1	//===- Simplex.cpp - MLIR Simplex Class -----------------------------------===//
2	//
3	// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4	// See https://llvm.org/LICENSE.txt for license information.
5	// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6	//
7	//===----------------------------------------------------------------------===//
8
9	#include "mlir/Analysis/Presburger/Simplex.h"
10	#include "mlir/Analysis/Presburger/Fraction.h"
11	#include "mlir/Analysis/Presburger/IntegerRelation.h"
12	#include "mlir/Analysis/Presburger/MPInt.h"
13	#include "mlir/Analysis/Presburger/Matrix.h"
14	#include "mlir/Analysis/Presburger/PresburgerSpace.h"
15	#include "mlir/Analysis/Presburger/Utils.h"
16	#include "mlir/Support/LLVM.h"
17	#include "mlir/Support/LogicalResult.h"
18	#include "llvm/ADT/STLExtras.h"
19	#include "llvm/ADT/SmallBitVector.h"
20	#include "llvm/ADT/SmallVector.h"
21	#include "llvm/Support/Compiler.h"
22	#include "llvm/Support/ErrorHandling.h"
23	#include "llvm/Support/raw_ostream.h"
24	#include <cassert>
25	#include <functional>
26	#include <limits>
27	#include <optional>
28	#include <tuple>
29	#include <utility>
30
31	using namespace mlir;
32	using namespace presburger;
33
34	using Direction = Simplex::Direction;
35
36	const int nullIndex = std::numeric_limits<int>::max();
37
38	// Return a + scale*b;
39	LLVM_ATTRIBUTE_UNUSED
40	static SmallVector<MPInt, 8>
41	scaleAndAddForAssert(ArrayRef<MPInt> a, const MPInt &scale, ArrayRef<MPInt> b) {
42	assert(a.size() == b.size());
43	SmallVector<MPInt, 8> res;
44	res.reserve(N: a.size());
45	for (unsigned i = 0, e = a.size(); i < e; ++i)
46	res.push_back(Elt: a[i] + scale * b[i]);
47	return res;
48	}
49
50	SimplexBase::SimplexBase(unsigned nVar, bool mustUseBigM)
51	: usingBigM(mustUseBigM), nRedundant(0), nSymbol(0),
52	tableau(0, getNumFixedCols() + nVar), empty(false) {
53	colUnknown.insert(I: colUnknown.begin(), NumToInsert: getNumFixedCols(), Elt: nullIndex);
54	for (unsigned i = 0; i < nVar; ++i) {
55	var.emplace_back(Args: Orientation::Column, /*restricted=*/Args: false,
56	/*pos=*/Args: getNumFixedCols() + i);
57	colUnknown.push_back(Elt: i);
58	}
59	}
60
61	SimplexBase::SimplexBase(unsigned nVar, bool mustUseBigM,
62	const llvm::SmallBitVector &isSymbol)
63	: SimplexBase(nVar, mustUseBigM) {
64	assert(isSymbol.size() == nVar && "invalid bitmask!");
65	// Invariant: nSymbol is the number of symbols that have been marked
66	// already and these occupy the columns
67	// [getNumFixedCols(), getNumFixedCols() + nSymbol).
68	for (unsigned symbolIdx : isSymbol.set_bits()) {
69	var[symbolIdx].isSymbol = true;
70	swapColumns(i: var[symbolIdx].pos, j: getNumFixedCols() + nSymbol);
71	++nSymbol;
72	}
73	}
74
75	const Simplex::Unknown &SimplexBase::unknownFromIndex(int index) const {
76	assert(index != nullIndex && "nullIndex passed to unknownFromIndex");
77	return index >= 0 ? var[index] : con[~index];
78	}
79
80	const Simplex::Unknown &SimplexBase::unknownFromColumn(unsigned col) const {
81	assert(col < getNumColumns() && "Invalid column");
82	return unknownFromIndex(index: colUnknown[col]);
83	}
84
85	const Simplex::Unknown &SimplexBase::unknownFromRow(unsigned row) const {
86	assert(row < getNumRows() && "Invalid row");
87	return unknownFromIndex(index: rowUnknown[row]);
88	}
89
90	Simplex::Unknown &SimplexBase::unknownFromIndex(int index) {
91	assert(index != nullIndex && "nullIndex passed to unknownFromIndex");
92	return index >= 0 ? var[index] : con[~index];
93	}
94
95	Simplex::Unknown &SimplexBase::unknownFromColumn(unsigned col) {
96	assert(col < getNumColumns() && "Invalid column");
97	return unknownFromIndex(index: colUnknown[col]);
98	}
99
100	Simplex::Unknown &SimplexBase::unknownFromRow(unsigned row) {
101	assert(row < getNumRows() && "Invalid row");
102	return unknownFromIndex(index: rowUnknown[row]);
103	}
104
105	unsigned SimplexBase::addZeroRow(bool makeRestricted) {
106	// Resize the tableau to accommodate the extra row.
107	unsigned newRow = tableau.appendExtraRow();
108	assert(getNumRows() == getNumRows() && "Inconsistent tableau size");
109	rowUnknown.push_back(Elt: ~con.size());
110	con.emplace_back(Args: Orientation::Row, Args&: makeRestricted, Args&: newRow);
111	undoLog.push_back(Elt: UndoLogEntry::RemoveLastConstraint);
112	tableau(newRow, 0) = 1;
113	return newRow;
114	}
115
116	/// Add a new row to the tableau corresponding to the given constant term and
117	/// list of coefficients. The coefficients are specified as a vector of
118	/// (variable index, coefficient) pairs.
119	unsigned SimplexBase::addRow(ArrayRef<MPInt> coeffs, bool makeRestricted) {
120	assert(coeffs.size() == var.size() + 1 &&
121	"Incorrect number of coefficients!");
122	assert(var.size() + getNumFixedCols() == getNumColumns() &&
123	"inconsistent column count!");
124
125	unsigned newRow = addZeroRow(makeRestricted);
126	tableau(newRow, 1) = coeffs.back();
127	if (usingBigM) {
128	// When the lexicographic pivot rule is used, instead of the variables
129	//
130	// x, y, z ...
131	//
132	// we internally use the variables
133	//
134	// M, M + x, M + y, M + z, ...
135	//
136	// where M is the big M parameter. As such, when the user tries to add
137	// a row ax + by + cz + d, we express it in terms of our internal variables
138	// as -(a + b + c)M + a(M + x) + b(M + y) + c(M + z) + d.
139	//
140	// Symbols don't use the big M parameter since they do not get lex
141	// optimized.
142	MPInt bigMCoeff(0);
143	for (unsigned i = 0; i < coeffs.size() - 1; ++i)
144	if (!var[i].isSymbol)
145	bigMCoeff -= coeffs[i];
146	// The coefficient to the big M parameter is stored in column 2.
147	tableau(newRow, 2) = bigMCoeff;
148	}
149
150	// Process each given variable coefficient.
151	for (unsigned i = 0; i < var.size(); ++i) {
152	unsigned pos = var[i].pos;
153	if (coeffs[i] == 0)
154	continue;
155
156	if (var[i].orientation == Orientation::Column) {
157	// If a variable is in column position at column col, then we just add the
158	// coefficient for that variable (scaled by the common row denominator) to
159	// the corresponding entry in the new row.
160	tableau(newRow, pos) += coeffs[i] * tableau(newRow, 0);
161	continue;
162	}
163
164	// If the variable is in row position, we need to add that row to the new
165	// row, scaled by the coefficient for the variable, accounting for the two
166	// rows potentially having different denominators. The new denominator is
167	// the lcm of the two.
168	MPInt lcm = presburger::lcm(a: tableau(newRow, 0), b: tableau(pos, 0));
169	MPInt nRowCoeff = lcm / tableau(newRow, 0);
170	MPInt idxRowCoeff = coeffs[i] * (lcm / tableau(pos, 0));
171	tableau(newRow, 0) = lcm;
172	for (unsigned col = 1, e = getNumColumns(); col < e; ++col)
173	tableau(newRow, col) =
174	nRowCoeff * tableau(newRow, col) + idxRowCoeff * tableau(pos, col);
175	}
176
177	tableau.normalizeRow(row: newRow);
178	// Push to undo log along with the index of the new constraint.
179	return con.size() - 1;
180	}
181
182	namespace {
183	bool signMatchesDirection(const MPInt &elem, Direction direction) {
184	assert(elem != 0 && "elem should not be 0");
185	return direction == Direction::Up ? elem > 0 : elem < 0;
186	}
187
188	Direction flippedDirection(Direction direction) {
189	return direction == Direction::Up ? Direction::Down : Simplex::Direction::Up;
190	}
191	} // namespace
192
193	/// We simply make the tableau consistent while maintaining a lexicopositive
194	/// basis transform, and then return the sample value. If the tableau becomes
195	/// empty, we return empty.
196	///
197	/// Let the variables be x = (x_1, ... x_n).
198	/// Let the basis unknowns be y = (y_1, ... y_n).
199	/// We have that x = A*y + b for some n x n matrix A and n x 1 column vector b.
200	///
201	/// As we will show below, A*y is either zero or lexicopositive.
202	/// Adding a lexicopositive vector to b will make it lexicographically
203	/// greater, so A*y + b is always equal to or lexicographically greater than b.
204	/// Thus, since we can attain x = b, that is the lexicographic minimum.
205	///
206	/// We have that every column in A is lexicopositive, i.e., has at least
207	/// one non-zero element, with the first such element being positive. Since for
208	/// the tableau to be consistent we must have non-negative sample values not
209	/// only for the constraints but also for the variables, we also have x >= 0 and
210	/// y >= 0, by which we mean every element in these vectors is non-negative.
211	///
212	/// Proof that if every column in A is lexicopositive, and y >= 0, then
213	/// A*y is zero or lexicopositive. Begin by considering A_1, the first row of A.
214	/// If this row is all zeros, then (A*y)_1 = (A_1)*y = 0; proceed to the next
215	/// row. If we run out of rows, A*y is zero and we are done; otherwise, we
216	/// encounter some row A_i that has a non-zero element. Every column is
217	/// lexicopositive and so has some positive element before any negative elements
218	/// occur, so the element in this row for any column, if non-zero, must be
219	/// positive. Consider (A*y)_i = (A_i)*y. All the elements in both vectors are
220	/// non-negative, so if this is non-zero then it must be positive. Then the
221	/// first non-zero element of A*y is positive so A*y is lexicopositive.
222	///
223	/// Otherwise, if (A_i)*y is zero, then for every column j that had a non-zero
224	/// element in A_i, y_j is zero. Thus these columns have no contribution to A*y
225	/// and we can completely ignore these columns of A. We now continue downwards,
226	/// looking for rows of A that have a non-zero element other than in the ignored
227	/// columns. If we find one, say A_k, once again these elements must be positive
228	/// since they are the first non-zero element in each of these columns, so if
229	/// (A_k)*y is not zero then we have that A*y is lexicopositive and if not we
230	/// add these to the set of ignored columns and continue to the next row. If we
231	/// run out of rows, then A*y is zero and we are done.
232	MaybeOptimum<SmallVector<Fraction, 8>> LexSimplex::findRationalLexMin() {
233	if (restoreRationalConsistency().failed()) {
234	markEmpty();
235	return OptimumKind::Empty;
236	}
237	return getRationalSample();
238	}
239
240	/// Given a row that has a non-integer sample value, add an inequality such
241	/// that this fractional sample value is cut away from the polytope. The added
242	/// inequality will be such that no integer points are removed. i.e., the
243	/// integer lexmin, if it exists, is the same with and without this constraint.
244	///
245	/// Let the row be
246	/// (c + coeffM*M + a_1*s_1 + ... + a_m*s_m + b_1*y_1 + ... + b_n*y_n)/d,
247	/// where s_1, ... s_m are the symbols and
248	/// y_1, ... y_n are the other basis unknowns.
249	///
250	/// For this to be an integer, we want
251	/// coeffM*M + a_1*s_1 + ... + a_m*s_m + b_1*y_1 + ... + b_n*y_n = -c (mod d)
252	/// Note that this constraint must always hold, independent of the basis,
253	/// becuse the row unknown's value always equals this expression, even if *we*
254	/// later compute the sample value from a different expression based on a
255	/// different basis.
256	///
257	/// Let us assume that M has a factor of d in it. Imposing this constraint on M
258	/// does not in any way hinder us from finding a value of M that is big enough.
259	/// Moreover, this function is only called when the symbolic part of the sample,
260	/// a_1*s_1 + ... + a_m*s_m, is known to be an integer.
261	///
262	/// Also, we can safely reduce the coefficients modulo d, so we have:
263	///
264	/// (b_1%d)y_1 + ... + (b_n%d)y_n = (-c%d) + k*d for some integer `k`
265	///
266	/// Note that all coefficient modulos here are non-negative. Also, all the
267	/// unknowns are non-negative here as both constraints and variables are
268	/// non-negative in LexSimplexBase. (We used the big M trick to make the
269	/// variables non-negative). Therefore, the LHS here is non-negative.
