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ortools-clone/ortools/sat/linear_programming_constraint.cc

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// Copyright 2010-2018 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
#include "ortools/sat/linear_programming_constraint.h"
#include <algorithm>
#include <cmath>
#include <iterator>
#include <limits>
#include <string>
#include <utility>
#include <vector>
#include "absl/container/flat_hash_map.h"
#include "absl/numeric/int128.h"
#include "ortools/base/commandlineflags.h"
#include "ortools/base/int_type_indexed_vector.h"
#include "ortools/base/integral_types.h"
#include "ortools/base/logging.h"
#include "ortools/base/map_util.h"
#include "ortools/base/mathutil.h"
#include "ortools/base/stl_util.h"
#include "ortools/glop/parameters.pb.h"
#include "ortools/glop/preprocessor.h"
#include "ortools/glop/status.h"
#include "ortools/graph/strongly_connected_components.h"
#include "ortools/lp_data/lp_types.h"
#include "ortools/sat/implied_bounds.h"
#include "ortools/sat/integer.h"
#include "ortools/util/saturated_arithmetic.h"
namespace operations_research {
namespace sat {
using glop::ColIndex;
using glop::Fractional;
using glop::RowIndex;
const double LinearProgrammingConstraint::kCpEpsilon = 1e-4;
const double LinearProgrammingConstraint::kLpEpsilon = 1e-6;
// TODO(user): make SatParameters singleton too, otherwise changing them after
// a constraint was added will have no effect on this class.
LinearProgrammingConstraint::LinearProgrammingConstraint(Model* model)
: constraint_manager_(model),
sat_parameters_(*(model->GetOrCreate<SatParameters>())),
model_(model),
time_limit_(model->GetOrCreate<TimeLimit>()),
integer_trail_(model->GetOrCreate<IntegerTrail>()),
trail_(model->GetOrCreate<Trail>()),
model_heuristics_(model->GetOrCreate<SearchHeuristicsVector>()),
integer_encoder_(model->GetOrCreate<IntegerEncoder>()),
random_(model->GetOrCreate<ModelRandomGenerator>()),
implied_bounds_processor_({}, integer_trail_,
model->GetOrCreate<ImpliedBounds>()),
dispatcher_(model->GetOrCreate<LinearProgrammingDispatcher>()),
expanded_lp_solution_(
*model->GetOrCreate<LinearProgrammingConstraintLpSolution>()) {
// Tweak the default parameters to make the solve incremental.
glop::GlopParameters parameters;
parameters.set_use_dual_simplex(true);
simplex_.SetParameters(parameters);
if (sat_parameters_.use_branching_in_lp() ||
sat_parameters_.search_branching() == SatParameters::LP_SEARCH) {
compute_reduced_cost_averages_ = true;
}
}
LinearProgrammingConstraint::~LinearProgrammingConstraint() {
VLOG(1) << "Total number of simplex iterations: "
<< total_num_simplex_iterations_;
}
void LinearProgrammingConstraint::AddLinearConstraint(
const LinearConstraint& ct) {
DCHECK(!lp_constraint_is_registered_);
constraint_manager_.Add(ct);
// We still create the mirror variable right away though.
//
// TODO(user): clean this up? Note that it is important that the variable
// in lp_data_ never changes though, so we can restart from the current
// lp solution and be incremental (even if the constraints changed).
