diff --git a/ortools/sat/cp_model.cc b/ortools/sat/cp_model.cc
index d033537a396..dbb9ca6f419 100644
--- a/ortools/sat/cp_model.cc
+++ b/ortools/sat/cp_model.cc
@@ -218,6 +218,12 @@ void CircuitConstraint::AddArc(int tail, int head, BoolVar literal) {
   proto_->mutable_circuit()->add_literals(literal.index_);
 }
 
+void MultipleCircuitConstraint::AddArc(int tail, int head, BoolVar literal) {
+  proto_->mutable_routes()->add_tails(tail);
+  proto_->mutable_routes()->add_heads(head);
+  proto_->mutable_routes()->add_literals(literal.index_);
+}
+
 void TableConstraint::AddTuple(absl::Span<const int64> tuple) {
   CHECK_EQ(tuple.size(), proto_->table().vars_size());
   for (const int64 t : tuple) {
@@ -551,6 +557,10 @@ CircuitConstraint CpModelBuilder::AddCircuitConstraint() {
   return CircuitConstraint(cp_model_.add_constraints());
 }
 
+MultipleCircuitConstraint CpModelBuilder::AddMultipleCircuitConstraint() {
+  return MultipleCircuitConstraint(cp_model_.add_constraints());
+}
+
 TableConstraint CpModelBuilder::AddAllowedAssignments(
     absl::Span<const IntVar> vars) {
   ConstraintProto* const proto = cp_model_.add_constraints();
diff --git a/ortools/sat/cp_model.h b/ortools/sat/cp_model.h
index 478b754ff33..38c6abe046b 100644
--- a/ortools/sat/cp_model.h
+++ b/ortools/sat/cp_model.h
@@ -113,6 +113,7 @@ class BoolVar {
   friend class CpModelBuilder;
   friend class IntVar;
   friend class IntervalVar;
+  friend class MultipleCircuitConstraint;
   friend class LinearExpr;
   friend class ReservoirConstraint;
   friend bool SolutionBooleanValue(const CpSolverResponse& r, BoolVar x);
@@ -458,6 +459,29 @@ class CircuitConstraint : public Constraint {
   using Constraint::Constraint;
 };
 
+/**
+ * Specialized circuit constraint.
+ *
+ * This constraint allows adding arcs to the multiple circuit constraint
+ * incrementally.
+ */
+class MultipleCircuitConstraint : public Constraint {
+ public:
+  /**
+   * Add an arc to the circuit.
+   *
+   * @param tail the index of the tail node.
+   * @param head the index of the head node.
+   * @param literal it will be set to true if the arc is selected.
+   */
+  void AddArc(int tail, int head, BoolVar literal);
+
+ private:
+  friend class CpModelBuilder;
+
+  using Constraint::Constraint;
+};
+
 /**
  * Specialized assignment constraint.
  *
@@ -651,9 +675,25 @@ class CpModelBuilder {
    *
    * It returns a circuit constraint that allows adding arcs incrementally after
    * construction.
+   *
    */
   CircuitConstraint AddCircuitConstraint();
 
+  /**
+   * Adds a multiple circuit constraint, aka the "VRP" (Vehicle Routing Problem)
+   * constraint.
+   *
+   * The direct graph where arc #i (from tails[i] to head[i]) is present iff
+   * literals[i] is true must satisfy this set of properties:
+   * - #incoming arcs == 1 except for node 0.
+   * - #outgoing arcs == 1 except for node 0.
+   * - for node zero, #incoming arcs == #outgoing arcs.
+   * - There are no duplicate arcs.
+   * - Self-arcs are allowed except for node 0.
+   * - There is no cycle in this graph, except through node 0.
+   */
+  MultipleCircuitConstraint AddMultipleCircuitConstraint();
+
   /**
    * Adds an allowed assignments constraint.
    *
diff --git a/ortools/sat/pseudo_costs.cc b/ortools/sat/pseudo_costs.cc
index 022b7fffa82..b5d8f35e0f0 100644
--- a/ortools/sat/pseudo_costs.cc
+++ b/ortools/sat/pseudo_costs.cc
@@ -44,7 +44,7 @@ void PseudoCosts::UpdateCostForVar(IntegerVariable var, double new_cost) {
   if (var >= pseudo_costs_.size()) {
     // Create space for new variable and its negation.
     const int new_size = std::max(var, NegationOf(var)).value() + 1;
-    pseudo_costs_.resize(new_size, IncrementalAverage(initial_cost_));
+    pseudo_costs_.resize(new_size, IncrementalAverage(0.0));
   }
   CHECK_LT(var, pseudo_costs_.size());
   pseudo_costs_[var].AddData(new_cost);
@@ -71,8 +71,7 @@ void PseudoCosts::UpdateCost(
       // Note that this initial value will be overwritten the first time
       // UpdateCostForVar() is called.
       if (!pseudo_costs_initialized_) {
-        initial_cost_ = current_pseudo_cost / 2.0;
-        InitializeCosts(initial_cost_);
+        InitializeCosts(0.0);
       }
       UpdateCostForVar(decision.var, current_pseudo_cost);
     }
@@ -101,8 +100,8 @@ IntegerVariable PseudoCosts::GetBestDecisionVar() {
 
     // TODO(user): Experiment with different ways to merge the costs.
     const double current_merged_cost =
-        std::min(GetCost(positive_var), epsilon) *
-        std::min(GetCost(negative_var), epsilon);
+        std::max(GetCost(positive_var), epsilon) *
+        std::max(GetCost(negative_var), epsilon);
 
     if (current_merged_cost > best_cost) {
       chosen_var = positive_var;
diff --git a/ortools/sat/pseudo_costs.h b/ortools/sat/pseudo_costs.h
index 868b011b61b..5170b694649 100644
--- a/ortools/sat/pseudo_costs.h
+++ b/ortools/sat/pseudo_costs.h
@@ -69,8 +69,6 @@ class PseudoCosts {
 
   bool pseudo_costs_initialized_ = false;
 
-  double initial_cost_ = 0.0;
-
   gtl::ITIVector<IntegerVariable, IncrementalAverage> pseudo_costs_;
 };