diff --git a/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h b/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
index ad9e75cf178..d5497788a02 100644
--- a/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
+++ b/Polyline_simplification_2/include/CGAL/Polyline_simplification_2/simplify.h
@@ -335,7 +335,7 @@ operator()()
       } else {
         (*u)->set_cost(*dist);
         if(mpq->contains(*u)){
-          mpq->update(*u, true);
+          mpq->update(*u);
         }
         else{
           mpq->push(*u);
@@ -353,7 +353,7 @@ operator()()
       } else {
         (*w)->set_cost(*dist);
         if(mpq->contains(*w)){
-          mpq->update(*w, true);
+          mpq->update(*w);
         }
         else{
           mpq->push(*w);
diff --git a/STL_Extension/include/CGAL/Modifiable_priority_queue.h b/STL_Extension/include/CGAL/Modifiable_priority_queue.h
index 4e765306d50..6ee9c4199a6 100644
--- a/STL_Extension/include/CGAL/Modifiable_priority_queue.h
+++ b/STL_Extension/include/CGAL/Modifiable_priority_queue.h
@@ -78,22 +78,17 @@ public:
   typedef typename Heap::value_type value_type;
   typedef typename Heap::size_type  size_type;
 
-  typedef bool handle ;
-
 public:
 
   Modifiable_priority_queue( size_type largest_ID, Compare const& c = Compare(), ID const& id = ID() ) : mHeap(largest_ID,c,id) {}
 
-  handle push ( value_type const& v ) { mHeap.push(v) ; return handle(true) ; }
-
-  handle update ( value_type const& v, handle h ) { mHeap.update(v); return h ; }
+  void push ( value_type const& v ) { mHeap.push(v) ; }
 
   void update ( value_type const& v ) { mHeap.update(v); }
 
-  handle erase ( value_type const& v, handle  ) { mHeap.remove(v); return null_handle() ; }
-  handle erase ( value_type const& v  ) { mHeap.remove(v); return null_handle() ; }
+  void erase ( value_type const& v  ) { mHeap.remove(v); }
 
-  value_type top() const { return mHeap.top() ; }
+  value_type top() const { return mHeap.top(); }
 
   void pop() { mHeap.pop(); }
 
@@ -121,8 +116,6 @@ public:
     return v;
   }
 
-  static handle null_handle() { return handle(false); }
-
 private:
 
   Heap mHeap ;
diff --git a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/internal/Edge_collapse.h b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/internal/Edge_collapse.h
index e327e86be6e..8db94cb744c 100644
--- a/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/internal/Edge_collapse.h
+++ b/Surface_mesh_simplification/include/CGAL/Surface_mesh_simplification/internal/Edge_collapse.h
@@ -174,27 +174,22 @@ public:
   };
 
   typedef Modifiable_priority_queue<halfedge_descriptor, Compare_cost, edge_id>     PQ;
-  typedef typename PQ::handle                                                       PQ_handle;
 
   // An Edge_data is associated with EVERY _ edge in the mesh (collapsable or not).
-  // It relates the edge with the PQ-handle needed to update the priority queue
+  // It contains the edge status wrt the priority queue
   // It also relates the edge with a policy-based cache
-  class Edge_data
+  struct Edge_data
   {
-  public :
-    Edge_data() : m_PQ_h() {}
-
     const Cost_type& cost() const { return m_cost; }
     Cost_type& cost() { return m_cost; }
 
-    PQ_handle queue_handle() const { return m_PQ_h;}
-    bool is_in_PQ() const { return m_PQ_h != PQ::null_handle(); }
-    void set_PQ_handle(PQ_handle h) { m_PQ_h = h; }
-    void reset_queue_handle() { m_PQ_h = PQ::null_handle(); }
+    bool is_in_PQ() const { return m_is_in_PQ; }
+    void set_is_in_PQ() { m_is_in_PQ=true; }
+    void reset_in_queue_status() { m_is_in_PQ = false; }
 
   private:
     Cost_type m_cost;
-    PQ_handle m_PQ_h;
+    bool m_is_in_PQ = false;
   };
 
   typedef Edge_data*                                                                Edge_data_ptr;
@@ -301,7 +296,8 @@ private:
     CGAL_expensive_assertion(!data.is_in_PQ());
     CGAL_expensive_assertion(!mPQ->contains(h));
 
-    data.set_PQ_handle(mPQ->push(h));
+    mPQ->push(h);
+    data.set_is_in_PQ();
 
     CGAL_expensive_assertion(data.is_in_PQ());
     CGAL_expensive_assertion(mPQ->contains(h));
@@ -313,7 +309,8 @@ private:
     CGAL_expensive_assertion(data.is_in_PQ());
     CGAL_expensive_assertion(mPQ->contains(h));
 
-    data.set_PQ_handle(mPQ->update(h, data.queue_handle()));
+    mPQ->update(h);
+    data.set_is_in_PQ();
 
     CGAL_assertion(data.is_in_PQ());
     CGAL_expensive_assertion(mPQ->contains(h));
@@ -325,7 +322,8 @@ private:
     CGAL_expensive_assertion(data.is_in_PQ());
     CGAL_expensive_assertion(mPQ->contains(h));
 
-    data.set_PQ_handle(mPQ->erase(h, data.queue_handle()));
+    mPQ->erase(h);
+    data.reset_in_queue_status();
 
     CGAL_expensive_assertion(!data.is_in_PQ());
     CGAL_expensive_assertion(!mPQ->contains(h));
@@ -339,7 +337,7 @@ private:
       CGAL_assertion(is_primary_edge(*opt_h));
       CGAL_expensive_assertion(get_data(*opt_h).is_in_PQ());
 
-      get_data(*opt_h).reset_queue_handle();
+      get_data(*opt_h).reset_in_queue_status();
 
       CGAL_expensive_assertion(!get_data(*opt_h).is_in_PQ());
       CGAL_expensive_assertion(!mPQ->contains(*opt_h));