precedences.h

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// Copyright 2010-2025 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_PRECEDENCES_H_
#define OR_TOOLS_SAT_PRECEDENCES_H_
 
#include <algorithm>
#include <cstdint>
#include <deque>
#include <functional>
#include <memory>
#include <utility>
#include <vector>
 
#include "absl/container/flat_hash_map.h"
#include "absl/container/inlined_vector.h"
#include "absl/log/check.h"
#include "absl/types/span.h"
#include "ortools/base/strong_vector.h"
#include "ortools/graph/graph.h"
#include "ortools/sat/integer.h"
#include "ortools/sat/integer_base.h"
#include "ortools/sat/model.h"
#include "ortools/sat/sat_base.h"
#include "ortools/sat/sat_solver.h"
#include "ortools/sat/synchronization.h"
#include "ortools/sat/util.h"
#include "ortools/util/bitset.h"
#include "ortools/util/rev.h"
#include "ortools/util/strong_integers.h"
 
namespace operations_research {
namespace sat {
 
struct FullIntegerPrecedence {
  IntegerVariable var;
  std::vector<int> indices;
  std::vector<IntegerValue> offsets;
};
 
// Stores all the precedences relation of the form "tail_X + offset <= head_X"
// that we could extract from the linear constraint of the model. These are
// stored in a directed graph.
//
// TODO(user): Support conditional relation.
// TODO(user): Support non-DAG like graph.
// TODO(user): Support variable offset that can be updated as search progress.
class PrecedenceRelations : public ReversibleInterface {
 public:
  explicit PrecedenceRelations(Model* model)
      : params_(*model->GetOrCreate<SatParameters>()),
        trail_(model->GetOrCreate<Trail>()),
        integer_trail_(model->GetOrCreate<IntegerTrail>()) {
    integer_trail_->RegisterReversibleClass(this);
  }
 
  void Resize(int num_variables) {
    graph_.ReserveNodes(num_variables);
    graph_.AddNode(num_variables - 1);
  }
 
  // Add a relation tail + offset <= head.
  // Returns true if it was added and is considered "new".
  bool Add(IntegerVariable tail, IntegerVariable head, IntegerValue offset);
 
  // Adds add relation (enf => a + b <= rhs) that is assumed to be true at
  // the current level.
  //
  // It will be automatically reverted via the SetLevel() functions that is
  // called before any integer propagations trigger.
  //
  // This is assumed to be called when a relation becomes true (enforcement are
  // assigned) and when it becomes false in reverse order (CHECKed).
  void PushConditionalRelation(absl::Span<const Literal> enforcements,
                               IntegerVariable a, IntegerVariable b,
                               IntegerValue rhs);
 
  // Called each time we change decision level.
  void SetLevel(int level) final;
 
  // Returns a set of relations var >= max_i(vars[index[i]] + offsets[i]).
  //
  // This currently only works if the precedence relation form a DAG.
  // If not we will just abort. TODO(user): generalize.
  //
  // TODO(user): Put some work limit in place, as this can be slow. Complexity
  // is in O(vars.size()) * num_arcs.
  //
  // TODO(user): Since we don't need ALL precedences, we could just work on a
  // sub-DAG of the full precedence graph instead of aborting. Or we can just
  // support the general non-DAG cases.
  //
  // TODO(user): Many relations can be redundant. Filter them.
  void ComputeFullPrecedences(absl::Span<const IntegerVariable> vars,
                              std::vector<FullIntegerPrecedence>* output);
 
  // Returns a set of precedences (var, index) such that var is after
  // vars[index]. All entries for the same variable will be contiguous and
  // sorted by index. We only list variable with at least two entries. The
  // offset can be retrieved via GetConditionalOffset(vars[index], var).
  struct PrecedenceData {
    IntegerVariable var;
    int index;
  };
  void CollectPrecedences(absl::Span<const IntegerVariable> vars,
                          std::vector<PrecedenceData>* output);
 
  // If we don't have too many variable, we compute the full transitive closure
  // and can query in O(1) if there is a relation between two variables.
  // This can be used to optimize some scheduling propagation and reasons.
  //
  // Warning: If there are too many, this will NOT contain all relations.
  //
  // Returns kMinIntegerValue if there are none.
  // Otherwise a + offset <= b.
  IntegerValue GetOffset(IntegerVariable a, IntegerVariable b) const;
 
