events, bool use_lifting, Model* model,
    LinearConstraintManager* manager) {
  TopNCuts top_n_cuts(15);
 
  // Sort by start min to bucketize by start_min.
  std::sort(events.begin(), events.end(),
            [](const CtEvent& e1, const CtEvent& e2) {
              return e1.x_start_min < e2.x_start_min;
            });
  for (int start = 0; start + 1 < events.size(); ++start) {
    // Skip to the next start_min value.
    if (start > 0 &&
        events[start].x_start_min == events[start - 1].x_start_min) {
      continue;
    }
 
    const IntegerValue sequence_start_min = events[start].x_start_min;
    std::vector<CtEvent> residual_tasks(events.begin() + start, events.end());
 
    // We look at event that start before sequence_start_min, but are forced
    // to cross this time point. In that case, we replace this event by a
    // truncated event starting at sequence_start_min. To do this, we reduce
    // the size_min, align the start_min with the sequence_start_min, and
    // scale the energy down accordingly.
    if (use_lifting) {
      for (int before = 0; before < start; ++before) {
        if (events[before].x_start_min + events[before].x_size_min >
            sequence_start_min) {
          // Build the vector of energies as the vector of sizes.
          CtEvent event = events[before];  // Copy.
          event.lifted = true;
          const IntegerValue old_size_min = event.x_size_min;
          event.x_size_min =
              event.x_size_min + event.x_start_min - sequence_start_min;
          event.x_start_min = sequence_start_min;
          // We can rescale the energy min correctly.
          //
          // Let's take the example of a rectangle of size 2 * 20 that overlaps
          // sequence start min by 1, and that can rotate by 90 degrees.
          // The energy min is 40, size min is 2, size_max is 20.
          // If the rectangle is horizontal, the lifted energy is:
          //     (20 - 1) * 2 = 38
          // If the rectangle is vertical, the lifted energy is:
          //     (2 - 1) * 20 = 20
          // The min of the two is always reached when size = size_min.
          event.energy_min = event.energy_min * event.x_size_min / old_size_min;
          residual_tasks.push_back(event);
        }
      }
    }
 
    std::sort(residual_tasks.begin(), residual_tasks.end(),
              [](const CtEvent& e1, const CtEvent& e2) {
                return e1.x_lp_end < e2.x_lp_end;
              });
 
    int best_end = -1;
    double best_efficacy = 0.01;
    IntegerValue best_min_contrib(0);
    IntegerValue sum_duration(0);
    IntegerValue sum_square_duration(0);
    IntegerValue best_size_divisor(0);
    double unscaled_lp_contrib = 0;
    IntegerValue current_start_min(kMaxIntegerValue);
    IntegerValue y_start_min = kMaxIntegerValue;
    IntegerValue y_end_max = kMinIntegerValue;
 
    for (int i = 0; i < residual_tasks.size(); ++i) {
      const CtEvent& event = residual_tasks[i];
      DCHECK_GE(event.x_start_min, sequence_start_min);
      const IntegerValue energy = event.energy_min;
      sum_duration += energy;
      sum_square_duration += energy * energy;
      unscaled_lp_contrib += event.x_lp_end * ToDouble(energy);
      current_start_min = std::min(current_start_min, event.x_start_min);
      y_start_min = std::min(y_start_min, event.y_start_min);
      y_end_max = std::max(y_end_max, event.y_end_max);
 
      const IntegerValue size_divisor = y_end_max - y_start_min;
 
      // We compute the cuts with all the sizes actually equal to
      //     size_min * demand_min / size_divisor
      // but to keep the computation in the integer domain, we multiply by
      // size_divisor where needed instead.
      const IntegerValue min_contrib =
          (sum_duration * sum_duration + sum_square_duration) / 2 +
          current_start_min * sum_duration * size_divisor;
      const double efficacy = (ToDouble(min_contrib) -
                               unscaled_lp_contrib * ToDouble(size_divisor)) /
                              std::sqrt(ToDouble(sum_square_duration));
      // TODO(user): Check overflow and ignore if too big.
      if (efficacy > best_efficacy) {
        best_efficacy = efficacy;
        best_end = i;
        best_min_contrib = min_contrib;
        best_size_divisor = size_divisor;
      }
    }
    if (best_end != -1) {
      LinearConstraintBuilder cut(model, best_min_contrib, kMaxIntegerValue);
      bool is_lifted = false;
      bool use_energy = false;
      for (int i = 0; i <= best_end; ++i) {
        const CtEvent& event = residual_tasks[i];
        is_lifted |= event.lifted;
        use_energy |= event.use_energy;
        cut.AddTerm(event.x_end, event.energy_min * best_size_divisor);
      }
      std::string full_name = cut_name;
      if (is_lifted) full_name.append("_lifted");
      if (use_energy) full_name.append("_energy");
      top_n_cuts.AddCut(cut.Build(), full_name, lp_values);
    }
  }
  top_n_cuts.TransferToManager(lp_values, manager);
}
 