270	/// Since 0 <= (-c%d) < d, k is the quotient of dividing the LHS by d and
271	/// is therefore non-negative as well.
272	///
273	/// So we have
274	/// ((b_1%d)y_1 + ... + (b_n%d)y_n - (-c%d))/d >= 0.
275	///
276	/// The constraint is violated when added (it would be useless otherwise)
277	/// so we immediately try to move it to a column.
278	LogicalResult LexSimplexBase::addCut(unsigned row) {
279	MPInt d = tableau(row, 0);
280	unsigned cutRow = addZeroRow(/*makeRestricted=*/true);
281	tableau(cutRow, 0) = d;
282	tableau(cutRow, 1) = -mod(lhs: -tableau(row, 1), rhs: d); // -c%d.
283	tableau(cutRow, 2) = 0;
284	for (unsigned col = 3 + nSymbol, e = getNumColumns(); col < e; ++col)
285	tableau(cutRow, col) = mod(lhs: tableau(row, col), rhs: d); // b_i%d.
286	return moveRowUnknownToColumn(row: cutRow);
287	}
288
289	std::optional<unsigned> LexSimplex::maybeGetNonIntegralVarRow() const {
290	for (const Unknown &u : var) {
291	if (u.orientation == Orientation::Column)
292	continue;
293	// If the sample value is of the form (a/d)M + b/d, we need b to be
294	// divisible by d. We assume M contains all possible
295	// factors and is divisible by everything.
296	unsigned row = u.pos;
297	if (tableau(row, 1) % tableau(row, 0) != 0)
298	return row;
299	}
300	return {};
301	}
302
303	MaybeOptimum<SmallVector<MPInt, 8>> LexSimplex::findIntegerLexMin() {
304	// We first try to make the tableau consistent.
305	if (restoreRationalConsistency().failed())
306	return OptimumKind::Empty;
307
308	// Then, if the sample value is integral, we are done.
309	while (std::optional<unsigned> maybeRow = maybeGetNonIntegralVarRow()) {
310	// Otherwise, for the variable whose row has a non-integral sample value,
311	// we add a cut, a constraint that remove this rational point
312	// while preserving all integer points, thus keeping the lexmin the same.
313	// We then again try to make the tableau with the new constraint
314	// consistent. This continues until the tableau becomes empty, in which
315	// case there is no integer point, or until there are no variables with
316	// non-integral sample values.
317	//
318	// Failure indicates that the tableau became empty, which occurs when the
319	// polytope is integer empty.
320	if (addCut(row: *maybeRow).failed())
321	return OptimumKind::Empty;
322	if (restoreRationalConsistency().failed())
323	return OptimumKind::Empty;
324	}
325
326	MaybeOptimum<SmallVector<Fraction, 8>> sample = getRationalSample();
327	assert(!sample.isEmpty() && "If we reached here the sample should exist!");
328	if (sample.isUnbounded())
329	return OptimumKind::Unbounded;
330	return llvm::to_vector<8>(
331	Range: llvm::map_range(C&: *sample, F: std::mem_fn(pm: &Fraction::getAsInteger)));
332	}
333
334	bool LexSimplex::isSeparateInequality(ArrayRef<MPInt> coeffs) {
335	SimplexRollbackScopeExit scopeExit(*this);
336	addInequality(coeffs);
337	return findIntegerLexMin().isEmpty();
338	}
339
340	bool LexSimplex::isRedundantInequality(ArrayRef<MPInt> coeffs) {
341	return isSeparateInequality(coeffs: getComplementIneq(ineq: coeffs));
342	}
343
344	SmallVector<MPInt, 8>
345	SymbolicLexSimplex::getSymbolicSampleNumerator(unsigned row) const {
346	SmallVector<MPInt, 8> sample;
347	sample.reserve(N: nSymbol + 1);
348	for (unsigned col = 3; col < 3 + nSymbol; ++col)
349	sample.push_back(Elt: tableau(row, col));
350	sample.push_back(Elt: tableau(row, 1));
351	return sample;
352	}
353
354	SmallVector<MPInt, 8>
355	SymbolicLexSimplex::getSymbolicSampleIneq(unsigned row) const {
356	SmallVector<MPInt, 8> sample = getSymbolicSampleNumerator(row);
357	// The inequality is equivalent to the GCD-normalized one.
358	normalizeRange(range: sample);
359	return sample;
360	}
361
362	void LexSimplexBase::appendSymbol() {
363	appendVariable();
364	swapColumns(i: 3 + nSymbol, j: getNumColumns() - 1);
365	var.back().isSymbol = true;
366	nSymbol++;
367	}
368
369	static bool isRangeDivisibleBy(ArrayRef<MPInt> range, const MPInt &divisor) {
370	assert(divisor > 0 && "divisor must be positive!");
371	return llvm::all_of(Range&: range,
372	P: [divisor](const MPInt &x) { return x % divisor == 0; });
373	}
374
375	bool SymbolicLexSimplex::isSymbolicSampleIntegral(unsigned row) const {
376	MPInt denom = tableau(row, 0);
377	return tableau(row, 1) % denom == 0 &&
378	isRangeDivisibleBy(range: tableau.getRow(row).slice(N: 3, M: nSymbol), divisor: denom);
379	}
380
381	/// This proceeds similarly to LexSimplexBase::addCut(). We are given a row that
382	/// has a symbolic sample value with fractional coefficients.
383	///
384	/// Let the row be
385	/// (c + coeffM*M + sum_i a_i*s_i + sum_j b_j*y_j)/d,
386	/// where s_1, ... s_m are the symbols and
387	/// y_1, ... y_n are the other basis unknowns.
388	///
389	/// As in LexSimplex::addCut, for this to be an integer, we want
390	///
391	/// coeffM*M + sum_j b_j*y_j = -c + sum_i (-a_i*s_i) (mod d)
392	///
393	/// This time, a_1*s_1 + ... + a_m*s_m may not be an integer. We find that
394	///
395	/// sum_i (b_i%d)y_i = ((-c%d) + sum_i (-a_i%d)s_i)%d + k*d for some integer k
396	///
397	/// where we take a modulo of the whole symbolic expression on the right to
398	/// bring it into the range [0, d - 1]. Therefore, as in addCut(),
399	/// k is the quotient on dividing the LHS by d, and since LHS >= 0, we have
400	/// k >= 0 as well. If all the a_i are divisible by d, then we can add the
401	/// constraint directly. Otherwise, we realize the modulo of the symbolic
402	/// expression by adding a division variable
403	///
404	/// q = ((-c%d) + sum_i (-a_i%d)s_i)/d
405	///
406	/// to the symbol domain, so the equality becomes
407	///
408	/// sum_i (b_i%d)y_i = (-c%d) + sum_i (-a_i%d)s_i - q*d + k*d for some integer k
409	///
410	/// So the cut is
411	/// (sum_i (b_i%d)y_i - (-c%d) - sum_i (-a_i%d)s_i + q*d)/d >= 0
412	/// This constraint is violated when added so we immediately try to move it to a
413	/// column.
414	LogicalResult SymbolicLexSimplex::addSymbolicCut(unsigned row) {
415	MPInt d = tableau(row, 0);
416	if (isRangeDivisibleBy(range: tableau.getRow(row).slice(N: 3, M: nSymbol), divisor: d)) {
417	// The coefficients of symbols in the symbol numerator are divisible
418	// by the denominator, so we can add the constraint directly,
419	// i.e., ignore the symbols and add a regular cut as in addCut().
420	return addCut(row);
421	}
422
423	// Construct the division variable `q = ((-c%d) + sum_i (-a_i%d)s_i)/d`.
424	SmallVector<MPInt, 8> divCoeffs;
425	divCoeffs.reserve(N: nSymbol + 1);
426	MPInt divDenom = d;
427	for (unsigned col = 3; col < 3 + nSymbol; ++col)
428	divCoeffs.push_back(Elt: mod(lhs: -tableau(row, col), rhs: divDenom)); // (-a_i%d)s_i
429	divCoeffs.push_back(Elt: mod(lhs: -tableau(row, 1), rhs: divDenom)); // -c%d.
430	normalizeDiv(num: divCoeffs, denom&: divDenom);
431
432	domainSimplex.addDivisionVariable(coeffs: divCoeffs, denom: divDenom);
433	domainPoly.addLocalFloorDiv(dividend: divCoeffs, divisor: divDenom);
434
435	// Update `this` to account for the additional symbol we just added.
436	appendSymbol();
437
438	// Add the cut (sum_i (b_i%d)y_i - (-c%d) + sum_i -(-a_i%d)s_i + q*d)/d >= 0.
439	unsigned cutRow = addZeroRow(/*makeRestricted=*/true);
440	tableau(cutRow, 0) = d;
441	tableau(cutRow, 2) = 0;
442
443	tableau(cutRow, 1) = -mod(lhs: -tableau(row, 1), rhs: d); // -(-c%d).
444	for (unsigned col = 3; col < 3 + nSymbol - 1; ++col)
445	tableau(cutRow, col) = -mod(lhs: -tableau(row, col), rhs: d); // -(-a_i%d)s_i.
446	tableau(cutRow, 3 + nSymbol - 1) = d; // q*d.
447
448	for (unsigned col = 3 + nSymbol, e = getNumColumns(); col < e; ++col)
449	tableau(cutRow, col) = mod(lhs: tableau(row, col), rhs: d); // (b_i%d)y_i.
450	return moveRowUnknownToColumn(row: cutRow);
451	}
452
453	void SymbolicLexSimplex::recordOutput(SymbolicLexOpt &result) const {
454	IntMatrix output(0, domainPoly.getNumVars() + 1);
455	output.reserveRows(rows: result.lexopt.getNumOutputs());
456	for (const Unknown &u : var) {
457	if (u.isSymbol)
458	continue;
459
460	if (u.orientation == Orientation::Column) {
461	// M + u has a sample value of zero so u has a sample value of -M, i.e,
462	// unbounded.
463	result.unboundedDomain.unionInPlace(disjunct: domainPoly);
464	return;
465	}
466
467	MPInt denom = tableau(u.pos, 0);
468	if (tableau(u.pos, 2) < denom) {
469	// M + u has a sample value of fM + something, where f < 1, so
470	// u = (f - 1)M + something, which has a negative coefficient for M,
471	// and so is unbounded.
472	result.unboundedDomain.unionInPlace(disjunct: domainPoly);
473	return;
474	}
475	assert(tableau(u.pos, 2) == denom &&
476	"Coefficient of M should not be greater than 1!");
477
478	SmallVector<MPInt, 8> sample = getSymbolicSampleNumerator(row: u.pos);
479	for (MPInt &elem : sample) {
480	assert(elem % denom == 0 && "coefficients must be integral!");
481	elem /= denom;
482	}
483	output.appendExtraRow(elems: sample);
484	}
485
486	// Store the output in a MultiAffineFunction and add it the result.
487	PresburgerSpace funcSpace = result.lexopt.getSpace();
488	funcSpace.insertVar(kind: VarKind::Local, pos: 0, num: domainPoly.getNumLocalVars());
489
490	result.lexopt.addPiece(
491	piece: {.domain: PresburgerSet(domainPoly),
492	.output: MultiAffineFunction(funcSpace, output, domainPoly.getLocalReprs())});
493	}
494
495	std::optional<unsigned> SymbolicLexSimplex::maybeGetAlwaysViolatedRow() {
496	// First look for rows that are clearly violated just from the big M
497	// coefficient, without needing to perform any simplex queries on the domain.
498	for (unsigned row = 0, e = getNumRows(); row < e; ++row)
499	if (tableau(row, 2) < 0)
500	return row;
501
502	for (unsigned row = 0, e = getNumRows(); row < e; ++row) {
503	if (tableau(row, 2) > 0)
504	continue;
505	if (domainSimplex.isSeparateInequality(coeffs: getSymbolicSampleIneq(row))) {
506	// Sample numerator always takes negative values in the symbol domain.
507	return row;
508	}
509	}
510	return {};
511	}
512
513	std::optional<unsigned> SymbolicLexSimplex::maybeGetNonIntegralVarRow() {
514	for (const Unknown &u : var) {
515	if (u.orientation == Orientation::Column)
516	continue;
517	assert(!u.isSymbol && "Symbol should not be in row orientation!");
518	if (!isSymbolicSampleIntegral(row: u.pos))
519	return u.pos;
520	}
521	return {};
522	}
523
524	/// The non-branching pivots are just the ones moving the rows
525	/// that are always violated in the symbol domain.
526	LogicalResult SymbolicLexSimplex::doNonBranchingPivots() {
527	while (std::optional<unsigned> row = maybeGetAlwaysViolatedRow())
528	if (moveRowUnknownToColumn(row: *row).failed())
529	return failure();
530	return success();
531	}
532
533	SymbolicLexOpt SymbolicLexSimplex::computeSymbolicIntegerLexMin() {
534	SymbolicLexOpt result(PresburgerSpace::getRelationSpace(
535	/*numDomain=*/domainPoly.getNumDimVars(),
536	/*numRange=*/var.size() - nSymbol,
537	/*numSymbols=*/domainPoly.getNumSymbolVars()));
538
539	/// The algorithm is more naturally expressed recursively, but we implement
540	/// it iteratively here to avoid potential issues with stack overflows in the
541	/// compiler. We explicitly maintain the stack frames in a vector.
