[MLIR][Presburger][Simplex] moveRowUnknownToColumn: support the row sample value being zero

When the sample value is zero, everything is the same except that failure to
pivot does not imply emptiness. So, leave it to the user to mark as empty if
necessary, if they know the sample value is strictly negative. This is needed
for an upcoming symbolic lexmin heuristic.

Reviewed By: Groverkss

Differential Revision: https://reviews.llvm.org/D123604

Author: Superty

mlir/include/mlir/Analysis/Presburger/Simplex.h

@@ -440,10 +440,9 @@ class LexSimplexBase : public SimplexBase {
   void appendSymbol();
 
   /// Try to move the specified row to column orientation while preserving the
-  /// lexicopositivity of the basis transform. The row must have a negative
-  /// sample value. If this is not possible, return failure. This only occurs
-  /// when the constraints have no solution; the tableau will be marked empty in
-  /// such a case.
+  /// lexicopositivity of the basis transform. The row must have a non-positive
+  /// sample value. If this is not possible, return failure. This occurs when
+  /// the constraints have no solution or the sample value is zero.
   LogicalResult moveRowUnknownToColumn(unsigned row);
 
   /// Given a row that has a non-integer sample value, add an inequality to cut

mlir/lib/Analysis/Presburger/Simplex.cpp

@@ -212,8 +212,10 @@ Direction flippedDirection(Direction direction) {
 /// add these to the set of ignored columns and continue to the next row. If we
 /// run out of rows, then A*y is zero and we are done.
 MaybeOptimum<SmallVector<Fraction, 8>> LexSimplex::findRationalLexMin() {
-  if (restoreRationalConsistency().failed())
+  if (restoreRationalConsistency().failed()) {
+    markEmpty();
     return OptimumKind::Empty;
+  }
   return getRationalSample();
 }
 
@@ -679,16 +681,16 @@ LogicalResult LexSimplex::restoreRationalConsistency() {
 }
 
 // Move the row unknown to column orientation while preserving lexicopositivity
-// of the basis transform. The sample value of the row must be negative.
+// of the basis transform. The sample value of the row must be non-positive.
 //
 // We only consider pivots where the pivot element is positive. Suppose no such
 // pivot exists, i.e., some violated row has no positive coefficient for any
 // basis unknown. The row can be represented as (s + c_1*u_1 + ... + c_n*u_n)/d,
 // where d is the denominator, s is the sample value and the c_i are the basis
-// coefficients. Since any feasible assignment of the basis satisfies u_i >= 0
-// for all i, and we have s < 0 as well as c_i < 0 for all i, any feasible
-// assignment would violate this row and therefore the constraints have no
-// solution.
+// coefficients. If s != 0, then since any feasible assignment of the basis
+// satisfies u_i >= 0 for all i, and we have s < 0 as well as c_i < 0 for all i,
+// any feasible assignment would violate this row and therefore the constraints
+// have no solution.
 //
 // We can preserve lexicopositivity by picking the pivot column with positive
 // pivot element that makes the lexicographically smallest change to the sample
@@ -726,10 +728,10 @@ LogicalResult LexSimplex::restoreRationalConsistency() {
 // B'.col(k) = B.col(k) - B(i,k) * B.col(j) / B(i,j) for k != j
 // and similarly, s' = s - s_i * B.col(j) / B(i,j).
 //
-// Since the row is violated, we have s_i < 0, so the change in sample value
-// when pivoting with column a is lexicographically smaller than that when
-// pivoting with column b iff B.col(a) / B(i, a) is lexicographically smaller
-// than B.col(b) / B(i, b).
+// If s_i == 0, then the sample value remains unchanged. Otherwise, if s_i < 0,
+// the change in sample value when pivoting with column a is lexicographically
+// smaller than that when pivoting with column b iff B.col(a) / B(i, a) is
+// lexicographically smaller than B.col(b) / B(i, b).
 //
 // Since B(i, j) > 0, column j remains lexicopositive.
 //
@@ -749,10 +751,8 @@ LogicalResult LexSimplexBase::moveRowUnknownToColumn(unsigned row) {
         !maybeColumn ? col : getLexMinPivotColumn(row, *maybeColumn, col);
   }
 
-  if (!maybeColumn) {
-    markEmpty();
+  if (!maybeColumn)
     return failure();
-  }
 
   pivot(row, *maybeColumn);
   return success();