  // Returns the minimum distance between a and b, and the reason for it (all
  // true). Note that we always check GetOffset() so if it is better, the
  // returned literal reason will be empty.
  //
  // We separate the two because usually the reason is only needed when we push,
  // which happen less often, so we don't mind doing two hash lookups, and we
  // really want to optimize the GetConditionalOffset() instead.
  //
  // Important: This doesn't contains the transitive closure.
  // Important: The span is only valid in a narrow scope.
  IntegerValue GetConditionalOffset(IntegerVariable a, IntegerVariable b) const;
  absl::Span<const Literal> GetConditionalEnforcements(IntegerVariable a,
                                                       IntegerVariable b) const;
 
  // The current code requires the internal data to be processed once all
  // relations are loaded.
  //
  // TODO(user): Be more dynamic as we start to add relations during search.
  void Build();
 
 private:
  void CreateLevelEntryIfNeeded();
 
  std::pair<IntegerVariable, IntegerVariable> GetKey(IntegerVariable a,
                                                     IntegerVariable b) const {
    return a <= b ? std::make_pair(a, b) : std::make_pair(b, a);
  }
 
  // tail + offset <= head.
  // Which is the same as tail - head <= -offset.
  bool AddInternal(IntegerVariable tail, IntegerVariable head,
                   IntegerValue offset) {
    const auto key = GetKey(tail, NegationOf(head));
    const auto [it, inserted] = root_relations_.insert({key, -offset});
    UpdateBestRelationIfBetter(key, -offset);
    if (inserted) {
      const int new_size = std::max(tail.value(), NegationOf(head).value()) + 1;
      if (new_size > after_.size()) after_.resize(new_size);
      after_[tail].push_back(head);
      after_[NegationOf(head)].push_back(NegationOf(tail));
      return true;
    }
    it->second = std::min(it->second, -offset);
    return false;
  }
 
  void UpdateBestRelationIfBetter(
      std::pair<IntegerVariable, IntegerVariable> key, IntegerValue rhs) {
    const auto [it, inserted] = best_relations_.insert({key, rhs});
    if (!inserted) {
      it->second = std::min(it->second, rhs);
    }
  }
 
  void UpdateBestRelation(std::pair<IntegerVariable, IntegerVariable> key,
                          IntegerValue rhs) {
    const auto it = root_relations_.find(key);
    if (it != root_relations_.end()) {
      rhs = std::min(rhs, it->second);
    }
    if (rhs == kMaxIntegerValue) {
      best_relations_.erase(key);
    } else {
      best_relations_[key] = rhs;
    }
  }
 
  const SatParameters& params_;
  Trail* trail_;
  IntegerTrail* integer_trail_;
 
  util::StaticGraph<> graph_;
  std::vector<IntegerValue> arc_offsets_;
 
  bool is_built_ = false;
  bool is_dag_ = false;
  std::vector<IntegerVariable> topological_order_;
 
  // Conditional stack for push/pop of conditional relations.
  //
  // TODO(user): this kind of reversible hash_map is already implemented in
  // other part of the code. Consolidate.
  struct ConditionalEntry {
    ConditionalEntry(int p, IntegerValue r,
                     std::pair<IntegerVariable, IntegerVariable> k,
                     absl::Span<const Literal> e)
        : prev_entry(p), rhs(r), key(k), enforcements(e.begin(), e.end()) {}
 
    int prev_entry;
    IntegerValue rhs;
    std::pair<IntegerVariable, IntegerVariable> key;
    absl::InlinedVector<Literal, 4> enforcements;
  };
  std::vector<ConditionalEntry> conditional_stack_;
  std::vector<std::pair<int, int>> level_to_stack_size_;
 
  // This is always stored in the form (a + b <= rhs).
  // The conditional relations contains indices in the conditional_stack_.
  absl::flat_hash_map<std::pair<IntegerVariable, IntegerVariable>, IntegerValue>
      root_relations_;
  absl::flat_hash_map<std::pair<IntegerVariable, IntegerVariable>, int>
      conditional_relations_;
 
  // Contains std::min() of the offset from root_relations_ and
  // conditional_relations_.
  absl::flat_hash_map<std::pair<IntegerVariable, IntegerVariable>, IntegerValue>
      best_relations_;
 