CutGenerator CreateNoOverlapCompletionTimeCutGenerator(
    const std::vector<IntervalVariable>& intervals, Model* model) {
  CutGenerator result;
 
  SchedulingConstraintHelper* helper =
      new SchedulingConstraintHelper(intervals, model);
  model->TakeOwnership(helper);
 
  AddIntegerVariableFromIntervals(helper, model, &result.vars);
  gtl::STLSortAndRemoveDuplicates(&result.vars);
 
  Trail* trail = model->GetOrCreate<Trail>();
 
  result.generate_cuts =
      [trail, helper, model](
          const absl::StrongVector<IntegerVariable, double>& lp_values,
          LinearConstraintManager* manager) {
        if (trail->CurrentDecisionLevel() > 0) return true;
 
        auto generate_cuts = [&lp_values, model, manager,
                              helper](const std::string& cut_name) {
          std::vector<CtEvent> events;
          for (int index = 0; index < helper->NumTasks(); ++index) {
            if (!helper->IsPresent(index)) continue;
            const IntegerValue size_min = helper->SizeMin(index);
            if (size_min > 0) {
              const AffineExpression end_expr = helper->Ends()[index];
              CtEvent event;
              event.x_start_min = helper->StartMin(index);
              event.x_size_min = size_min;
              event.x_end = end_expr;
              event.x_lp_end = end_expr.LpValue(lp_values);
              event.y_start_min = IntegerValue(0);
              event.y_end_max = IntegerValue(1);
              event.energy_min = size_min;
              events.push_back(event);
            }
          }
          GenerateCompletionTimeCuts(cut_name, lp_values, std::move(events),
                                     /*use_lifting=*/false, model, manager);
        };
        if (!helper->SynchronizeAndSetTimeDirection(true)) return false;
        generate_cuts("NoOverlapCompletionTime");
        if (!helper->SynchronizeAndSetTimeDirection(false)) return false;
        generate_cuts("NoOverlapCompletionTimeMirror");
        return true;
      };
  return result;
}
 
CutGenerator CreateCumulativeCompletionTimeCutGenerator(
    const std::vector<IntervalVariable>& intervals,
    const AffineExpression& capacity,
    const std::vector<AffineExpression>& demands,
    const std::vector<LinearExpression>& energies, Model* model) {
  CutGenerator result;
 
  SchedulingConstraintHelper* helper =
      new SchedulingConstraintHelper(intervals, model);
  model->TakeOwnership(helper);
 
  IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
  AppendVariablesToCumulativeCut(capacity, demands, integer_trail, &result);
 
  AddIntegerVariableFromIntervals(helper, model, &result.vars);
  gtl::STLSortAndRemoveDuplicates(&result.vars);
 
  Trail* trail = model->GetOrCreate<Trail>();
 
  result.generate_cuts =
      [trail, integer_trail, helper, demands, energies, capacity, model](
          const absl::StrongVector<IntegerVariable, double>& lp_values,
          LinearConstraintManager* manager) {
        if (trail->CurrentDecisionLevel() > 0) return true;
 
        const IntegerValue capacity_max = integer_trail->UpperBound(capacity);
        auto generate_cuts = [&lp_values, model, manager, helper, capacity_max,
                              integer_trail, &demands,
                              &energies](const std::string& cut_name) {
          std::vector<CtEvent> events;
          for (int index = 0; index < helper->NumTasks(); ++index) {
            if (!helper->IsPresent(index)) continue;
            if (helper->SizeMin(index) > 0 &&
                integer_trail->LowerBound(demands[index]) > 0) {
              const AffineExpression end_expr = helper->Ends()[index];
              const IntegerValue size_min = helper->SizeMin(index);
              const IntegerValue demand_min =
                  integer_trail->LowerBound(demands[index]);
 
              CtEvent event;
              event.x_start_min = helper->StartMin(index);
              event.x_size_min = size_min;
              event.x_end = end_expr;
              event.x_lp_end = end_expr.LpValue(lp_values);
              event.y_start_min = IntegerValue(0);
              event.y_end_max = IntegerValue(capacity_max);
              event.energy_min = size_min * demand_min;
              // TODO(user): Investigate and re-enable a correct version.
              if (/*DISABLES_CODE*/ (false) &&
                  ProductIsLinearized(energies[index])) {
                const IntegerValue linearized_energy =
                    LinExprLowerBound(energies[index], *integer_trail);
                if (linearized_energy > event.energy_min) {
                  event.energy_min = linearized_energy;
                  event.use_energy = true;
                }
              }
              events.push_back(event);
            }
          }
          GenerateCompletionTimeCuts(cut_name, lp_values, std::move(events),
                                     /*use_lifting=*/true, model, manager);
        };
        if (!helper->SynchronizeAndSetTimeDirection(true)) return false;
        generate_cuts("CumulativeCompletionTime");
        if (!helper->SynchronizeAndSetTimeDirection(false)) return false;
        generate_cuts("CumulativeCompletionTimeMirror");
        return true;
      };
  return result;
}
 