542	///
543	/// To "recurse", we store the current "stack frame", i.e., state variables
544	/// that we will need when we "return", into `stack`, increment `level`, and
545	/// `continue`. To "tail recurse", we just `continue`.
546	/// To "return", we decrement `level` and `continue`.
547	///
548	/// When there is no stack frame for the current `level`, this indicates that
549	/// we have just "recursed" or "tail recursed". When there does exist one,
550	/// this indicates that we have just "returned" from recursing. There is only
551	/// one point at which non-tail calls occur so we always "return" there.
552	unsigned level = 1;
553	struct StackFrame {
554	int splitIndex;
555	unsigned snapshot;
556	unsigned domainSnapshot;
557	IntegerRelation::CountsSnapshot domainPolyCounts;
558	};
559	SmallVector<StackFrame, 8> stack;
560
561	while (level > 0) {
562	assert(level >= stack.size());
563	if (level > stack.size()) {
564	if (empty || domainSimplex.findIntegerLexMin().isEmpty()) {
565	// No integer points; return.
566	--level;
567	continue;
568	}
569
570	if (doNonBranchingPivots().failed()) {
571	// Could not find pivots for violated constraints; return.
572	--level;
573	continue;
574	}
575
576	SmallVector<MPInt, 8> symbolicSample;
577	unsigned splitRow = 0;
578	for (unsigned e = getNumRows(); splitRow < e; ++splitRow) {
579	if (tableau(splitRow, 2) > 0)
580	continue;
581	assert(tableau(splitRow, 2) == 0 &&
582	"Non-branching pivots should have been handled already!");
583
584	symbolicSample = getSymbolicSampleIneq(row: splitRow);
585	if (domainSimplex.isRedundantInequality(coeffs: symbolicSample))
586	continue;
587
588	// It's neither redundant nor separate, so it takes both positive and
589	// negative values, and hence constitutes a row for which we need to
590	// split the domain and separately run each case.
591	assert(!domainSimplex.isSeparateInequality(symbolicSample) &&
592	"Non-branching pivots should have been handled already!");
593	break;
594	}
595
596	if (splitRow < getNumRows()) {
597	unsigned domainSnapshot = domainSimplex.getSnapshot();
598	IntegerRelation::CountsSnapshot domainPolyCounts =
599	domainPoly.getCounts();
600
601	// First, we consider the part of the domain where the row is not
602	// violated. We don't have to do any pivots for the row in this case,
603	// but we record the additional constraint that defines this part of
604	// the domain.
605	domainSimplex.addInequality(coeffs: symbolicSample);
606	domainPoly.addInequality(inEq: symbolicSample);
607
608	// Recurse.
609	//
610	// On return, the basis as a set is preserved but not the internal
611	// ordering within rows or columns. Thus, we take note of the index of
612	// the Unknown that caused the split, which may be in a different
613	// row when we come back from recursing. We will need this to recurse
614	// on the other part of the split domain, where the row is violated.
615	//
616	// Note that we have to capture the index above and not a reference to
617	// the Unknown itself, since the array it lives in might get
618	// reallocated.
619	int splitIndex = rowUnknown[splitRow];
620	unsigned snapshot = getSnapshot();
621	stack.push_back(
622	Elt: {.splitIndex: splitIndex, .snapshot: snapshot, .domainSnapshot: domainSnapshot, .domainPolyCounts: domainPolyCounts});
623	++level;
624	continue;
625	}
626
627	// The tableau is rationally consistent for the current domain.
628	// Now we look for non-integral sample values and add cuts for them.
629	if (std::optional<unsigned> row = maybeGetNonIntegralVarRow()) {
630	if (addSymbolicCut(row: *row).failed()) {
631	// No integral points; return.
632	--level;
633	continue;
634	}
635
636	// Rerun this level with the added cut constraint (tail recurse).
637	continue;
638	}
639
640	// Record output and return.
641	recordOutput(result);
642	--level;
643	continue;
644	}
645
646	if (level == stack.size()) {
647	// We have "returned" from "recursing".
648	const StackFrame &frame = stack.back();
649	domainPoly.truncate(counts: frame.domainPolyCounts);
650	domainSimplex.rollback(snapshot: frame.domainSnapshot);
651	rollback(snapshot: frame.snapshot);
652	const Unknown &u = unknownFromIndex(index: frame.splitIndex);
653
654	// Drop the frame. We don't need it anymore.
655	stack.pop_back();
656
657	// Now we consider the part of the domain where the unknown `splitIndex`
658	// was negative.
659	assert(u.orientation == Orientation::Row &&
660	"The split row should have been returned to row orientation!");
661	SmallVector<MPInt, 8> splitIneq =
662	getComplementIneq(ineq: getSymbolicSampleIneq(row: u.pos));
663	normalizeRange(range: splitIneq);
664	if (moveRowUnknownToColumn(row: u.pos).failed()) {
665	// The unknown can't be made non-negative; return.
666	--level;
667	continue;
668	}
669
670	// The unknown can be made negative; recurse with the corresponding domain
671	// constraints.
672	domainSimplex.addInequality(coeffs: splitIneq);
673	domainPoly.addInequality(inEq: splitIneq);
674
675	// We are now taking care of the second half of the domain and we don't
676	// need to do anything else here after returning, so it's a tail recurse.
677	continue;
678	}
679	}
680
681	return result;
682	}
683
684	bool LexSimplex::rowIsViolated(unsigned row) const {
685	if (tableau(row, 2) < 0)
686	return true;
687	if (tableau(row, 2) == 0 && tableau(row, 1) < 0)
688	return true;
689	return false;
690	}
691
692	std::optional<unsigned> LexSimplex::maybeGetViolatedRow() const {
693	for (unsigned row = 0, e = getNumRows(); row < e; ++row)
694	if (rowIsViolated(row))
695	return row;
696	return {};
697	}
698
699	/// We simply look for violated rows and keep trying to move them to column
700	/// orientation, which always succeeds unless the constraints have no solution
701	/// in which case we just give up and return.
702	LogicalResult LexSimplex::restoreRationalConsistency() {
703	if (empty)
704	return failure();
705	while (std::optional<unsigned> maybeViolatedRow = maybeGetViolatedRow())
706	if (moveRowUnknownToColumn(row: *maybeViolatedRow).failed())
707	return failure();
708	return success();
709	}
710
711	// Move the row unknown to column orientation while preserving lexicopositivity
712	// of the basis transform. The sample value of the row must be non-positive.
713	//
714	// We only consider pivots where the pivot element is positive. Suppose no such
715	// pivot exists, i.e., some violated row has no positive coefficient for any
716	// basis unknown. The row can be represented as (s + c_1*u_1 + ... + c_n*u_n)/d,
717	// where d is the denominator, s is the sample value and the c_i are the basis
718	// coefficients. If s != 0, then since any feasible assignment of the basis
719	// satisfies u_i >= 0 for all i, and we have s < 0 as well as c_i < 0 for all i,
720	// any feasible assignment would violate this row and therefore the constraints
721	// have no solution.
722	//
723	// We can preserve lexicopositivity by picking the pivot column with positive
724	// pivot element that makes the lexicographically smallest change to the sample
725	// point.
726	//
727	// Proof. Let
728	// x = (x_1, ... x_n) be the variables,
729	// z = (z_1, ... z_m) be the constraints,
730	// y = (y_1, ... y_n) be the current basis, and
731	// define w = (x_1, ... x_n, z_1, ... z_m) = B*y + s.
732	// B is basically the simplex tableau of our implementation except that instead
733	// of only describing the transform to get back the non-basis unknowns, it
734	// defines the values of all the unknowns in terms of the basis unknowns.
735	// Similarly, s is the column for the sample value.
736	//
737	// Our goal is to show that each column in B, restricted to the first n
738	// rows, is lexicopositive after the pivot if it is so before. This is
739	// equivalent to saying the columns in the whole matrix are lexicopositive;
740	// there must be some non-zero element in every column in the first n rows since
741	// the n variables cannot be spanned without using all the n basis unknowns.
742	//
743	// Consider a pivot where z_i replaces y_j in the basis. Recall the pivot
744	// transform for the tableau derived for SimplexBase::pivot:
745	//
746	// pivot col other col pivot col other col
747	// pivot row a b -> pivot row 1/a -b/a
748	// other row c d other row c/a d - bc/a
749	//
750	// Similarly, a pivot results in B changing to B' and c to c'; the difference
751	// between the tableau and these matrices B and B' is that there is no special
752	// case for the pivot row, since it continues to represent the same unknown. The
753	// same formula applies for all rows:
754	//
755	// B'.col(j) = B.col(j) / B(i,j)
756	// B'.col(k) = B.col(k) - B(i,k) * B.col(j) / B(i,j) for k != j
757	// and similarly, s' = s - s_i * B.col(j) / B(i,j).
758	//
759	// If s_i == 0, then the sample value remains unchanged. Otherwise, if s_i < 0,
760	// the change in sample value when pivoting with column a is lexicographically
761	// smaller than that when pivoting with column b iff B.col(a) / B(i, a) is
762	// lexicographically smaller than B.col(b) / B(i, b).
763	//
764	// Since B(i, j) > 0, column j remains lexicopositive.
765	//
766	// For the other columns, suppose C.col(k) is not lexicopositive.
767	// This means that for some p, for all t < p,
768	// C(t,k) = 0 => B(t,k) = B(t,j) * B(i,k) / B(i,j) and
769	// C(t,k) < 0 => B(p,k) < B(t,j) * B(i,k) / B(i,j),
770	// which is in contradiction to the fact that B.col(j) / B(i,j) must be
771	// lexicographically smaller than B.col(k) / B(i,k), since it lexicographically
772	// minimizes the change in sample value.
773	LogicalResult LexSimplexBase::moveRowUnknownToColumn(unsigned row) {
774	std::optional<unsigned> maybeColumn;
775	for (unsigned col = 3 + nSymbol, e = getNumColumns(); col < e; ++col) {
776	if (tableau(row, col) <= 0)
777	continue;
778	maybeColumn =
779	!maybeColumn ? col : getLexMinPivotColumn(row, colA: *maybeColumn, colB: col);
780	}
781
782	if (!maybeColumn)
783	return failure();
784
785	pivot(row, col: *maybeColumn);
786	return success();
787	}
788
789	unsigned LexSimplexBase::getLexMinPivotColumn(unsigned row, unsigned colA,
790	unsigned colB) const {
791	// First, let's consider the non-symbolic case.
792	// A pivot causes the following change. (in the diagram the matrix elements
793	// are shown as rationals and there is no common denominator used)
794	//
795	// pivot col big M col const col
796	// pivot row a p b
797	// other row c q d
798	// |
799	// v
800	//
801	// pivot col big M col const col
802	// pivot row 1/a -p/a -b/a
803	// other row c/a q - pc/a d - bc/a
804	//
805	// Let the sample value of the pivot row be s = pM + b before the pivot. Since
806	// the pivot row represents a violated constraint we know that s < 0.
807	//
808	// If the variable is a non-pivot column, its sample value is zero before and
809	// after the pivot.
810	//
811	// If the variable is the pivot column, then its sample value goes from 0 to
812	// (-p/a)M + (-b/a), i.e. 0 to -(pM + b)/a. Thus the change in the sample
813	// value is -s/a.
814	//
815	// If the variable is the pivot row, its sample value goes from s to 0, for a
816	// change of -s.
817	//
818	// If the variable is a non-pivot row, its sample value changes from
819	// qM + d to qM + d + (-pc/a)M + (-bc/a). Thus the change in sample value
820	// is -(pM + b)(c/a) = -sc/a.
821	//
822	// Thus the change in sample value is either 0, -s/a, -s, or -sc/a. Here -s is
823	// fixed for all calls to this function since the row and tableau are fixed.
824	// The callee just wants to compare the return values with the return value of
825	// other invocations of the same function. So the -s is common for all
826	// comparisons involved and can be ignored, since -s is strictly positive.
827	//
828	// Thus we take away this common factor and just return 0, 1/a, 1, or c/a as
829	// appropriate. This allows us to run the entire algorithm treating M
830	// symbolically, as the pivot to be performed does not depend on the value
831	// of M, so long as the sample value s is negative. Note that this is not
832	// because of any special feature of M; by the same argument, we ignore the
833	// symbols too. The caller ensure that the sample value s is negative for
834	// all possible values of the symbols.
835	auto getSampleChangeCoeffForVar = [this, row](unsigned col,
836	const Unknown &u) -> Fraction {
837	MPInt a = tableau(row, col);
838	if (u.orientation == Orientation::Column) {
839	// Pivot column case.
840	if (u.pos == col)
841	return {1, a};
842
843	// Non-pivot column case.