  // Store for each variable x, the variables y that appears in GetOffset(x, y)
  // or GetConditionalOffset(x, y). That is the variable that are after x with
  // an offset. Note that conditional_after_ is updated on dive/backtrack.
  util_intops::StrongVector<IntegerVariable, std::vector<IntegerVariable>>
      after_;
  util_intops::StrongVector<IntegerVariable, std::vector<IntegerVariable>>
      conditional_after_;
 
  // Temp data for CollectPrecedences.
  std::vector<IntegerVariable> var_with_positive_degree_;
  util_intops::StrongVector<IntegerVariable, int> var_to_degree_;
  util_intops::StrongVector<IntegerVariable, int> var_to_last_index_;
  std::vector<PrecedenceData> tmp_precedences_;
};
 
// This class implement a propagator on simple inequalities between integer
// variables of the form (i1 + offset <= i2). The offset can be constant or
// given by the value of a third integer variable. Offsets can also be negative.
//
// The algorithm works by mapping the problem onto a graph where the edges carry
// the offset and the nodes correspond to one of the two bounds of an integer
// variable (lower_bound or -upper_bound). It then find the fixed point using an
// incremental variant of the Bellman-Ford(-Tarjan) algorithm.
//
// This is also known as an "integer difference logic theory" in the SMT world.
// Another word is "separation logic".
//
// TODO(user): We could easily generalize the code to support any relation of
// the form a*X + b*Y + c*Z >= rhs (or <=). Do that since this class should be
// a lot faster at propagating small linear inequality than the generic
// propagator and the overhead of supporting coefficient should not be too bad.
class PrecedencesPropagator : public SatPropagator, PropagatorInterface {
 public:
  explicit PrecedencesPropagator(Model* model)
      : SatPropagator("PrecedencesPropagator"),
        relations_(model->GetOrCreate<PrecedenceRelations>()),
        trail_(model->GetOrCreate<Trail>()),
        integer_trail_(model->GetOrCreate<IntegerTrail>()),
        shared_stats_(model->Mutable<SharedStatistics>()),
        watcher_(model->GetOrCreate<GenericLiteralWatcher>()),
        watcher_id_(watcher_->Register(this)) {
    model->GetOrCreate<SatSolver>()->AddPropagator(this);
    integer_trail_->RegisterWatcher(&modified_vars_);
    watcher_->SetPropagatorPriority(watcher_id_, 0);
  }
 
  // This type is neither copyable nor movable.
  PrecedencesPropagator(const PrecedencesPropagator&) = delete;
  PrecedencesPropagator& operator=(const PrecedencesPropagator&) = delete;
  ~PrecedencesPropagator() override;
 
  bool Propagate() final;
  bool Propagate(Trail* trail) final;
  void Untrail(const Trail& trail, int trail_index) final;
 
  // Propagates all the outgoing arcs of the given variable (and only those). It
  // is more efficient to do all these propagation in one go by calling
  // Propagate(), but for scheduling problem, we wants to propagate right away
  // the end of an interval when its start moved.
  bool PropagateOutgoingArcs(IntegerVariable var);
 
  // Add a precedence relation (i1 + offset <= i2) between integer variables.
  //
  // Important: The optionality of the variable should be marked BEFORE this
  // is called.
  void AddPrecedence(IntegerVariable i1, IntegerVariable i2);
  void AddPrecedenceWithOffset(IntegerVariable i1, IntegerVariable i2,
                               IntegerValue offset);
  void AddPrecedenceWithVariableOffset(IntegerVariable i1, IntegerVariable i2,
                                       IntegerVariable offset_var);
 
  // Same as above, but the relation is only true when the given literal is.
  void AddConditionalPrecedence(IntegerVariable i1, IntegerVariable i2,
                                Literal l);
  void AddConditionalPrecedenceWithOffset(IntegerVariable i1,
                                          IntegerVariable i2,
                                          IntegerValue offset, Literal l);
 
  // Generic function that cover all of the above case and more.
  void AddPrecedenceWithAllOptions(IntegerVariable i1, IntegerVariable i2,
                                   IntegerValue offset,
                                   IntegerVariable offset_var,
                                   absl::Span<const Literal> presence_literals);
 
  // This version check current precedence. It is however "slow".
  bool AddPrecedenceWithOffsetIfNew(IntegerVariable i1, IntegerVariable i2,
                                    IntegerValue offset);
 
 private:
  DEFINE_STRONG_INDEX_TYPE(ArcIndex);
  DEFINE_STRONG_INDEX_TYPE(OptionalArcIndex);
 
  // Information about an individual arc.
  struct ArcInfo {
    IntegerVariable tail_var;
    IntegerVariable head_var;
 
    IntegerValue offset;
    IntegerVariable offset_var;  // kNoIntegerVariable if none.
 