for (const IntegerVariable var : ct.vars) {
GetOrCreateMirrorVariable(PositiveVariable(var));
}
}
glop::ColIndex LinearProgrammingConstraint::GetOrCreateMirrorVariable(
IntegerVariable positive_variable) {
DCHECK(VariableIsPositive(positive_variable));
const auto it = mirror_lp_variable_.find(positive_variable);
if (it == mirror_lp_variable_.end()) {
const glop::ColIndex col(integer_variables_.size());
implied_bounds_processor_.AddLpVariable(positive_variable);
mirror_lp_variable_[positive_variable] = col;
integer_variables_.push_back(positive_variable);
lp_solution_.push_back(std::numeric_limits<double>::infinity());
lp_reduced_cost_.push_back(0.0);
(*dispatcher_)[positive_variable] = this;
const int index = std::max(positive_variable.value(),
NegationOf(positive_variable).value());
if (index >= expanded_lp_solution_.size()) {
expanded_lp_solution_.resize(index + 1, 0.0);
}
return col;
}
return it->second;
}
void LinearProgrammingConstraint::SetObjectiveCoefficient(IntegerVariable ivar,
IntegerValue coeff) {
CHECK(!lp_constraint_is_registered_);
objective_is_defined_ = true;
IntegerVariable pos_var = VariableIsPositive(ivar) ? ivar : NegationOf(ivar);
if (ivar != pos_var) coeff = -coeff;
constraint_manager_.SetObjectiveCoefficient(pos_var, coeff);
const glop::ColIndex col = GetOrCreateMirrorVariable(pos_var);
integer_objective_.push_back({col, coeff});
objective_infinity_norm_ =
std::max(objective_infinity_norm_, IntTypeAbs(coeff));
}
// TODO(user): As the search progress, some variables might get fixed. Exploit
// this to reduce the number of variables in the LP and in the
// ConstraintManager? We might also detect during the search that two variable
// are equivalent.
//
// TODO(user): On TSP/VRP with a lot of cuts, this can take 20% of the overall
// running time. We should be able to almost remove most of this from the
// profile by being more incremental (modulo LP scaling).
//
// TODO(user): A longer term idea for LP with a lot of variables is to not
// add all variables to each LP solve and do some "sifting". That can be useful
// for TSP for instance where the number of edges is large, but only a small
// fraction will be used in the optimal solution.
bool LinearProgrammingConstraint::CreateLpFromConstraintManager() {
// Fill integer_lp_.
integer_lp_.clear();
infinity_norms_.clear();
const auto& all_constraints = constraint_manager_.AllConstraints();
for (const auto index : constraint_manager_.LpConstraints()) {
const LinearConstraint& ct = all_constraints[index].constraint;
integer_lp_.push_back(LinearConstraintInternal());
LinearConstraintInternal& new_ct = integer_lp_.back();
new_ct.lb = ct.lb;
new_ct.ub = ct.ub;
const int size = ct.vars.size();
IntegerValue infinity_norm(0);
if (ct.lb > ct.ub) {
LOG(INFO) << "Trivial infeasible bound in an LP constraint";
return false;
}
if (ct.lb > kMinIntegerValue) {
infinity_norm = std::max(infinity_norm, IntTypeAbs(ct.lb));
}
if (ct.ub < kMaxIntegerValue) {
infinity_norm = std::max(infinity_norm, IntTypeAbs(ct.ub));
}
for (int i = 0; i < size; ++i) {
// We only use positive variable inside this class.
IntegerVariable var = ct.vars[i];
IntegerValue coeff = ct.coeffs[i];
if (!VariableIsPositive(var)) {
var = NegationOf(var);
coeff = -coeff;
}
infinity_norm = std::max(infinity_norm, IntTypeAbs(coeff));
new_ct.terms.push_back({GetOrCreateMirrorVariable(var), coeff});
}
infinity_norms_.push_back(infinity_norm);
// Important to keep lp_data_ "clean".
std::sort(new_ct.terms.begin(), new_ct.terms.end());
}
// Copy the integer_lp_ into lp_data_.
lp_data_.Clear();
for (int i = 0; i < integer_variables_.size(); ++i) {
CHECK_EQ(glop::ColIndex(i), lp_data_.CreateNewVariable());
}
for (const auto entry : integer_objective_) {
lp_data_.SetObjectiveCoefficient(entry.first, ToDouble(entry.second));
}
for (const LinearConstraintInternal& ct : integer_lp_) {
const ConstraintIndex row = lp_data_.CreateNewConstraint();
lp_data_.SetConstraintBounds(row, ToDouble(ct.lb), ToDouble(ct.ub));
for (const auto& term : ct.terms) {
lp_data_.SetCoefficient(row, term.first, ToDouble(term.second));
}
}
lp_data_.NotifyThatColumnsAreClean();
// We scale the LP using the level zero bounds that we later override
// with the current ones.