844	return {0, 1};
845	}
846
847	// Pivot row case.
848	if (u.pos == row)
849	return {1, 1};
850
851	// Non-pivot row case.
852	MPInt c = tableau(u.pos, col);
853	return {c, a};
854	};
855
856	for (const Unknown &u : var) {
857	Fraction changeA = getSampleChangeCoeffForVar(colA, u);
858	Fraction changeB = getSampleChangeCoeffForVar(colB, u);
859	if (changeA < changeB)
860	return colA;
861	if (changeA > changeB)
862	return colB;
863	}
864
865	// If we reached here, both result in exactly the same changes, so it
866	// doesn't matter which we return.
867	return colA;
868	}
869
870	/// Find a pivot to change the sample value of the row in the specified
871	/// direction. The returned pivot row will involve `row` if and only if the
872	/// unknown is unbounded in the specified direction.
873	///
874	/// To increase (resp. decrease) the value of a row, we need to find a live
875	/// column with a non-zero coefficient. If the coefficient is positive, we need
876	/// to increase (decrease) the value of the column, and if the coefficient is
877	/// negative, we need to decrease (increase) the value of the column. Also,
878	/// we cannot decrease the sample value of restricted columns.
879	///
880	/// If multiple columns are valid, we break ties by considering a lexicographic
881	/// ordering where we prefer unknowns with lower index.
882	std::optional<SimplexBase::Pivot>
883	Simplex::findPivot(int row, Direction direction) const {
884	std::optional<unsigned> col;
885	for (unsigned j = 2, e = getNumColumns(); j < e; ++j) {
886	MPInt elem = tableau(row, j);
887	if (elem == 0)
888	continue;
889
890	if (unknownFromColumn(col: j).restricted &&
891	!signMatchesDirection(elem, direction))
892	continue;
893	if (!col || colUnknown[j] < colUnknown[*col])
894	col = j;
895	}
896
897	if (!col)
898	return {};
899
900	Direction newDirection =
901	tableau(row, *col) < 0 ? flippedDirection(direction) : direction;
902	std::optional<unsigned> maybePivotRow = findPivotRow(skipRow: row, direction: newDirection, col: *col);
903	return Pivot{.row: maybePivotRow.value_or(u&: row), .column: *col};
904	}
905
906	/// Swap the associated unknowns for the row and the column.
907	///
908	/// First we swap the index associated with the row and column. Then we update
909	/// the unknowns to reflect their new position and orientation.
910	void SimplexBase::swapRowWithCol(unsigned row, unsigned col) {
911	std::swap(a&: rowUnknown[row], b&: colUnknown[col]);
912	Unknown &uCol = unknownFromColumn(col);
913	Unknown &uRow = unknownFromRow(row);
914	uCol.orientation = Orientation::Column;
915	uRow.orientation = Orientation::Row;
916	uCol.pos = col;
917	uRow.pos = row;
918	}
919
920	void SimplexBase::pivot(Pivot pair) { pivot(row: pair.row, col: pair.column); }
921
922	/// Pivot pivotRow and pivotCol.
923	///
924	/// Let R be the pivot row unknown and let C be the pivot col unknown.
925	/// Since initially R = a*C + sum b_i * X_i
926	/// (where the sum is over the other column's unknowns, x_i)
927	/// C = (R - (sum b_i * X_i))/a
928	///
929	/// Let u be some other row unknown.
930	/// u = c*C + sum d_i * X_i
931	/// So u = c*(R - sum b_i * X_i)/a + sum d_i * X_i
932	///
933	/// This results in the following transform:
934	/// pivot col other col pivot col other col
935	/// pivot row a b -> pivot row 1/a -b/a
936	/// other row c d other row c/a d - bc/a
937	///
938	/// Taking into account the common denominators p and q:
939	///
940	/// pivot col other col pivot col other col
941	/// pivot row a/p b/p -> pivot row p/a -b/a
942	/// other row c/q d/q other row cp/aq (da - bc)/aq
943	///
944	/// The pivot row transform is accomplished be swapping a with the pivot row's
945	/// common denominator and negating the pivot row except for the pivot column
946	/// element.
947	void SimplexBase::pivot(unsigned pivotRow, unsigned pivotCol) {
948	assert(pivotCol >= getNumFixedCols() && "Refusing to pivot invalid column");
949	assert(!unknownFromColumn(pivotCol).isSymbol);
950
951	swapRowWithCol(row: pivotRow, col: pivotCol);
952	std::swap(a&: tableau(pivotRow, 0), b&: tableau(pivotRow, pivotCol));
953	// We need to negate the whole pivot row except for the pivot column.
954	if (tableau(pivotRow, 0) < 0) {
955	// If the denominator is negative, we negate the row by simply negating the
956	// denominator.
957	tableau(pivotRow, 0) = -tableau(pivotRow, 0);
958	tableau(pivotRow, pivotCol) = -tableau(pivotRow, pivotCol);
959	} else {
960	for (unsigned col = 1, e = getNumColumns(); col < e; ++col) {
961	if (col == pivotCol)
962	continue;
963	tableau(pivotRow, col) = -tableau(pivotRow, col);
964	}
965	}
966	tableau.normalizeRow(row: pivotRow);
967
968	for (unsigned row = 0, numRows = getNumRows(); row < numRows; ++row) {
969	if (row == pivotRow)
970	continue;
971	if (tableau(row, pivotCol) == 0) // Nothing to do.
972	continue;
973	tableau(row, 0) *= tableau(pivotRow, 0);
974	for (unsigned col = 1, numCols = getNumColumns(); col < numCols; ++col) {
975	if (col == pivotCol)
976	continue;
977	// Add rather than subtract because the pivot row has been negated.
978	tableau(row, col) = tableau(row, col) * tableau(pivotRow, 0) +
979	tableau(row, pivotCol) * tableau(pivotRow, col);
980	}
981	tableau(row, pivotCol) *= tableau(pivotRow, pivotCol);
982	tableau.normalizeRow(row);
983	}
984	}
985
986	/// Perform pivots until the unknown has a non-negative sample value or until
987	/// no more upward pivots can be performed. Return success if we were able to
988	/// bring the row to a non-negative sample value, and failure otherwise.
989	LogicalResult Simplex::restoreRow(Unknown &u) {
990	assert(u.orientation == Orientation::Row &&
991	"unknown should be in row position");
992
993	while (tableau(u.pos, 1) < 0) {
994	std::optional<Pivot> maybePivot = findPivot(row: u.pos, direction: Direction::Up);
995	if (!maybePivot)
996	break;
997
998	pivot(pair: *maybePivot);
999	if (u.orientation == Orientation::Column)
1000	return success(); // the unknown is unbounded above.
1001	}
1002	return success(isSuccess: tableau(u.pos, 1) >= 0);
1003	}
1004
1005	/// Find a row that can be used to pivot the column in the specified direction.
1006	/// This returns an empty optional if and only if the column is unbounded in the
1007	/// specified direction (ignoring skipRow, if skipRow is set).
1008	///
1009	/// If skipRow is set, this row is not considered, and (if it is restricted) its
1010	/// restriction may be violated by the returned pivot. Usually, skipRow is set
1011	/// because we don't want to move it to column position unless it is unbounded,
1012	/// and we are either trying to increase the value of skipRow or explicitly
1013	/// trying to make skipRow negative, so we are not concerned about this.
1014	///
1015	/// If the direction is up (resp. down) and a restricted row has a negative
1016	/// (positive) coefficient for the column, then this row imposes a bound on how
1017	/// much the sample value of the column can change. Such a row with constant
1018	/// term c and coefficient f for the column imposes a bound of c/|f| on the
1019	/// change in sample value (in the specified direction). (note that c is
1020	/// non-negative here since the row is restricted and the tableau is consistent)
1021	///
1022	/// We iterate through the rows and pick the row which imposes the most
1023	/// stringent bound, since pivoting with a row changes the row's sample value to
1024	/// 0 and hence saturates the bound it imposes. We break ties between rows that
1025	/// impose the same bound by considering a lexicographic ordering where we
1026	/// prefer unknowns with lower index value.
1027	std::optional<unsigned> Simplex::findPivotRow(std::optional<unsigned> skipRow,
1028	Direction direction,
1029	unsigned col) const {
1030	std::optional<unsigned> retRow;
1031	// Initialize these to zero in order to silence a warning about retElem and
1032	// retConst being used uninitialized in the initialization of `diff` below. In
1033	// reality, these are always initialized when that line is reached since these
1034	// are set whenever retRow is set.
1035	MPInt retElem, retConst;
1036	for (unsigned row = nRedundant, e = getNumRows(); row < e; ++row) {
1037	if (skipRow && row == *skipRow)
1038	continue;
1039	MPInt elem = tableau(row, col);
1040	if (elem == 0)
1041	continue;
1042	if (!unknownFromRow(row).restricted)
1043	continue;
1044	if (signMatchesDirection(elem, direction))
1045	continue;
1046	MPInt constTerm = tableau(row, 1);
1047
1048	if (!retRow) {
1049	retRow = row;
1050	retElem = elem;
1051	retConst = constTerm;
1052	continue;
1053	}
1054
1055	MPInt diff = retConst * elem - constTerm * retElem;
1056	if ((diff == 0 && rowUnknown[row] < rowUnknown[*retRow]) ||
1057	(diff != 0 && !signMatchesDirection(elem: diff, direction))) {
1058	retRow = row;
1059	retElem = elem;
1060	retConst = constTerm;
1061	}
1062	}
1063	return retRow;
1064	}
1065
1066	bool SimplexBase::isEmpty() const { return empty; }
1067
1068	void SimplexBase::swapRows(unsigned i, unsigned j) {
1069	if (i == j)
1070	return;
1071	tableau.swapRows(row: i, otherRow: j);
1072	std::swap(a&: rowUnknown[i], b&: rowUnknown[j]);
1073	unknownFromRow(row: i).pos = i;
1074	unknownFromRow(row: j).pos = j;
1075	}
1076
1077	void SimplexBase::swapColumns(unsigned i, unsigned j) {
1078	assert(i < getNumColumns() && j < getNumColumns() &&
1079	"Invalid columns provided!");
1080	if (i == j)
1081	return;
1082	tableau.swapColumns(column: i, otherColumn: j);
1083	std::swap(a&: colUnknown[i], b&: colUnknown[j]);
1084	unknownFromColumn(col: i).pos = i;
1085	unknownFromColumn(col: j).pos = j;
1086	}
1087
1088	/// Mark this tableau empty and push an entry to the undo stack.
1089	void SimplexBase::markEmpty() {
1090	// If the set is already empty, then we shouldn't add another UnmarkEmpty log
1091	// entry, since in that case the Simplex will be erroneously marked as
1092	// non-empty when rolling back past this point.
1093	if (empty)
1094	return;
1095	undoLog.push_back(Elt: UndoLogEntry::UnmarkEmpty);
1096	empty = true;
1097	}
1098
1099	/// Add an inequality to the tableau. If coeffs is c_0, c_1, ... c_n, where n
1100	/// is the current number of variables, then the corresponding inequality is
1101	/// c_n + c_0*x_0 + c_1*x_1 + ... + c_{n-1}*x_{n-1} >= 0.
1102	///
1103	/// We add the inequality and mark it as restricted. We then try to make its
1104	/// sample value non-negative. If this is not possible, the tableau has become
1105	/// empty and we mark it as such.
1106	void Simplex::addInequality(ArrayRef<MPInt> coeffs) {
1107	unsigned conIndex = addRow(coeffs, /*makeRestricted=*/true);
1108	LogicalResult result = restoreRow(u&: con[conIndex]);
1109	if (failed(result))
1110	markEmpty();
1111	}
1112
1113	/// Add an equality to the tableau. If coeffs is c_0, c_1, ... c_n, where n
1114	/// is the current number of variables, then the corresponding equality is
1115	/// c_n + c_0*x_0 + c_1*x_1 + ... + c_{n-1}*x_{n-1} == 0.
1116	///
1117	/// We simply add two opposing inequalities, which force the expression to
1118	/// be zero.
1119	void SimplexBase::addEquality(ArrayRef<MPInt> coeffs) {
1120	addInequality(coeffs);
1121	SmallVector<MPInt, 8> negatedCoeffs;
1122	for (const MPInt &coeff : coeffs)
1123	negatedCoeffs.emplace_back(Args: -coeff);
1124	addInequality(coeffs: negatedCoeffs);
1125	}
1126
1127	unsigned SimplexBase::getNumVariables() const { return var.size(); }
1128	unsigned SimplexBase::getNumConstraints() const { return con.size(); }
1129
1130	/// Return a snapshot of the current state. This is just the current size of the
1131	/// undo log.