    // This arc is "present" iff all these literals are true.
    absl::InlinedVector<Literal, 6> presence_literals;
 
    // Used temporarily by our implementation of the Bellman-Ford algorithm. It
    // should be false at the beginning of BellmanFordTarjan().
    mutable bool is_marked;
  };
 
  // Internal functions to add new precedence relations.
  //
  // Note that internally, we only propagate lower bounds, so each time we add
  // an arc, we actually create two of them: one on the given variables, and one
  // on their negation.
  void AdjustSizeFor(IntegerVariable i);
  void AddArc(IntegerVariable tail, IntegerVariable head, IntegerValue offset,
              IntegerVariable offset_var,
              absl::Span<const Literal> presence_literals);
 
  // Enqueue a new lower bound for the variable arc.head_lb that was deduced
  // from the current value of arc.tail_lb and the offset of this arc.
  bool EnqueueAndCheck(const ArcInfo& arc, IntegerValue new_head_lb,
                       Trail* trail);
  IntegerValue ArcOffset(const ArcInfo& arc) const;
 
  // Inspect all the optional arcs that needs inspection (to stay sparse) and
  // check if their presence literal can be propagated to false.
  void PropagateOptionalArcs(Trail* trail);
 
  // The core algorithm implementation is split in these functions. One must
  // first call InitializeBFQueueWithModifiedNodes() that will push all the
  // IntegerVariable whose lower bound has been modified since the last call.
  // Then, BellmanFordTarjan() will take care of all the propagation and returns
  // false in case of conflict. Internally, it uses DisassembleSubtree() which
  // is the Tarjan variant to detect a possible positive cycle. Before exiting,
  // it will call CleanUpMarkedArcsAndParents().
  //
  // The Tarjan version of the Bellam-Ford algorithm is really nice in our
  // context because it was really easy to make it incremental. Moreover, it
  // supports batch increment!
  //
  // This implementation is kind of unique because of our context and the fact
  // that it is incremental, but a good reference is "Negative-cycle detection
  // algorithms", Boris V. Cherkassky, Andrew V. Goldberg, 1996,
  // http://people.cs.nctu.edu.tw/~tjshen/doc/ne.pdf
  void InitializeBFQueueWithModifiedNodes();
  bool BellmanFordTarjan(Trail* trail);
  bool DisassembleSubtree(int source, int target,
                          std::vector<bool>* can_be_skipped);
  void AnalyzePositiveCycle(ArcIndex first_arc, Trail* trail,
                            std::vector<Literal>* must_be_all_true,
                            std::vector<Literal>* literal_reason,
                            std::vector<IntegerLiteral>* integer_reason);
  void CleanUpMarkedArcsAndParents();
 
  // Loops over all the arcs and verify that there is no propagation left.
  // This is only meant to be used in a DCHECK() and is not optimized.
  bool NoPropagationLeft(const Trail& trail) const;
 
  // Update relations_.
  void PushConditionalRelations(const ArcInfo& arc);
 
  // External class needed to get the IntegerVariable lower bounds and Enqueue
  // new ones.
  PrecedenceRelations* relations_;
  Trail* trail_;
  IntegerTrail* integer_trail_;
  SharedStatistics* shared_stats_ = nullptr;
  GenericLiteralWatcher* watcher_;
  int watcher_id_;
 
  // The key to our incrementality. This will be cleared once the propagation
  // is done, and automatically updated by the integer_trail_ with all the
  // IntegerVariable that changed since the last clear.
  SparseBitset<IntegerVariable> modified_vars_;
 