//
// TODO(user): As part of the scaling, we may also want to shift the initial
// variable bounds so that each variable contain the value zero in their
// domain. Maybe just once and for all at the beginning.
const int num_vars = integer_variables_.size();
for (int i = 0; i < num_vars; i++) {
const IntegerVariable cp_var = integer_variables_[i];
const double lb = ToDouble(integer_trail_->LevelZeroLowerBound(cp_var));
const double ub = ToDouble(integer_trail_->LevelZeroUpperBound(cp_var));
lp_data_.SetVariableBounds(glop::ColIndex(i), lb, ub);
}
scaler_.Scale(&lp_data_);
UpdateBoundsOfLpVariables();
lp_data_.NotifyThatColumnsAreClean();
lp_data_.AddSlackVariablesWhereNecessary(false);
VLOG(1) << "LP relaxation: " << lp_data_.GetDimensionString() << ". "
<< constraint_manager_.AllConstraints().size()
<< " Managed constraints.";
return true;
}
LPSolveInfo LinearProgrammingConstraint::SolveLpForBranching() {
LPSolveInfo info;
glop::BasisState basis_state = simplex_.GetState();
const glop::Status status = simplex_.Solve(lp_data_, time_limit_);
total_num_simplex_iterations_ += simplex_.GetNumberOfIterations();
simplex_.LoadStateForNextSolve(basis_state);
if (!status.ok()) {
VLOG(1) << "The LP solver encountered an error: " << status.error_message();
info.status = glop::ProblemStatus::ABNORMAL;
return info;
}
info.status = simplex_.GetProblemStatus();
if (info.status == glop::ProblemStatus::OPTIMAL ||
info.status == glop::ProblemStatus::DUAL_FEASIBLE) {
// Record the objective bound.
info.lp_objective = simplex_.GetObjectiveValue();
info.new_obj_bound = IntegerValue(
static_cast<int64>(std::ceil(info.lp_objective - kCpEpsilon)));
}
return info;
}
void LinearProgrammingConstraint::FillReducedCostReasonIn(
const glop::DenseRow& reduced_costs,
std::vector<IntegerLiteral>* integer_reason) {
integer_reason->clear();
const int num_vars = integer_variables_.size();
for (int i = 0; i < num_vars; i++) {
const double rc = reduced_costs[glop::ColIndex(i)];
if (rc > kLpEpsilon) {
integer_reason->push_back(
integer_trail_->LowerBoundAsLiteral(integer_variables_[i]));
} else if (rc < -kLpEpsilon) {
integer_reason->push_back(
integer_trail_->UpperBoundAsLiteral(integer_variables_[i]));
}
}
integer_trail_->RemoveLevelZeroBounds(integer_reason);
}
bool LinearProgrammingConstraint::BranchOnVar(IntegerVariable positive_var) {
// From the current LP solution, branch on the given var if fractional.
DCHECK(lp_solution_is_set_);
const double current_value = GetSolutionValue(positive_var);
DCHECK_GT(std::abs(current_value - std::round(current_value)), kCpEpsilon);
// Used as empty reason in this method.
integer_reason_.clear();
bool deductions_were_made = false;
UpdateBoundsOfLpVariables();
const IntegerValue current_obj_lb = integer_trail_->LowerBound(objective_cp_);
// This will try to branch in both direction around the LP value of the
// given variable and push any deduction done this way.
const glop::ColIndex lp_var = GetOrCreateMirrorVariable(positive_var);
const double current_lb = ToDouble(integer_trail_->LowerBound(positive_var));
const double current_ub = ToDouble(integer_trail_->UpperBound(positive_var));
const double factor = scaler_.VariableScalingFactor(lp_var);
if (current_value < current_lb || current_value > current_ub) {
return false;
}
// Form LP1 var <= floor(current_value)
const double new_ub = std::floor(current_value);
lp_data_.SetVariableBounds(lp_var, current_lb * factor, new_ub * factor);
LPSolveInfo lower_branch_info = SolveLpForBranching();
if (lower_branch_info.status != glop::ProblemStatus::OPTIMAL &&
lower_branch_info.status != glop::ProblemStatus::DUAL_FEASIBLE &&
lower_branch_info.status != glop::ProblemStatus::DUAL_UNBOUNDED) {
return false;
}
if (lower_branch_info.status == glop::ProblemStatus::DUAL_UNBOUNDED) {
// Push the other branch.