1132	unsigned SimplexBase::getSnapshot() const { return undoLog.size(); }
1133
1134	unsigned SimplexBase::getSnapshotBasis() {
1135	SmallVector<int, 8> basis;
1136	for (int index : colUnknown) {
1137	if (index != nullIndex)
1138	basis.push_back(Elt: index);
1139	}
1140	savedBases.push_back(Elt: std::move(basis));
1141
1142	undoLog.emplace_back(Args: UndoLogEntry::RestoreBasis);
1143	return undoLog.size() - 1;
1144	}
1145
1146	void SimplexBase::removeLastConstraintRowOrientation() {
1147	assert(con.back().orientation == Orientation::Row);
1148
1149	// Move this unknown to the last row and remove the last row from the
1150	// tableau.
1151	swapRows(i: con.back().pos, j: getNumRows() - 1);
1152	// It is not strictly necessary to shrink the tableau, but for now we
1153	// maintain the invariant that the tableau has exactly getNumRows()
1154	// rows.
1155	tableau.resizeVertically(newNRows: getNumRows() - 1);
1156	rowUnknown.pop_back();
1157	con.pop_back();
1158	}
1159
1160	// This doesn't find a pivot row only if the column has zero
1161	// coefficients for every row.
1162	//
1163	// If the unknown is a constraint, this can't happen, since it was added
1164	// initially as a row. Such a row could never have been pivoted to a column. So
1165	// a pivot row will always be found if we have a constraint.
1166	//
1167	// If we have a variable, then the column has zero coefficients for every row
1168	// iff no constraints have been added with a non-zero coefficient for this row.
1169	std::optional<unsigned> SimplexBase::findAnyPivotRow(unsigned col) {
1170	for (unsigned row = nRedundant, e = getNumRows(); row < e; ++row)
1171	if (tableau(row, col) != 0)
1172	return row;
1173	return {};
1174	}
1175
1176	// It's not valid to remove the constraint by deleting the column since this
1177	// would result in an invalid basis.
1178	void Simplex::undoLastConstraint() {
1179	if (con.back().orientation == Orientation::Column) {
1180	// We try to find any pivot row for this column that preserves tableau
1181	// consistency (except possibly the column itself, which is going to be
1182	// deallocated anyway).
1183	//
1184	// If no pivot row is found in either direction, then the unknown is
1185	// unbounded in both directions and we are free to perform any pivot at
1186	// all. To do this, we just need to find any row with a non-zero
1187	// coefficient for the column. findAnyPivotRow will always be able to
1188	// find such a row for a constraint.
1189	unsigned column = con.back().pos;
1190	if (std::optional<unsigned> maybeRow =
1191	findPivotRow(skipRow: {}, direction: Direction::Up, col: column)) {
1192	pivot(pivotRow: *maybeRow, pivotCol: column);
1193	} else if (std::optional<unsigned> maybeRow =
1194	findPivotRow(skipRow: {}, direction: Direction::Down, col: column)) {
1195	pivot(pivotRow: *maybeRow, pivotCol: column);
1196	} else {
1197	std::optional<unsigned> row = findAnyPivotRow(col: column);
1198	assert(row && "Pivot should always exist for a constraint!");
1199	pivot(pivotRow: *row, pivotCol: column);
1200	}
1201	}
1202	removeLastConstraintRowOrientation();
1203	}
1204
1205	// It's not valid to remove the constraint by deleting the column since this
1206	// would result in an invalid basis.
1207	void LexSimplexBase::undoLastConstraint() {
1208	if (con.back().orientation == Orientation::Column) {
1209	// When removing the last constraint during a rollback, we just need to find
1210	// any pivot at all, i.e., any row with non-zero coefficient for the
1211	// column, because when rolling back a lexicographic simplex, we always
1212	// end by restoring the exact basis that was present at the time of the
1213	// snapshot, so what pivots we perform while undoing doesn't matter as
1214	// long as we get the unknown to row orientation and remove it.
1215	unsigned column = con.back().pos;
1216	std::optional<unsigned> row = findAnyPivotRow(col: column);
1217	assert(row && "Pivot should always exist for a constraint!");
1218	pivot(pivotRow: *row, pivotCol: column);
1219	}
1220	removeLastConstraintRowOrientation();
1221	}
1222
1223	void SimplexBase::undo(UndoLogEntry entry) {
1224	if (entry == UndoLogEntry::RemoveLastConstraint) {
1225	// Simplex and LexSimplex handle this differently, so we call out to a
1226	// virtual function to handle this.
1227	undoLastConstraint();
1228	} else if (entry == UndoLogEntry::RemoveLastVariable) {
1229	// Whenever we are rolling back the addition of a variable, it is guaranteed
1230	// that the variable will be in column position.
1231	//
1232	// We can see this as follows: any constraint that depends on this variable
1233	// was added after this variable was added, so the addition of such
1234	// constraints should already have been rolled back by the time we get to
1235	// rolling back the addition of the variable. Therefore, no constraint
1236	// currently has a component along the variable, so the variable itself must
1237	// be part of the basis.
1238	assert(var.back().orientation == Orientation::Column &&
1239	"Variable to be removed must be in column orientation!");
1240
1241	if (var.back().isSymbol)
1242	nSymbol--;
1243
1244	// Move this variable to the last column and remove the column from the
1245	// tableau.
1246	swapColumns(i: var.back().pos, j: getNumColumns() - 1);
1247	tableau.resizeHorizontally(newNColumns: getNumColumns() - 1);
1248	var.pop_back();
1249	colUnknown.pop_back();
1250	} else if (entry == UndoLogEntry::UnmarkEmpty) {
1251	empty = false;
1252	} else if (entry == UndoLogEntry::UnmarkLastRedundant) {
1253	nRedundant--;
1254	} else if (entry == UndoLogEntry::RestoreBasis) {
1255	assert(!savedBases.empty() && "No bases saved!");
1256
1257	SmallVector<int, 8> basis = std::move(savedBases.back());
1258	savedBases.pop_back();
1259
1260	for (int index : basis) {
1261	Unknown &u = unknownFromIndex(index);
1262	if (u.orientation == Orientation::Column)
1263	continue;
1264	for (unsigned col = getNumFixedCols(), e = getNumColumns(); col < e;
1265	col++) {
1266	assert(colUnknown[col] != nullIndex &&
1267	"Column should not be a fixed column!");
1268	if (llvm::is_contained(Range&: basis, Element: colUnknown[col]))
1269	continue;
1270	if (tableau(u.pos, col) == 0)
1271	continue;
1272	pivot(pivotRow: u.pos, pivotCol: col);
1273	break;
1274	}
1275
1276	assert(u.orientation == Orientation::Column && "No pivot found!");
1277	}
1278	}
1279	}
1280
1281	/// Rollback to the specified snapshot.
1282	///
1283	/// We undo all the log entries until the log size when the snapshot was taken
1284	/// is reached.
1285	void SimplexBase::rollback(unsigned snapshot) {
1286	while (undoLog.size() > snapshot) {
1287	undo(entry: undoLog.back());
1288	undoLog.pop_back();
1289	}
1290	}
1291
1292	/// We add the usual floor division constraints:
1293	/// `0 <= coeffs - denom*q <= denom - 1`, where `q` is the new division
1294	/// variable.
1295	///
1296	/// This constrains the remainder `coeffs - denom*q` to be in the
1297	/// range `[0, denom - 1]`, which fixes the integer value of the quotient `q`.
1298	void SimplexBase::addDivisionVariable(ArrayRef<MPInt> coeffs,
1299	const MPInt &denom) {
1300	assert(denom > 0 && "Denominator must be positive!");
1301	appendVariable();
1302
1303	SmallVector<MPInt, 8> ineq(coeffs.begin(), coeffs.end());
1304	MPInt constTerm = ineq.back();
1305	ineq.back() = -denom;
1306	ineq.push_back(Elt: constTerm);
1307	addInequality(coeffs: ineq);
1308
1309	for (MPInt &coeff : ineq)
1310	coeff = -coeff;
1311	ineq.back() += denom - 1;
1312	addInequality(coeffs: ineq);
1313	}
1314
1315	void SimplexBase::appendVariable(unsigned count) {
1316	if (count == 0)
1317	return;
1318	var.reserve(N: var.size() + count);
1319	colUnknown.reserve(N: colUnknown.size() + count);
1320	for (unsigned i = 0; i < count; ++i) {
1321	var.emplace_back(Args: Orientation::Column, /*restricted=*/Args: false,
1322	/*pos=*/Args: getNumColumns() + i);
1323	colUnknown.push_back(Elt: var.size() - 1);
1324	}
1325	tableau.resizeHorizontally(newNColumns: getNumColumns() + count);
1326	undoLog.insert(I: undoLog.end(), NumToInsert: count, Elt: UndoLogEntry::RemoveLastVariable);
1327	}
1328
1329	/// Add all the constraints from the given IntegerRelation.
1330	void SimplexBase::intersectIntegerRelation(const IntegerRelation &rel) {
1331	assert(rel.getNumVars() == getNumVariables() &&
1332	"IntegerRelation must have same dimensionality as simplex");
1333	for (unsigned i = 0, e = rel.getNumInequalities(); i < e; ++i)
1334	addInequality(coeffs: rel.getInequality(idx: i));
1335	for (unsigned i = 0, e = rel.getNumEqualities(); i < e; ++i)
1336	addEquality(coeffs: rel.getEquality(idx: i));
1337	}
1338
1339	MaybeOptimum<Fraction> Simplex::computeRowOptimum(Direction direction,
1340	unsigned row) {
1341	// Keep trying to find a pivot for the row in the specified direction.
1342	while (std::optional<Pivot> maybePivot = findPivot(row, direction)) {
1343	// If findPivot returns a pivot involving the row itself, then the optimum
1344	// is unbounded, so we return std::nullopt.
1345	if (maybePivot->row == row)
1346	return OptimumKind::Unbounded;
1347	pivot(pair: *maybePivot);
1348	}
1349
1350	// The row has reached its optimal sample value, which we return.
1351	// The sample value is the entry in the constant column divided by the common
1352	// denominator for this row.
1353	return Fraction(tableau(row, 1), tableau(row, 0));
1354	}
1355
1356	/// Compute the optimum of the specified expression in the specified direction,
1357	/// or std::nullopt if it is unbounded.
1358	MaybeOptimum<Fraction> Simplex::computeOptimum(Direction direction,
1359	ArrayRef<MPInt> coeffs) {
1360	if (empty)
1361	return OptimumKind::Empty;
1362
1363	SimplexRollbackScopeExit scopeExit(*this);
1364	unsigned conIndex = addRow(coeffs);
1365	unsigned row = con[conIndex].pos;
1366	return computeRowOptimum(direction, row);
1367	}
1368
1369	MaybeOptimum<Fraction> Simplex::computeOptimum(Direction direction,
1370	Unknown &u) {
1371	if (empty)
1372	return OptimumKind::Empty;
1373	if (u.orientation == Orientation::Column) {
1374	unsigned column = u.pos;
1375	std::optional<unsigned> pivotRow = findPivotRow(skipRow: {}, direction, col: column);
1376	// If no pivot is returned, the constraint is unbounded in the specified
1377	// direction.
1378	if (!pivotRow)
1379	return OptimumKind::Unbounded;
1380	pivot(pivotRow: *pivotRow, pivotCol: column);
1381	}
1382
1383	unsigned row = u.pos;
1384	MaybeOptimum<Fraction> optimum = computeRowOptimum(direction, row);
1385	if (u.restricted && direction == Direction::Down &&
1386	(optimum.isUnbounded() || *optimum < Fraction(0, 1))) {
1387	if (failed(result: restoreRow(u)))
1388	llvm_unreachable("Could not restore row!");
1389	}
1390	return optimum;
1391	}
1392
1393	bool Simplex::isBoundedAlongConstraint(unsigned constraintIndex) {
1394	assert(!empty && "It is not meaningful to ask whether a direction is bounded "
1395	"in an empty set.");
1396	// The constraint's perpendicular is already bounded below, since it is a
1397	// constraint. If it is also bounded above, we can return true.
1398	return computeOptimum(direction: Direction::Up, u&: con[constraintIndex]).isBounded();
1399	}
1400
1401	/// Redundant constraints are those that are in row orientation and lie in
1402	/// rows 0 to nRedundant - 1.
1403	bool Simplex::isMarkedRedundant(unsigned constraintIndex) const {
1404	const Unknown &u = con[constraintIndex];
1405	return u.orientation == Orientation::Row && u.pos < nRedundant;
1406	}
1407
1408	/// Mark the specified row redundant.
1409	///
1410	/// This is done by moving the unknown to the end of the block of redundant
1411	/// rows (namely, to row nRedundant) and incrementing nRedundant to
1412	/// accomodate the new redundant row.
1413	void Simplex::markRowRedundant(Unknown &u) {
1414	assert(u.orientation == Orientation::Row &&
1415	"Unknown should be in row position!");
1416	assert(u.pos >= nRedundant && "Unknown is already marked redundant!");
1417	swapRows(i: u.pos, j: nRedundant);
1418	++nRedundant;
1419	undoLog.emplace_back(Args: UndoLogEntry::UnmarkLastRedundant);
1420	}
1421
1422	/// Find a subset of constraints that is redundant and mark them redundant.