  // An arc needs to be inspected for propagation (i.e. is impacted) if its
  // tail_var changed. If an arc has 3 variables (tail, offset, head), it will
  // appear as 6 different entries in the arcs_ vector, one for each variable
  // and its negation, each time with a different tail.
  //
  // TODO(user): rearranging the index so that the arc of the same node are
  // consecutive like in StaticGraph should have a big performance impact.
  //
  // TODO(user): We do not need to store ArcInfo.tail_var here.
  util_intops::StrongVector<IntegerVariable, absl::InlinedVector<ArcIndex, 6>>
      impacted_arcs_;
  util_intops::StrongVector<ArcIndex, ArcInfo> arcs_;
 
  // This is similar to impacted_arcs_/arcs_ but it is only used to propagate
  // one of the presence literals when the arc cannot be present. An arc needs
  // to appear only once in potential_arcs_, but it will be referenced by
  // all its variable in impacted_potential_arcs_.
  util_intops::StrongVector<IntegerVariable,
                            absl::InlinedVector<OptionalArcIndex, 6>>
      impacted_potential_arcs_;
  util_intops::StrongVector<OptionalArcIndex, ArcInfo> potential_arcs_;
 
  // Each time a literal becomes true, this list the set of arcs for which we
  // need to decrement their count. When an arc count reach zero, it must be
  // added to the set of impacted_arcs_. Note that counts never becomes
  // negative.
  //
  // TODO(user): Try a one-watcher approach instead. Note that in most cases
  // arc should be controlled by 1 or 2 literals, so not sure it is worth it.
  util_intops::StrongVector<LiteralIndex, absl::InlinedVector<ArcIndex, 6>>
      literal_to_new_impacted_arcs_;
  util_intops::StrongVector<ArcIndex, int> arc_counts_;
 
  // Temp vectors to hold the reason of an assignment.
  std::vector<Literal> literal_reason_;
  std::vector<IntegerLiteral> integer_reason_;
 
  // Temp vectors for the Bellman-Ford algorithm. The graph in which this
  // algorithm works is in one to one correspondence with the IntegerVariable in
  // impacted_arcs_.
  std::deque<int> bf_queue_;
  std::vector<bool> bf_in_queue_;
  std::vector<bool> bf_can_be_skipped_;
  std::vector<ArcIndex> bf_parent_arc_of_;
 
  // Temp vector used by the tree traversal in DisassembleSubtree().
  std::vector<int> tmp_vector_;
 
  // Stats.
  int64_t num_cycles_ = 0;
  int64_t num_pushes_ = 0;
  int64_t num_enforcement_pushes_ = 0;
};
 
// Similar to AffineExpression, but with a zero constant.
// If coeff is zero, then this is always zero and var is ignored.
struct LinearTerm {
  IntegerVariable var = kNoIntegerVariable;
  IntegerValue coeff = IntegerValue(0);
 
  void MakeCoeffPositive() {
    if (coeff < 0) {
      coeff = -coeff;
      var = NegationOf(var);
    }
  }
};
 
// A relation of the form enforcement => a + b \in [lhs, rhs].
// Note that the [lhs, rhs] interval should always be within [min_activity,
// max_activity] where the activity is the value of a + b.
struct Relation {
  Literal enforcement;
  LinearTerm a;
  LinearTerm b;
  IntegerValue lhs;
  IntegerValue rhs;
};
 
// A repository of all the enforced linear constraints of size 1 or 2.
//
// TODO(user): This is not always needed, find a way to clean this once we
// don't need it.
class BinaryRelationRepository {
 public:
  int size() const { return relations_.size(); }
  const Relation& relation(int index) const { return relations_[index]; }
 
  absl::Span<const int> relation_indices(LiteralIndex lit) const {
    if (lit >= lit_to_relations_.size()) return {};
    return lit_to_relations_[lit];
  }
 
  // Adds a relation lit => a + b \in [lhs, rhs].
  void Add(Literal lit, LinearTerm a, LinearTerm b, IntegerValue lhs,
           IntegerValue rhs);
 
  // Builds the literal to relations mapping. This should be called once all the
  // relations have been added.
  void Build();
 