const IntegerLiteral deduction = IntegerLiteral::GreaterOrEqual(
positive_var, IntegerValue(std::ceil(current_value)));
if (!integer_trail_->Enqueue(deduction, {}, integer_reason_)) {
return false;
}
deductions_were_made = true;
} else if (lower_branch_info.new_obj_bound <= current_obj_lb) {
return false;
}
// Form LP2 var >= ceil(current_value)
const double new_lb = std::ceil(current_value);
lp_data_.SetVariableBounds(lp_var, new_lb * factor, current_ub * factor);
LPSolveInfo upper_branch_info = SolveLpForBranching();
if (upper_branch_info.status != glop::ProblemStatus::OPTIMAL &&
upper_branch_info.status != glop::ProblemStatus::DUAL_FEASIBLE &&
upper_branch_info.status != glop::ProblemStatus::DUAL_UNBOUNDED) {
return deductions_were_made;
}
if (upper_branch_info.status == glop::ProblemStatus::DUAL_UNBOUNDED) {
// Push the other branch if not infeasible.
if (lower_branch_info.status != glop::ProblemStatus::DUAL_UNBOUNDED) {
const IntegerLiteral deduction = IntegerLiteral::LowerOrEqual(
positive_var, IntegerValue(std::floor(current_value)));
if (!integer_trail_->Enqueue(deduction, {}, integer_reason_)) {
return deductions_were_made;
}
deductions_were_made = true;
}
} else if (upper_branch_info.new_obj_bound <= current_obj_lb) {
return deductions_were_made;
}
IntegerValue approximate_obj_lb = kMinIntegerValue;
if (lower_branch_info.status == glop::ProblemStatus::DUAL_UNBOUNDED &&
upper_branch_info.status == glop::ProblemStatus::DUAL_UNBOUNDED) {
return integer_trail_->ReportConflict(integer_reason_);
} else if (lower_branch_info.status == glop::ProblemStatus::DUAL_UNBOUNDED) {
approximate_obj_lb = upper_branch_info.new_obj_bound;
} else if (upper_branch_info.status == glop::ProblemStatus::DUAL_UNBOUNDED) {
approximate_obj_lb = lower_branch_info.new_obj_bound;
} else {
approximate_obj_lb = std::min(lower_branch_info.new_obj_bound,
upper_branch_info.new_obj_bound);
}
// NOTE: On some problems, the approximate_obj_lb could be inexact which add
// some tolerance to CP-SAT where currently there is none.
if (approximate_obj_lb <= current_obj_lb) return deductions_were_made;
// Push the bound to the trail.
const IntegerLiteral deduction =
IntegerLiteral::GreaterOrEqual(objective_cp_, approximate_obj_lb);
if (!integer_trail_->Enqueue(deduction, {}, integer_reason_)) {
return deductions_were_made;
}
return true;
}
void LinearProgrammingConstraint::RegisterWith(Model* model) {
DCHECK(!lp_constraint_is_registered_);
lp_constraint_is_registered_ = true;
model->GetOrCreate<LinearProgrammingConstraintCollection>()->push_back(this);
// Note fdid, this is not really needed by should lead to better cache
// locality.
std::sort(integer_objective_.begin(), integer_objective_.end());
// Set the LP to its initial content.
if (!sat_parameters_.add_lp_constraints_lazily()) {
constraint_manager_.AddAllConstraintsToLp();
}
if (!CreateLpFromConstraintManager()) {
model->GetOrCreate<SatSolver>()->NotifyThatModelIsUnsat();
return;
}
GenericLiteralWatcher* watcher = model->GetOrCreate<GenericLiteralWatcher>();
const int watcher_id = watcher->Register(this);
const int num_vars = integer_variables_.size();
for (int i = 0; i < num_vars; i++) {
watcher->WatchIntegerVariable(integer_variables_[i], watcher_id, i);
}
if (objective_is_defined_) {
watcher->WatchUpperBound(objective_cp_, watcher_id);
}
watcher->SetPropagatorPriority(watcher_id, 2);
watcher->AlwaysCallAtLevelZero(watcher_id);
if (integer_variables_.size() >= 20) { // Do not use on small subparts.