1423	void Simplex::detectRedundant(unsigned offset, unsigned count) {
1424	assert(offset + count <= con.size() && "invalid range!");
1425	// It is not meaningful to talk about redundancy for empty sets.
1426	if (empty)
1427	return;
1428
1429	// Iterate through the constraints and check for each one if it can attain
1430	// negative sample values. If it can, it's not redundant. Otherwise, it is.
1431	// We mark redundant constraints redundant.
1432	//
1433	// Constraints that get marked redundant in one iteration are not respected
1434	// when checking constraints in later iterations. This prevents, for example,
1435	// two identical constraints both being marked redundant since each is
1436	// redundant given the other one. In this example, only the first of the
1437	// constraints that is processed will get marked redundant, as it should be.
1438	for (unsigned i = 0; i < count; ++i) {
1439	Unknown &u = con[offset + i];
1440	if (u.orientation == Orientation::Column) {
1441	unsigned column = u.pos;
1442	std::optional<unsigned> pivotRow =
1443	findPivotRow(skipRow: {}, direction: Direction::Down, col: column);
1444	// If no downward pivot is returned, the constraint is unbounded below
1445	// and hence not redundant.
1446	if (!pivotRow)
1447	continue;
1448	pivot(pivotRow: *pivotRow, pivotCol: column);
1449	}
1450
1451	unsigned row = u.pos;
1452	MaybeOptimum<Fraction> minimum = computeRowOptimum(direction: Direction::Down, row);
1453	if (minimum.isUnbounded() || *minimum < Fraction(0, 1)) {
1454	// Constraint is unbounded below or can attain negative sample values and
1455	// hence is not redundant.
1456	if (failed(result: restoreRow(u)))
1457	llvm_unreachable("Could not restore non-redundant row!");
1458	continue;
1459	}
1460
1461	markRowRedundant(u);
1462	}
1463	}
1464
1465	bool Simplex::isUnbounded() {
1466	if (empty)
1467	return false;
1468
1469	SmallVector<MPInt, 8> dir(var.size() + 1);
1470	for (unsigned i = 0; i < var.size(); ++i) {
1471	dir[i] = 1;
1472
1473	if (computeOptimum(direction: Direction::Up, coeffs: dir).isUnbounded())
1474	return true;
1475
1476	if (computeOptimum(direction: Direction::Down, coeffs: dir).isUnbounded())
1477	return true;
1478
1479	dir[i] = 0;
1480	}
1481	return false;
1482	}
1483
1484	/// Make a tableau to represent a pair of points in the original tableau.
1485	///
1486	/// The product constraints and variables are stored as: first A's, then B's.
1487	///
1488	/// The product tableau has row layout:
1489	/// A's redundant rows, B's redundant rows, A's other rows, B's other rows.
1490	///
1491	/// It has column layout:
1492	/// denominator, constant, A's columns, B's columns.
1493	Simplex Simplex::makeProduct(const Simplex &a, const Simplex &b) {
1494	unsigned numVar = a.getNumVariables() + b.getNumVariables();
1495	unsigned numCon = a.getNumConstraints() + b.getNumConstraints();
1496	Simplex result(numVar);
1497
1498	result.tableau.reserveRows(rows: numCon);
1499	result.empty = a.empty || b.empty;
1500
1501	auto concat = [](ArrayRef<Unknown> v, ArrayRef<Unknown> w) {
1502	SmallVector<Unknown, 8> result;
1503	result.reserve(N: v.size() + w.size());
1504	result.insert(I: result.end(), From: v.begin(), To: v.end());
1505	result.insert(I: result.end(), From: w.begin(), To: w.end());
1506	return result;
1507	};
1508	result.con = concat(a.con, b.con);
1509	result.var = concat(a.var, b.var);
1510
1511	auto indexFromBIndex = [&](int index) {
1512	return index >= 0 ? a.getNumVariables() + index
1513	: ~(a.getNumConstraints() + ~index);
1514	};
1515
1516	result.colUnknown.assign(NumElts: 2, Elt: nullIndex);
1517	for (unsigned i = 2, e = a.getNumColumns(); i < e; ++i) {
1518	result.colUnknown.push_back(Elt: a.colUnknown[i]);
1519	result.unknownFromIndex(index: result.colUnknown.back()).pos =
1520	result.colUnknown.size() - 1;
1521	}
1522	for (unsigned i = 2, e = b.getNumColumns(); i < e; ++i) {
1523	result.colUnknown.push_back(Elt: indexFromBIndex(b.colUnknown[i]));
1524	result.unknownFromIndex(index: result.colUnknown.back()).pos =
1525	result.colUnknown.size() - 1;
1526	}
1527
1528	auto appendRowFromA = [&](unsigned row) {
1529	unsigned resultRow = result.tableau.appendExtraRow();
1530	for (unsigned col = 0, e = a.getNumColumns(); col < e; ++col)
1531	result.tableau(resultRow, col) = a.tableau(row, col);
1532	result.rowUnknown.push_back(Elt: a.rowUnknown[row]);
1533	result.unknownFromIndex(index: result.rowUnknown.back()).pos =
1534	result.rowUnknown.size() - 1;
1535	};
1536
1537	// Also fixes the corresponding entry in rowUnknown and var/con (as the case
1538	// may be).
1539	auto appendRowFromB = [&](unsigned row) {
1540	unsigned resultRow = result.tableau.appendExtraRow();
1541	result.tableau(resultRow, 0) = b.tableau(row, 0);
1542	result.tableau(resultRow, 1) = b.tableau(row, 1);
1543
1544	unsigned offset = a.getNumColumns() - 2;
1545	for (unsigned col = 2, e = b.getNumColumns(); col < e; ++col)
1546	result.tableau(resultRow, offset + col) = b.tableau(row, col);
1547	result.rowUnknown.push_back(Elt: indexFromBIndex(b.rowUnknown[row]));
1548	result.unknownFromIndex(index: result.rowUnknown.back()).pos =
1549	result.rowUnknown.size() - 1;
1550	};
1551
1552	result.nRedundant = a.nRedundant + b.nRedundant;
1553	for (unsigned row = 0; row < a.nRedundant; ++row)
1554	appendRowFromA(row);
1555	for (unsigned row = 0; row < b.nRedundant; ++row)
1556	appendRowFromB(row);
1557	for (unsigned row = a.nRedundant, e = a.getNumRows(); row < e; ++row)
1558	appendRowFromA(row);
1559	for (unsigned row = b.nRedundant, e = b.getNumRows(); row < e; ++row)
1560	appendRowFromB(row);
1561
1562	return result;
1563	}
1564
1565	std::optional<SmallVector<Fraction, 8>> Simplex::getRationalSample() const {
1566	if (empty)
1567	return {};
1568
1569	SmallVector<Fraction, 8> sample;
1570	sample.reserve(N: var.size());
1571	// Push the sample value for each variable into the vector.
1572	for (const Unknown &u : var) {
1573	if (u.orientation == Orientation::Column) {
1574	// If the variable is in column position, its sample value is zero.
1575	sample.emplace_back(Args: 0, Args: 1);
1576	} else {
1577	// If the variable is in row position, its sample value is the
1578	// entry in the constant column divided by the denominator.
1579	MPInt denom = tableau(u.pos, 0);
1580	sample.emplace_back(Args: tableau(u.pos, 1), Args&: denom);
1581	}
1582	}
1583	return sample;
1584	}
1585
1586	void LexSimplexBase::addInequality(ArrayRef<MPInt> coeffs) {
1587	addRow(coeffs, /*makeRestricted=*/true);
1588	}
1589
1590	MaybeOptimum<SmallVector<Fraction, 8>> LexSimplex::getRationalSample() const {
1591	if (empty)
1592	return OptimumKind::Empty;
1593
1594	SmallVector<Fraction, 8> sample;
1595	sample.reserve(N: var.size());
1596	// Push the sample value for each variable into the vector.
1597	for (const Unknown &u : var) {
1598	// When the big M parameter is being used, each variable x is represented
1599	// as M + x, so its sample value is finite if and only if it is of the
1600	// form 1*M + c. If the coefficient of M is not one then the sample value
1601	// is infinite, and we return an empty optional.
1602
1603	if (u.orientation == Orientation::Column) {
1604	// If the variable is in column position, the sample value of M + x is
1605	// zero, so x = -M which is unbounded.
1606	return OptimumKind::Unbounded;
1607	}
1608
1609	// If the variable is in row position, its sample value is the
1610	// entry in the constant column divided by the denominator.
1611	MPInt denom = tableau(u.pos, 0);
1612	if (usingBigM)
1613	if (tableau(u.pos, 2) != denom)
1614	return OptimumKind::Unbounded;
1615	sample.emplace_back(Args: tableau(u.pos, 1), Args&: denom);
1616	}
1617	return sample;
1618	}
1619
1620	std::optional<SmallVector<MPInt, 8>> Simplex::getSamplePointIfIntegral() const {
1621	// If the tableau is empty, no sample point exists.
1622	if (empty)
1623	return {};
1624
1625	// The value will always exist since the Simplex is non-empty.
1626	SmallVector<Fraction, 8> rationalSample = *getRationalSample();
1627	SmallVector<MPInt, 8> integerSample;
1628	integerSample.reserve(N: var.size());
1629	for (const Fraction &coord : rationalSample) {
1630	// If the sample is non-integral, return std::nullopt.
1631	if (coord.num % coord.den != 0)
1632	return {};
1633	integerSample.push_back(Elt: coord.num / coord.den);
1634	}
1635	return integerSample;
1636	}
1637
1638	/// Given a simplex for a polytope, construct a new simplex whose variables are
1639	/// identified with a pair of points (x, y) in the original polytope. Supports
1640	/// some operations needed for generalized basis reduction. In what follows,
1641	/// dotProduct(x, y) = x_1 * y_1 + x_2 * y_2 + ... x_n * y_n where n is the
1642	/// dimension of the original polytope.
1643	///
1644	/// This supports adding equality constraints dotProduct(dir, x - y) == 0. It
1645	/// also supports rolling back this addition, by maintaining a snapshot stack
1646	/// that contains a snapshot of the Simplex's state for each equality, just
1647	/// before that equality was added.
1648	class presburger::GBRSimplex {
1649	using Orientation = Simplex::Orientation;
1650
1651	public:
1652	GBRSimplex(const Simplex &originalSimplex)
1653	: simplex(Simplex::makeProduct(a: originalSimplex, b: originalSimplex)),
1654	simplexConstraintOffset(simplex.getNumConstraints()) {}
1655
1656	/// Add an equality dotProduct(dir, x - y) == 0.
1657	/// First pushes a snapshot for the current simplex state to the stack so
1658	/// that this can be rolled back later.
1659	void addEqualityForDirection(ArrayRef<MPInt> dir) {
1660	assert(llvm::any_of(dir, [](const MPInt &x) { return x != 0; }) &&
1661	"Direction passed is the zero vector!");
1662	snapshotStack.push_back(Elt: simplex.getSnapshot());
1663	simplex.addEquality(coeffs: getCoeffsForDirection(dir));
1664	}
1665	/// Compute max(dotProduct(dir, x - y)).
1666	Fraction computeWidth(ArrayRef<MPInt> dir) {
1667	MaybeOptimum<Fraction> maybeWidth =
1668	simplex.computeOptimum(direction: Direction::Up, coeffs: getCoeffsForDirection(dir));
1669	assert(maybeWidth.isBounded() && "Width should be bounded!");
1670	return *maybeWidth;
1671	}
1672
1673	/// Compute max(dotProduct(dir, x - y)) and save the dual variables for only
1674	/// the direction equalities to `dual`.
1675	Fraction computeWidthAndDuals(ArrayRef<MPInt> dir,
1676	SmallVectorImpl<MPInt> &dual,
1677	MPInt &dualDenom) {
1678	// We can't just call into computeWidth or computeOptimum since we need to
1679	// access the state of the tableau after computing the optimum, and these
1680	// functions rollback the insertion of the objective function into the
1681	// tableau before returning. We instead add a row for the objective function
1682	// ourselves, call into computeOptimum, compute the duals from the tableau
1683	// state, and finally rollback the addition of the row before returning.
1684	SimplexRollbackScopeExit scopeExit(simplex);
1685	unsigned conIndex = simplex.addRow(coeffs: getCoeffsForDirection(dir));
1686	unsigned row = simplex.con[conIndex].pos;
1687	MaybeOptimum<Fraction> maybeWidth =
1688	simplex.computeRowOptimum(direction: Simplex::Direction::Up, row);
1689	assert(maybeWidth.isBounded() && "Width should be bounded!");
1690	dualDenom = simplex.tableau(row, 0);
1691	dual.clear();
1692
1693	// The increment is i += 2 because equalities are added as two inequalities,
1694	// one positive and one negative. Each iteration processes one equality.
1695	for (unsigned i = simplexConstraintOffset; i < conIndex; i += 2) {
1696	// The dual variable for an inequality in column orientation is the
1697	// negative of its coefficient at the objective row. If the inequality is
1698	// in row orientation, the corresponding dual variable is zero.