  // Assuming level-zero bounds + any (var >= value) in the input map,
  // fills "output" with a "propagated" set of bounds assuming lit is true (by
  // using the relations enforced by lit). Note that we will only fill bounds >
  // level-zero ones in output.
  //
  // Returns false if the new bounds are infeasible at level zero.
  bool PropagateLocalBounds(
      const IntegerTrail& integer_trail, Literal lit,
      const absl::flat_hash_map<IntegerVariable, IntegerValue>& input,
      absl::flat_hash_map<IntegerVariable, IntegerValue>* output) const;
 
 private:
  bool is_built_ = false;
  std::vector<Relation> relations_;
  CompactVectorVector<LiteralIndex, int> lit_to_relations_;
};
 
// Detects if at least one of a subset of linear of size 2 or 1, touching the
// same variable, must be true. When this is the case we add a new propagator to
// propagate that fact.
//
// TODO(user): Shall we do that on the main thread before the workers are
// spawned? note that the probing version need the model to be loaded though.
class GreaterThanAtLeastOneOfDetector {
 public:
  explicit GreaterThanAtLeastOneOfDetector(Model* model)
      : repository_(*model->GetOrCreate<BinaryRelationRepository>()) {}
 
  // Advanced usage. To be called once all the constraints have been added to
  // the model. This will detect GreaterThanAtLeastOneOfConstraint().
  // Returns the number of added constraint.
  //
  // TODO(user): This can be quite slow, add some kind of deterministic limit
  // so that we can use it all the time.
  int AddGreaterThanAtLeastOneOfConstraints(Model* model,
                                            bool auto_detect_clauses = false);
 
 private:
  // Given an existing clause, sees if it can be used to add "greater than at
  // least one of" type of constraints. Returns the number of such constraint
  // added.
  int AddGreaterThanAtLeastOneOfConstraintsFromClause(
      absl::Span<const Literal> clause, Model* model);
 
  // Another approach for AddGreaterThanAtLeastOneOfConstraints(), this one
  // might be a bit slow as it relies on the propagation engine to detect
  // clauses between incoming arcs presence literals.
  // Returns the number of added constraints.
  int AddGreaterThanAtLeastOneOfConstraintsWithClauseAutoDetection(
      Model* model);
 
  // Once we identified a clause and relevant indices, this build the
  // constraint. Returns true if we actually add it.
  bool AddRelationFromIndices(IntegerVariable var,
                              absl::Span<const Literal> clause,
                              absl::Span<const int> indices, Model* model);
 
  BinaryRelationRepository& repository_;
};
 
// =============================================================================
// Implementation of the small API functions below.
// =============================================================================
 
inline void PrecedencesPropagator::AddPrecedence(IntegerVariable i1,
                                                 IntegerVariable i2) {
  AddArc(i1, i2, /*offset=*/IntegerValue(0), /*offset_var=*/kNoIntegerVariable,
         {});
}
 
inline void PrecedencesPropagator::AddPrecedenceWithOffset(
    IntegerVariable i1, IntegerVariable i2, IntegerValue offset) {
  AddArc(i1, i2, offset, /*offset_var=*/kNoIntegerVariable, {});
}
 
inline void PrecedencesPropagator::AddConditionalPrecedence(IntegerVariable i1,
                                                            IntegerVariable i2,
                                                            Literal l) {
  AddArc(i1, i2, /*offset=*/IntegerValue(0), /*offset_var=*/kNoIntegerVariable,
         {l});
}
 
inline void PrecedencesPropagator::AddConditionalPrecedenceWithOffset(
    IntegerVariable i1, IntegerVariable i2, IntegerValue offset, Literal l) {
  AddArc(i1, i2, offset, /*offset_var=*/kNoIntegerVariable, {l});
}
 
inline void PrecedencesPropagator::AddPrecedenceWithVariableOffset(
    IntegerVariable i1, IntegerVariable i2, IntegerVariable offset_var) {
  AddArc(i1, i2, /*offset=*/IntegerValue(0), offset_var, {});
}
 
inline void PrecedencesPropagator::AddPrecedenceWithAllOptions(
    IntegerVariable i1, IntegerVariable i2, IntegerValue offset,
    IntegerVariable offset_var, absl::Span<const Literal> presence_literals) {
  AddArc(i1, i2, offset, offset_var, presence_literals);
}
 
// =============================================================================
// Model based functions.
// =============================================================================
 