auto* container = model->GetOrCreate<SearchHeuristicsVector>();
container->push_back(HeuristicLPPseudoCostBinary(model));
container->push_back(HeuristicLPMostInfeasibleBinary(model));
}
// Registering it with the trail make sure this class is always in sync when
// it is used in the decision heuristics.
integer_trail_->RegisterReversibleClass(this);
watcher->RegisterReversibleInt(watcher_id, &rev_optimal_constraints_size_);
}
void LinearProgrammingConstraint::SetLevel(int level) {
optimal_constraints_.resize(rev_optimal_constraints_size_);
if (lp_solution_is_set_ && level < lp_solution_level_) {
lp_solution_is_set_ = false;
}
// Special case for level zero, we "reload" any previously known optimal
// solution from that level.
//
// TODO(user): Keep all optimal solution in the current branch?
// TODO(user): Still try to add cuts/constraints though!
if (level == 0 && !level_zero_lp_solution_.empty()) {
lp_solution_is_set_ = true;
lp_solution_ = level_zero_lp_solution_;
lp_solution_level_ = 0;
for (int i = 0; i < lp_solution_.size(); i++) {
expanded_lp_solution_[integer_variables_[i]] = lp_solution_[i];
expanded_lp_solution_[NegationOf(integer_variables_[i])] =
-lp_solution_[i];
}
}
}
void LinearProgrammingConstraint::AddCutGenerator(CutGenerator generator) {
for (const IntegerVariable var : generator.vars) {
GetOrCreateMirrorVariable(VariableIsPositive(var) ? var : NegationOf(var));
}
cut_generators_.push_back(std::move(generator));
}
bool LinearProgrammingConstraint::IncrementalPropagate(
const std::vector<int>& watch_indices) {
if (!lp_solution_is_set_) return Propagate();
// At level zero, if there is still a chance to add cuts or lazy constraints,
// we re-run the LP.
if (trail_->CurrentDecisionLevel() == 0 && !lp_at_level_zero_is_final_) {
return Propagate();
}
// Check whether the change breaks the current LP solution. If it does, call
// Propagate() on the current LP.
for (const int index : watch_indices) {
const double lb =
ToDouble(integer_trail_->LowerBound(integer_variables_[index]));
const double ub =
ToDouble(integer_trail_->UpperBound(integer_variables_[index]));
const double value = lp_solution_[index];
if (value < lb - kCpEpsilon || value > ub + kCpEpsilon) return Propagate();
}
// TODO(user): The saved lp solution is still valid given the current variable
// bounds, so the LP optimal didn't change. However we might still want to add
// new cuts or new lazy constraints?
//
// TODO(user): Propagate the last optimal_constraint? Note that we need
// to be careful since the reversible int in IntegerSumLE are not registered.
// However, because we delete "optimalconstraints" on backtrack, we might not
// care.