1699	//
1700	// We want the dual for the original equality, which corresponds to two
1701	// inequalities: a positive inequality, which has the same coefficients as
1702	// the equality, and a negative equality, which has negated coefficients.
1703	//
1704	// Note that at most one of these inequalities can be in column
1705	// orientation because the column unknowns should form a basis and hence
1706	// must be linearly independent. If the positive inequality is in column
1707	// position, its dual is the dual corresponding to the equality. If the
1708	// negative inequality is in column position, the negation of its dual is
1709	// the dual corresponding to the equality. If neither is in column
1710	// position, then that means that this equality is redundant, and its dual
1711	// is zero.
1712	//
1713	// Note that it is NOT valid to perform pivots during the computation of
1714	// the duals. This entire dual computation must be performed on the same
1715	// tableau configuration.
1716	assert(!(simplex.con[i].orientation == Orientation::Column &&
1717	simplex.con[i + 1].orientation == Orientation::Column) &&
1718	"Both inequalities for the equality cannot be in column "
1719	"orientation!");
1720	if (simplex.con[i].orientation == Orientation::Column)
1721	dual.push_back(Elt: -simplex.tableau(row, simplex.con[i].pos));
1722	else if (simplex.con[i + 1].orientation == Orientation::Column)
1723	dual.push_back(Elt: simplex.tableau(row, simplex.con[i + 1].pos));
1724	else
1725	dual.emplace_back(Args: 0);
1726	}
1727	return *maybeWidth;
1728	}
1729
1730	/// Remove the last equality that was added through addEqualityForDirection.
1731	///
1732	/// We do this by rolling back to the snapshot at the top of the stack, which
1733	/// should be a snapshot taken just before the last equality was added.
1734	void removeLastEquality() {
1735	assert(!snapshotStack.empty() && "Snapshot stack is empty!");
1736	simplex.rollback(snapshot: snapshotStack.back());
1737	snapshotStack.pop_back();
1738	}
1739
1740	private:
1741	/// Returns coefficients of the expression 'dot_product(dir, x - y)',
1742	/// i.e., dir_1 * x_1 + dir_2 * x_2 + ... + dir_n * x_n
1743	/// - dir_1 * y_1 - dir_2 * y_2 - ... - dir_n * y_n,
1744	/// where n is the dimension of the original polytope.
1745	SmallVector<MPInt, 8> getCoeffsForDirection(ArrayRef<MPInt> dir) {
1746	assert(2 * dir.size() == simplex.getNumVariables() &&
1747	"Direction vector has wrong dimensionality");
1748	SmallVector<MPInt, 8> coeffs(dir.begin(), dir.end());
1749	coeffs.reserve(N: 2 * dir.size());
1750	for (const MPInt &coeff : dir)
1751	coeffs.push_back(Elt: -coeff);
1752	coeffs.emplace_back(Args: 0); // constant term
1753	return coeffs;
1754	}
1755
1756	Simplex simplex;
1757	/// The first index of the equality constraints, the index immediately after
1758	/// the last constraint in the initial product simplex.
1759	unsigned simplexConstraintOffset;
1760	/// A stack of snapshots, used for rolling back.
1761	SmallVector<unsigned, 8> snapshotStack;
1762	};
1763
1764	/// Reduce the basis to try and find a direction in which the polytope is
1765	/// "thin". This only works for bounded polytopes.
1766	///
1767	/// This is an implementation of the algorithm described in the paper
1768	/// "An Implementation of Generalized Basis Reduction for Integer Programming"
1769	/// by W. Cook, T. Rutherford, H. E. Scarf, D. Shallcross.
1770	///
1771	/// Let b_{level}, b_{level + 1}, ... b_n be the current basis.
1772	/// Let width_i(v) = max <v, x - y> where x and y are points in the original
1773	/// polytope such that <b_j, x - y> = 0 is satisfied for all level <= j < i.
1774	///
1775	/// In every iteration, we first replace b_{i+1} with b_{i+1} + u*b_i, where u
1776	/// is the integer such that width_i(b_{i+1} + u*b_i) is minimized. Let dual_i
1777	/// be the dual variable associated with the constraint <b_i, x - y> = 0 when
1778	/// computing width_{i+1}(b_{i+1}). It can be shown that dual_i is the
1779	/// minimizing value of u, if it were allowed to be fractional. Due to
1780	/// convexity, the minimizing integer value is either floor(dual_i) or
1781	/// ceil(dual_i), so we just need to check which of these gives a lower
1782	/// width_{i+1} value. If dual_i turned out to be an integer, then u = dual_i.
1783	///
1784	/// Now if width_i(b_{i+1}) < 0.75 * width_i(b_i), we swap b_i and (the new)
1785	/// b_{i + 1} and decrement i (unless i = level, in which case we stay at the
1786	/// same i). Otherwise, we increment i.
1787	///
1788	/// We keep f values and duals cached and invalidate them when necessary.
1789	/// Whenever possible, we use them instead of recomputing them. We implement the
1790	/// algorithm as follows.
1791	///
1792	/// In an iteration at i we need to compute:
1793	/// a) width_i(b_{i + 1})
1794	/// b) width_i(b_i)
1795	/// c) the integer u that minimizes width_i(b_{i + 1} + u*b_i)
1796	///
1797	/// If width_i(b_i) is not already cached, we compute it.
1798	///
1799	/// If the duals are not already cached, we compute width_{i+1}(b_{i+1}) and
1800	/// store the duals from this computation.
1801	///
1802	/// We call updateBasisWithUAndGetFCandidate, which finds the minimizing value
1803	/// of u as explained before, caches the duals from this computation, sets
1804	/// b_{i+1} to b_{i+1} + u*b_i, and returns the new value of width_i(b_{i+1}).
1805	///
1806	/// Now if width_i(b_{i+1}) < 0.75 * width_i(b_i), we swap b_i and b_{i+1} and
1807	/// decrement i, resulting in the basis
1808	/// ... b_{i - 1}, b_{i + 1} + u*b_i, b_i, b_{i+2}, ...
1809	/// with corresponding f values
1810	/// ... width_{i-1}(b_{i-1}), width_i(b_{i+1} + u*b_i), width_{i+1}(b_i), ...
1811	/// The values up to i - 1 remain unchanged. We have just gotten the middle
1812	/// value from updateBasisWithUAndGetFCandidate, so we can update that in the
1813	/// cache. The value at width_{i+1}(b_i) is unknown, so we evict this value from
1814	/// the cache. The iteration after decrementing needs exactly the duals from the
1815	/// computation of width_i(b_{i + 1} + u*b_i), so we keep these in the cache.
1816	///
1817	/// When incrementing i, no cached f values get invalidated. However, the cached
1818	/// duals do get invalidated as the duals for the higher levels are different.
1819	void Simplex::reduceBasis(IntMatrix &basis, unsigned level) {
1820	const Fraction epsilon(3, 4);
1821
1822	if (level == basis.getNumRows() - 1)
1823	return;
1824
1825	GBRSimplex gbrSimplex(*this);
1826	SmallVector<Fraction, 8> width;
1827	SmallVector<MPInt, 8> dual;
1828	MPInt dualDenom;
1829
1830	// Finds the value of u that minimizes width_i(b_{i+1} + u*b_i), caches the
1831	// duals from this computation, sets b_{i+1} to b_{i+1} + u*b_i, and returns
1832	// the new value of width_i(b_{i+1}).
1833	//
1834	// If dual_i is not an integer, the minimizing value must be either
1835	// floor(dual_i) or ceil(dual_i). We compute the expression for both and
1836	// choose the minimizing value.
1837	//
1838	// If dual_i is an integer, we don't need to perform these computations. We
1839	// know that in this case,
1840	// a) u = dual_i.
1841	// b) one can show that dual_j for j < i are the same duals we would have
1842	// gotten from computing width_i(b_{i + 1} + u*b_i), so the correct duals
1843	// are the ones already in the cache.
1844	// c) width_i(b_{i+1} + u*b_i) = min_{alpha} width_i(b_{i+1} + alpha * b_i),
1845	// which
1846	// one can show is equal to width_{i+1}(b_{i+1}). The latter value must
1847	// be in the cache, so we get it from there and return it.
1848	auto updateBasisWithUAndGetFCandidate = [&](unsigned i) -> Fraction {
1849	assert(i < level + dual.size() && "dual_i is not known!");
1850
1851	MPInt u = floorDiv(lhs: dual[i - level], rhs: dualDenom);
1852	basis.addToRow(sourceRow: i, targetRow: i + 1, scale: u);
1853	if (dual[i - level] % dualDenom != 0) {
1854	SmallVector<MPInt, 8> candidateDual[2];
1855	MPInt candidateDualDenom[2];
1856	Fraction widthI[2];
1857
1858	// Initially u is floor(dual) and basis reflects this.
1859	widthI[0] = gbrSimplex.computeWidthAndDuals(
1860	dir: basis.getRow(row: i + 1), dual&: candidateDual[0], dualDenom&: candidateDualDenom[0]);
1861
1862	// Now try ceil(dual), i.e. floor(dual) + 1.
1863	++u;
1864	basis.addToRow(sourceRow: i, targetRow: i + 1, scale: 1);
1865	widthI[1] = gbrSimplex.computeWidthAndDuals(
1866	dir: basis.getRow(row: i + 1), dual&: candidateDual[1], dualDenom&: candidateDualDenom[1]);
1867
1868	unsigned j = widthI[0] < widthI[1] ? 0 : 1;
1869	if (j == 0)
1870	// Subtract 1 to go from u = ceil(dual) back to floor(dual).
1871	basis.addToRow(sourceRow: i, targetRow: i + 1, scale: -1);
1872
1873	// width_i(b{i+1} + u*b_i) should be minimized at our value of u.
1874	// We assert that this holds by checking that the values of width_i at
1875	// u - 1 and u + 1 are greater than or equal to the value at u. If the
1876	// width is lesser at either of the adjacent values, then our computed
1877	// value of u is clearly not the minimizer. Otherwise by convexity the
1878	// computed value of u is really the minimizer.
1879
1880	// Check the value at u - 1.
1881	assert(gbrSimplex.computeWidth(scaleAndAddForAssert(
1882	basis.getRow(i + 1), MPInt(-1), basis.getRow(i))) >=
1883	widthI[j] &&
1884	"Computed u value does not minimize the width!");
1885	// Check the value at u + 1.
1886	assert(gbrSimplex.computeWidth(scaleAndAddForAssert(
1887	basis.getRow(i + 1), MPInt(+1), basis.getRow(i))) >=
1888	widthI[j] &&
1889	"Computed u value does not minimize the width!");
1890
1891	dual = std::move(candidateDual[j]);
1892	dualDenom = candidateDualDenom[j];
1893	return widthI[j];
1894	}
1895
1896	assert(i + 1 - level < width.size() && "width_{i+1} wasn't saved");
1897	// f_i(b_{i+1} + dual*b_i) == width_{i+1}(b_{i+1}) when `dual` minimizes the
1898	// LHS. (note: the basis has already been updated, so b_{i+1} + dual*b_i in
1899	// the above expression is equal to basis.getRow(i+1) below.)
1900	assert(gbrSimplex.computeWidth(basis.getRow(i + 1)) ==
1901	width[i + 1 - level]);
1902	return width[i + 1 - level];
1903	};
1904
1905	// In the ith iteration of the loop, gbrSimplex has constraints for directions
1906	// from `level` to i - 1.
1907	unsigned i = level;
1908	while (i < basis.getNumRows() - 1) {
1909	if (i >= level + width.size()) {
1910	// We don't even know the value of f_i(b_i), so let's find that first.
1911	// We have to do this first since later we assume that width already
1912	// contains values up to and including i.
1913
1914	assert((i == 0 || i - 1 < level + width.size()) &&
1915	"We are at level i but we don't know the value of width_{i-1}");
1916
1917	// We don't actually use these duals at all, but it doesn't matter
1918	// because this case should only occur when i is level, and there are no
1919	// duals in that case anyway.
1920	assert(i == level && "This case should only occur when i == level");
1921	width.push_back(
1922	Elt: gbrSimplex.computeWidthAndDuals(dir: basis.getRow(row: i), dual, dualDenom));
1923	}
1924
1925	if (i >= level + dual.size()) {
1926	assert(i + 1 >= level + width.size() &&
1927	"We don't know dual_i but we know width_{i+1}");
1928	// We don't know dual for our level, so let's find it.
1929	gbrSimplex.addEqualityForDirection(dir: basis.getRow(row: i));
1930	width.push_back(Elt: gbrSimplex.computeWidthAndDuals(dir: basis.getRow(row: i + 1), dual,
1931	dualDenom));
1932	gbrSimplex.removeLastEquality();
1933	}
1934
1935	// This variable stores width_i(b_{i+1} + u*b_i).