// a <= b.
inline std::function<void(Model*)> LowerOrEqual(IntegerVariable a,
                                                IntegerVariable b) {
  return [=](Model* model) {
    return model->GetOrCreate<PrecedencesPropagator>()->AddPrecedence(a, b);
  };
}
 
// a + offset <= b.
inline std::function<void(Model*)> LowerOrEqualWithOffset(IntegerVariable a,
                                                          IntegerVariable b,
                                                          int64_t offset) {
  return [=](Model* model) {
    model->GetOrCreate<PrecedenceRelations>()->Add(a, b, IntegerValue(offset));
    model->GetOrCreate<PrecedencesPropagator>()->AddPrecedenceWithOffset(
        a, b, IntegerValue(offset));
  };
}
 
// a + offset <= b. (when a and b are of the form 1 * var + offset).
inline std::function<void(Model*)> AffineCoeffOneLowerOrEqualWithOffset(
    AffineExpression a, AffineExpression b, int64_t offset) {
  CHECK_NE(a.var, kNoIntegerVariable);
  CHECK_EQ(a.coeff, 1);
  CHECK_NE(b.var, kNoIntegerVariable);
  CHECK_EQ(b.coeff, 1);
  return [=](Model* model) {
    model->GetOrCreate<PrecedenceRelations>()->Add(
        a.var, b.var, a.constant - b.constant + offset);
    model->GetOrCreate<PrecedencesPropagator>()->AddPrecedenceWithOffset(
        a.var, b.var, a.constant - b.constant + offset);
  };
}
 
// l => (a + b <= ub).
inline void AddConditionalSum2LowerOrEqual(
    absl::Span<const Literal> enforcement_literals, IntegerVariable a,
    IntegerVariable b, int64_t ub, Model* model) {
  // TODO(user): Refactor to be sure we do not miss any level zero relations.
  if (enforcement_literals.empty()) {
    model->GetOrCreate<PrecedenceRelations>()->Add(a, NegationOf(b),
                                                   IntegerValue(-ub));
  }
 
  PrecedencesPropagator* p = model->GetOrCreate<PrecedencesPropagator>();
  p->AddPrecedenceWithAllOptions(a, NegationOf(b), IntegerValue(-ub),
                                 kNoIntegerVariable, enforcement_literals);
}
 
// l => (a + b + c <= ub).
//
// TODO(user): Use level zero bounds to infer binary precedence relations?
inline void AddConditionalSum3LowerOrEqual(
    absl::Span<const Literal> enforcement_literals, IntegerVariable a,
    IntegerVariable b, IntegerVariable c, int64_t ub, Model* model) {
  PrecedencesPropagator* p = model->GetOrCreate<PrecedencesPropagator>();
  p->AddPrecedenceWithAllOptions(a, NegationOf(c), IntegerValue(-ub), b,
                                 enforcement_literals);
}
 
// a >= b.
inline std::function<void(Model*)> GreaterOrEqual(IntegerVariable a,
                                                  IntegerVariable b) {
  return [=](Model* model) {
    return model->GetOrCreate<PrecedencesPropagator>()->AddPrecedence(b, a);
  };
}
 
// a == b.
inline std::function<void(Model*)> Equality(IntegerVariable a,
                                            IntegerVariable b) {
  return [=](Model* model) {
    model->Add(LowerOrEqual(a, b));
    model->Add(LowerOrEqual(b, a));
  };
}
 
// a + offset == b.
inline std::function<void(Model*)> EqualityWithOffset(IntegerVariable a,
                                                      IntegerVariable b,
                                                      int64_t offset) {
  return [=](Model* model) {
    model->Add(LowerOrEqualWithOffset(a, b, offset));
    model->Add(LowerOrEqualWithOffset(b, a, -offset));
  };
}
 
// is_le => (a + offset <= b).
inline std::function<void(Model*)> ConditionalLowerOrEqualWithOffset(
    IntegerVariable a, IntegerVariable b, int64_t offset, Literal is_le) {
  return [=](Model* model) {
    PrecedencesPropagator* p = model->GetOrCreate<PrecedencesPropagator>();
    p->AddConditionalPrecedenceWithOffset(a, b, IntegerValue(offset), is_le);
  };
}
 
}  // namespace sat
}  // namespace operations_research
 
#endif  // OR_TOOLS_SAT_PRECEDENCES_H_