CutGenerator CreateNoOverlap2dCompletionTimeCutGenerator(
    const std::vector<IntervalVariable>& x_intervals,
    const std::vector<IntervalVariable>& y_intervals, Model* model) {
  CutGenerator result;
 
  SchedulingConstraintHelper* x_helper =
      new SchedulingConstraintHelper(x_intervals, model);
  model->TakeOwnership(x_helper);
 
  SchedulingConstraintHelper* y_helper =
      new SchedulingConstraintHelper(y_intervals, model);
  model->TakeOwnership(y_helper);
 
  AddIntegerVariableFromIntervals(x_helper, model, &result.vars);
  AddIntegerVariableFromIntervals(y_helper, model, &result.vars);
  gtl::STLSortAndRemoveDuplicates(&result.vars);
 
  Trail* trail = model->GetOrCreate<Trail>();
 
  result.generate_cuts =
      [trail, x_helper, y_helper, model](
          const absl::StrongVector<IntegerVariable, double>& lp_values,
          LinearConstraintManager* manager) {
        if (trail->CurrentDecisionLevel() > 0) return true;
 
        if (!x_helper->SynchronizeAndSetTimeDirection(true)) return false;
        if (!y_helper->SynchronizeAndSetTimeDirection(true)) return false;
 
        const int num_rectangles = x_helper->NumTasks();
        std::vector<int> active_rectangles;
        std::vector<IntegerValue> cached_areas(num_rectangles);
        std::vector<Rectangle> cached_rectangles(num_rectangles);
        for (int rect = 0; rect < num_rectangles; ++rect) {
          if (!y_helper->IsPresent(rect) || !y_helper->IsPresent(rect))
            continue;
 
          cached_areas[rect] =
              x_helper->SizeMin(rect) * y_helper->SizeMin(rect);
          if (cached_areas[rect] == 0) continue;
 
          // TODO(user): It might be possible/better to use some shifted value
          // here, but for now this code is not in the hot spot, so better be
          // defensive and only do connected components on really disjoint
          // rectangles.
          Rectangle& rectangle = cached_rectangles[rect];
          rectangle.x_min = x_helper->StartMin(rect);
          rectangle.x_max = x_helper->EndMax(rect);
          rectangle.y_min = y_helper->StartMin(rect);
          rectangle.y_max = y_helper->EndMax(rect);
 
          active_rectangles.push_back(rect);
        }
 
        if (active_rectangles.size() <= 1) return true;
 
        std::vector<absl::Span<int>> components =
            GetOverlappingRectangleComponents(
                cached_rectangles, absl::MakeSpan(active_rectangles));
        for (absl::Span<int> rectangles : components) {
          if (rectangles.size() <= 1) continue;
 
          auto generate_cuts = [&lp_values, model, manager, &rectangles,
                                &cached_areas](
                                   const std::string& cut_name,
                                   SchedulingConstraintHelper* x_helper,
                                   SchedulingConstraintHelper* y_helper) {
            std::vector<CtEvent> events;
 
            for (const int rect : rectangles) {
              const AffineExpression x_end_expr = x_helper->Ends()[rect];
              CtEvent event;
              event.x_start_min = x_helper->ShiftedStartMin(rect);
              event.x_size_min = x_helper->SizeMin(rect);
              event.x_end = x_end_expr;
              event.x_lp_end = x_end_expr.LpValue(lp_values);
              event.y_start_min = y_helper->ShiftedStartMin(rect);
              event.y_end_max = y_helper->ShiftedEndMax(rect);
              event.energy_min =
                  x_helper->SizeMin(rect) * y_helper->SizeMin(rect);
              events.push_back(event);
            }
 
            GenerateCompletionTimeCuts(cut_name, lp_values, std::move(events),
                                       /*use_lifting=*/true, model, manager);
          };
 
          if (!x_helper->SynchronizeAndSetTimeDirection(true)) return false;
          if (!y_helper->SynchronizeAndSetTimeDirection(true)) return false;
          generate_cuts("NoOverlap2dXCompletionTime", x_helper, y_helper);
          generate_cuts("NoOverlap2dYCompletionTime", y_helper, x_helper);
          if (!x_helper->SynchronizeAndSetTimeDirection(false)) return false;
          if (!y_helper->SynchronizeAndSetTimeDirection(false)) return false;
          generate_cuts("NoOverlap2dXCompletionTimeMirror", x_helper, y_helper);
          generate_cuts("NoOverlap2dYCompletionTimeMirror", y_helper, x_helper);
        }
        return true;
      };
  return result;
}
 
}  // namespace sat
}  // namespace operations_research