diff --git a/ortools/glop/revised_simplex.h b/ortools/glop/revised_simplex.h
index 277fb854d0..095f9e1244 100644
--- a/ortools/glop/revised_simplex.h
+++ b/ortools/glop/revised_simplex.h
@@ -50,9 +50,6 @@
 //  s1 is an m1-vector (m1 being the height of A1),
 //  s2 is an m2-vector (m2 being the height of A2).
 //
-// See Design Document at
-// or search for "Google Linear Programming" in Docs
-//
 // The following are very good references for terminology, data structures,
 // and algorithms. They all contain a wealth of references.
 //
diff --git a/ortools/lp_data/lp_data.cc b/ortools/lp_data/lp_data.cc
index e9e1a7ab2a..04194b3dbf 100644
--- a/ortools/lp_data/lp_data.cc
+++ b/ortools/lp_data/lp_data.cc
@@ -641,7 +641,7 @@ void LinearProgram::AddSlackVariablesWhereNecessary(
   // Clean up the matrix. We're going to add entries, but we'll only be adding
   // them to new columns, and only one entry per column, which does not
   // invalidate the "cleanness" of the matrix.
-  matrix_.CleanUp();
+  CleanUp();
 
   // Detect which constraints produce an integer slack variable. A constraint
   // has an integer slack variable, if it contains only integer variables with
@@ -682,6 +682,8 @@ void LinearProgram::AddSlackVariablesWhereNecessary(
     SetCoefficient(row, slack_col, 1.0);
     SetConstraintBounds(row, 0.0, 0.0);
   }
+
+  columns_are_known_to_be_clean_ = true;
   transpose_matrix_is_consistent_ = false;
   if (first_slack_variable_ == kInvalidCol) {
     first_slack_variable_ = original_num_variables;
diff --git a/ortools/lp_data/lp_data.h b/ortools/lp_data/lp_data.h
index 9eb310c032..eddd7e00de 100644
--- a/ortools/lp_data/lp_data.h
+++ b/ortools/lp_data/lp_data.h
@@ -521,6 +521,16 @@ class LinearProgram {
   // integer bounds.
   bool BoundsOfIntegerConstraintsAreInteger(Fractional tolerance) const;
 
+  // Advanced usage. Bypass the costly call to CleanUp() when we known that the
+  // change we made kept the matrix columns "clean" (see the comment of
+  // CleanUp()). This is unsafe but can save a big chunk of the running time
+  // when one does a small amount of incremental changes to the problem (like
+  // adding a new row with no duplicates or zero entries).
+  void NotifyThatColumnsAreClean() {
+    DCHECK(matrix_.IsCleanedUp());
+    columns_are_known_to_be_clean_ = true;
+  }
+
  private:
   // A helper function that updates the vectors integer_variables_list_,
   // binary_variables_list_, and non_binary_variables_list_.
diff --git a/ortools/sat/linear_programming_constraint.cc b/ortools/sat/linear_programming_constraint.cc
index 3106784399..13531432e8 100644
--- a/ortools/sat/linear_programming_constraint.cc
+++ b/ortools/sat/linear_programming_constraint.cc
@@ -239,6 +239,7 @@ bool LinearProgrammingConstraint::Propagate() {
     if (num_new_cuts > 0) {
       num_cuts_ += num_new_cuts;
       VLOG(1) << "#cuts " << num_cuts_;
+      lp_data_.NotifyThatColumnsAreClean();
       lp_data_.AddSlackVariablesWhereNecessary(false);
       const auto status = simplex_.Solve(lp_data_, time_limit_);
       CHECK(status.ok()) << "LinearProgrammingConstraint encountered an error: "