1936	Fraction widthICandidate = updateBasisWithUAndGetFCandidate(i);
1937	if (widthICandidate < epsilon * width[i - level]) {
1938	basis.swapRows(row: i, otherRow: i + 1);
1939	width[i - level] = widthICandidate;
1940	// The values of width_{i+1}(b_{i+1}) and higher may change after the
1941	// swap, so we remove the cached values here.
1942	width.resize(N: i - level + 1);
1943	if (i == level) {
1944	dual.clear();
1945	continue;
1946	}
1947
1948	gbrSimplex.removeLastEquality();
1949	i--;
1950	continue;
1951	}
1952
1953	// Invalidate duals since the higher level needs to recompute its own duals.
1954	dual.clear();
1955	gbrSimplex.addEqualityForDirection(dir: basis.getRow(row: i));
1956	i++;
1957	}
1958	}
1959
1960	/// Search for an integer sample point using a branch and bound algorithm.
1961	///
1962	/// Each row in the basis matrix is a vector, and the set of basis vectors
1963	/// should span the space. Initially this is the identity matrix,
1964	/// i.e., the basis vectors are just the variables.
1965	///
1966	/// In every level, a value is assigned to the level-th basis vector, as
1967	/// follows. Compute the minimum and maximum rational values of this direction.
1968	/// If only one integer point lies in this range, constrain the variable to
1969	/// have this value and recurse to the next variable.
1970	///
1971	/// If the range has multiple values, perform generalized basis reduction via
1972	/// reduceBasis and then compute the bounds again. Now we try constraining
1973	/// this direction in the first value in this range and "recurse" to the next
1974	/// level. If we fail to find a sample, we try assigning the direction the next
1975	/// value in this range, and so on.
1976	///
1977	/// If no integer sample is found from any of the assignments, or if the range
1978	/// contains no integer value, then of course the polytope is empty for the
1979	/// current assignment of the values in previous levels, so we return to
1980	/// the previous level.
1981	///
1982	/// If we reach the last level where all the variables have been assigned values
1983	/// already, then we simply return the current sample point if it is integral,
1984	/// and go back to the previous level otherwise.
1985	///
1986	/// To avoid potentially arbitrarily large recursion depths leading to stack
1987	/// overflows, this algorithm is implemented iteratively.
1988	std::optional<SmallVector<MPInt, 8>> Simplex::findIntegerSample() {
1989	if (empty)
1990	return {};
1991
1992	unsigned nDims = var.size();
1993	IntMatrix basis = IntMatrix::identity(dimension: nDims);
1994
1995	unsigned level = 0;
1996	// The snapshot just before constraining a direction to a value at each level.
1997	SmallVector<unsigned, 8> snapshotStack;
1998	// The maximum value in the range of the direction for each level.
1999	SmallVector<MPInt, 8> upperBoundStack;
2000	// The next value to try constraining the basis vector to at each level.
2001	SmallVector<MPInt, 8> nextValueStack;
2002
2003	snapshotStack.reserve(N: basis.getNumRows());
2004	upperBoundStack.reserve(N: basis.getNumRows());
2005	nextValueStack.reserve(N: basis.getNumRows());
2006	while (level != -1u) {
2007	if (level == basis.getNumRows()) {
2008	// We've assigned values to all variables. Return if we have a sample,
2009	// or go back up to the previous level otherwise.
2010	if (auto maybeSample = getSamplePointIfIntegral())
2011	return maybeSample;
2012	level--;
2013	continue;
2014	}
2015
2016	if (level >= upperBoundStack.size()) {
2017	// We haven't populated the stack values for this level yet, so we have
2018	// just come down a level ("recursed"). Find the lower and upper bounds.
2019	// If there is more than one integer point in the range, perform
2020	// generalized basis reduction.
2021	SmallVector<MPInt, 8> basisCoeffs =
2022	llvm::to_vector<8>(Range: basis.getRow(row: level));
2023	basisCoeffs.emplace_back(Args: 0);
2024
2025	auto [minRoundedUp, maxRoundedDown] = computeIntegerBounds(coeffs: basisCoeffs);
2026
2027	// We don't have any integer values in the range.
2028	// Pop the stack and return up a level.
2029	if (minRoundedUp.isEmpty() || maxRoundedDown.isEmpty()) {
2030	assert((minRoundedUp.isEmpty() && maxRoundedDown.isEmpty()) &&
2031	"If one bound is empty, both should be.");
2032	snapshotStack.pop_back();
2033	nextValueStack.pop_back();
2034	upperBoundStack.pop_back();
2035	level--;
2036	continue;
2037	}
2038
2039	// We already checked the empty case above.
2040	assert((minRoundedUp.isBounded() && maxRoundedDown.isBounded()) &&
2041	"Polyhedron should be bounded!");
2042
2043	// Heuristic: if the sample point is integral at this point, just return
2044	// it.
2045	if (auto maybeSample = getSamplePointIfIntegral())
2046	return *maybeSample;
2047
2048	if (*minRoundedUp < *maxRoundedDown) {
2049	reduceBasis(basis, level);
2050	basisCoeffs = llvm::to_vector<8>(Range: basis.getRow(row: level));
2051	basisCoeffs.emplace_back(Args: 0);
2052	std::tie(args&: minRoundedUp, args&: maxRoundedDown) =
2053	computeIntegerBounds(coeffs: basisCoeffs);
2054	}
2055
2056	snapshotStack.push_back(Elt: getSnapshot());
2057	// The smallest value in the range is the next value to try.
2058	// The values in the optionals are guaranteed to exist since we know the
2059	// polytope is bounded.
2060	nextValueStack.push_back(Elt: *minRoundedUp);
2061	upperBoundStack.push_back(Elt: *maxRoundedDown);
2062	}
2063
2064	assert((snapshotStack.size() - 1 == level &&
2065	nextValueStack.size() - 1 == level &&
2066	upperBoundStack.size() - 1 == level) &&
2067	"Mismatched variable stack sizes!");
2068
2069	// Whether we "recursed" or "returned" from a lower level, we rollback
2070	// to the snapshot of the starting state at this level. (in the "recursed"
2071	// case this has no effect)
2072	rollback(snapshot: snapshotStack.back());
2073	MPInt nextValue = nextValueStack.back();
2074	++nextValueStack.back();
2075	if (nextValue > upperBoundStack.back()) {
2076	// We have exhausted the range and found no solution. Pop the stack and
2077	// return up a level.
2078	snapshotStack.pop_back();
2079	nextValueStack.pop_back();
2080	upperBoundStack.pop_back();
2081	level--;
2082	continue;
2083	}
2084
2085	// Try the next value in the range and "recurse" into the next level.
2086	SmallVector<MPInt, 8> basisCoeffs(basis.getRow(row: level).begin(),
2087	basis.getRow(row: level).end());
2088	basisCoeffs.push_back(Elt: -nextValue);
2089	addEquality(coeffs: basisCoeffs);
2090	level++;
2091	}
2092
2093	return {};
2094	}
2095
2096	/// Compute the minimum and maximum integer values the expression can take. We
2097	/// compute each separately.
2098	std::pair<MaybeOptimum<MPInt>, MaybeOptimum<MPInt>>
2099	Simplex::computeIntegerBounds(ArrayRef<MPInt> coeffs) {
2100	MaybeOptimum<MPInt> minRoundedUp(
2101	computeOptimum(direction: Simplex::Direction::Down, coeffs).map(f&: ceil));
2102	MaybeOptimum<MPInt> maxRoundedDown(
2103	computeOptimum(direction: Simplex::Direction::Up, coeffs).map(f&: floor));
2104	return {minRoundedUp, maxRoundedDown};
2105	}
2106
2107	bool Simplex::isFlatAlong(ArrayRef<MPInt> coeffs) {
2108	assert(!isEmpty() && "cannot check for flatness of empty simplex!");
2109	auto upOpt = computeOptimum(direction: Simplex::Direction::Up, coeffs);
2110	auto downOpt = computeOptimum(direction: Simplex::Direction::Down, coeffs);
2111
2112	if (!upOpt.isBounded())
2113	return false;
2114	if (!downOpt.isBounded())
2115	return false;
2116
2117	return *upOpt == *downOpt;
2118	}
2119
2120	void SimplexBase::print(raw_ostream &os) const {
2121	os << "rows = " << getNumRows() << ", columns = " << getNumColumns() << "\n";
2122	if (empty)
2123	os << "Simplex marked empty!\n";
2124	os << "var: ";
2125	for (unsigned i = 0; i < var.size(); ++i) {
2126	if (i > 0)
2127	os << ", ";
2128	var[i].print(os);
2129	}
2130	os << "\ncon: ";
2131	for (unsigned i = 0; i < con.size(); ++i) {
2132	if (i > 0)
2133	os << ", ";
2134	con[i].print(os);
2135	}
2136	os << '\n';
2137	for (unsigned row = 0, e = getNumRows(); row < e; ++row) {
2138	if (row > 0)
2139	os << ", ";
2140	os << "r" << row << ": " << rowUnknown[row];
2141	}
2142	os << '\n';
2143	os << "c0: denom, c1: const";
2144	for (unsigned col = 2, e = getNumColumns(); col < e; ++col)
2145	os << ", c" << col << ": " << colUnknown[col];
2146	os << '\n';
2147	for (unsigned row = 0, numRows = getNumRows(); row < numRows; ++row) {
2148	for (unsigned col = 0, numCols = getNumColumns(); col < numCols; ++col)
2149	os << tableau(row, col) << '\t';
2150	os << '\n';
2151	}
2152	os << '\n';
2153	}
2154
2155	void SimplexBase::dump() const { print(os&: llvm::errs()); }
2156
2157	bool Simplex::isRationalSubsetOf(const IntegerRelation &rel) {
2158	if (isEmpty())
2159	return true;
2160
2161	for (unsigned i = 0, e = rel.getNumInequalities(); i < e; ++i)
2162	if (findIneqType(coeffs: rel.getInequality(idx: i)) != IneqType::Redundant)
2163	return false;
2164
2165	for (unsigned i = 0, e = rel.getNumEqualities(); i < e; ++i)
2166	if (!isRedundantEquality(coeffs: rel.getEquality(idx: i)))
2167	return false;
2168
2169	return true;
2170	}
2171
2172	/// Returns the type of the inequality with coefficients `coeffs`.
2173	/// Possible types are:
2174	/// Redundant The inequality is satisfied by all points in the polytope
2175	/// Cut The inequality is satisfied by some points, but not by others
2176	/// Separate The inequality is not satisfied by any point
2177	///
2178	/// Internally, this computes the minimum and the maximum the inequality with
2179	/// coefficients `coeffs` can take. If the minimum is >= 0, the inequality holds
2180	/// for all points in the polytope, so it is redundant. If the minimum is <= 0
2181	/// and the maximum is >= 0, the points in between the minimum and the
2182	/// inequality do not satisfy it, the points in between the inequality and the
2183	/// maximum satisfy it. Hence, it is a cut inequality. If both are < 0, no
2184	/// points of the polytope satisfy the inequality, which means it is a separate
2185	/// inequality.
2186	Simplex::IneqType Simplex::findIneqType(ArrayRef<MPInt> coeffs) {
2187	MaybeOptimum<Fraction> minimum = computeOptimum(direction: Direction::Down, coeffs);
2188	if (minimum.isBounded() && *minimum >= Fraction(0, 1)) {
2189	return IneqType::Redundant;
2190	}
2191	MaybeOptimum<Fraction> maximum = computeOptimum(direction: Direction::Up, coeffs);
2192	if ((!minimum.isBounded() || *minimum <= Fraction(0, 1)) &&
2193	(!maximum.isBounded() || *maximum >= Fraction(0, 1))) {
2194	return IneqType::Cut;
2195	}
2196	return IneqType::Separate;
2197	}
2198
2199	/// Checks whether the type of the inequality with coefficients `coeffs`
2200	/// is Redundant.
2201	bool Simplex::isRedundantInequality(ArrayRef<MPInt> coeffs) {
2202	assert(!empty &&
2203	"It is not meaningful to ask about redundancy in an empty set!");
2204	return findIneqType(coeffs) == IneqType::Redundant;
2205	}
2206
2207	/// Check whether the equality given by `coeffs == 0` is redundant given
2208	/// the existing constraints. This is redundant when `coeffs` is already
2209	/// always zero under the existing constraints. `coeffs` is always zero
2210	/// when the minimum and maximum value that `coeffs` can take are both zero.
2211	bool Simplex::isRedundantEquality(ArrayRef<MPInt> coeffs) {
2212	assert(!empty &&
2213	"It is not meaningful to ask about redundancy in an empty set!");
2214	MaybeOptimum<Fraction> minimum = computeOptimum(direction: Direction::Down, coeffs);
2215	MaybeOptimum<Fraction> maximum = computeOptimum(direction: Direction::Up, coeffs);
2216	assert((!minimum.isEmpty() && !maximum.isEmpty()) &&
2217	"Optima should be non-empty for a non-empty set");
2218	return minimum.isBounded() && maximum.isBounded() &&
2219	*maximum == Fraction(0, 1) && *minimum == Fraction(0, 1);
2220	}
2221