return true;
}
glop::Fractional LinearProgrammingConstraint::GetVariableValueAtCpScale(
glop::ColIndex var) {
return scaler_.UnscaleVariableValue(var, simplex_.GetVariableValue(var));
}
double LinearProgrammingConstraint::GetSolutionValue(
IntegerVariable variable) const {
return lp_solution_[gtl::FindOrDie(mirror_lp_variable_, variable).value()];
}
double LinearProgrammingConstraint::GetSolutionReducedCost(
IntegerVariable variable) const {
return lp_reduced_cost_[gtl::FindOrDie(mirror_lp_variable_, variable)
.value()];
}
void LinearProgrammingConstraint::UpdateBoundsOfLpVariables() {
const int num_vars = integer_variables_.size();
for (int i = 0; i < num_vars; i++) {
const IntegerVariable cp_var = integer_variables_[i];
const double lb = ToDouble(integer_trail_->LowerBound(cp_var));
const double ub = ToDouble(integer_trail_->UpperBound(cp_var));
const double factor = scaler_.VariableScalingFactor(glop::ColIndex(i));
lp_data_.SetVariableBounds(glop::ColIndex(i), lb * factor, ub * factor);
}
}
bool LinearProgrammingConstraint::SolveLp() {
if (trail_->CurrentDecisionLevel() == 0) {
lp_at_level_zero_is_final_ = false;
}
const auto status = simplex_.Solve(lp_data_, time_limit_);
total_num_simplex_iterations_ += simplex_.GetNumberOfIterations();
if (!status.ok()) {
VLOG(1) << "The LP solver encountered an error: " << status.error_message();
simplex_.ClearStateForNextSolve();
return false;
}
average_degeneracy_.AddData(CalculateDegeneracy());
if (average_degeneracy_.CurrentAverage() >= 1000.0) {
VLOG(2) << "High average degeneracy: "
<< average_degeneracy_.CurrentAverage();
}
if (simplex_.GetProblemStatus() == glop::ProblemStatus::OPTIMAL) {
lp_solution_is_set_ = true;
lp_solution_level_ = trail_->CurrentDecisionLevel();
const int num_vars = integer_variables_.size();
for (int i = 0; i < num_vars; i++) {
const glop::Fractional value =
GetVariableValueAtCpScale(glop::ColIndex(i));
lp_solution_[i] = value;
expanded_lp_solution_[integer_variables_[i]] = value;
expanded_lp_solution_[NegationOf(integer_variables_[i])] = -value;
}
if (lp_solution_level_ == 0) {
level_zero_lp_solution_ = lp_solution_;
}
}
return true;
}
LinearConstraint LinearProgrammingConstraint::ConvertToLinearConstraint(
const gtl::ITIVector<ColIndex, IntegerValue>& dense_vector,
IntegerValue upper_bound) {
LinearConstraint result;
for (ColIndex col(0); col < dense_vector.size(); ++col) {
const IntegerValue coeff = dense_vector[col];
if (coeff == 0) continue;
const IntegerVariable var = integer_variables_[col.value()];
result.vars.push_back(var);
result.coeffs.push_back(coeff);
}
result.lb = kMinIntegerValue;
result.ub = upper_bound;
return result;
}
namespace {
// Returns false in case of overflow
bool AddLinearExpressionMultiple(
IntegerValue multiplier,
const std::vector<std::pair<ColIndex, IntegerValue>>& terms,
gtl::ITIVector<ColIndex, IntegerValue>* dense_vector) {
for (const std::pair<ColIndex, IntegerValue> term : terms) {
if (!AddProductTo(multiplier, term.second, &(*dense_vector)[term.first])) {
return false;
}
}
return true;
}
} // namespace
bool LinearProgrammingConstraint::AddCutFromConstraints(
const std::string& name,
const std::vector<std::pair<RowIndex, IntegerValue>>& integer_multipliers) {
// This is initialized to a valid linear contraint (by taking linear
// combination of the LP rows) and will be transformed into a cut if
// possible.
//
// TODO(user): For CG cuts, Ideally this linear combination should have only
// one fractional variable (basis_col). But because of imprecision, we get a
// bunch of fractional entry with small coefficient (relative to the one of
// basis_col). We try to handle that in IntegerRoundingCut(), but it might be
// better to add small multiple of the involved rows to get rid of them.
LinearConstraint cut;
{
gtl::ITIVector<ColIndex, IntegerValue> dense_cut;
IntegerValue cut_ub;
if (!ComputeNewLinearConstraint(integer_multipliers, &dense_cut, &cut_ub)) {
VLOG(1) << "Issue, overflow!";
return false;
}
// Important: because we use integer_multipliers below, we cannot just
// divide by GCD or call PreventOverflow() here.
cut = ConvertToLinearConstraint(dense_cut, cut_ub);
}
// This should be tight.
const double norm = ToDouble(ComputeInfinityNorm(cut));
if (std::abs(ComputeActivity(cut, expanded_lp_solution_) - ToDouble(cut.ub)) /
norm >
1e-4) {
VLOG(1) << "Cut not tight " << ComputeActivity(cut, expanded_lp_solution_)
<< " " << ToDouble(cut.ub);
return false;
}
// Unlike for the knapsack cuts, it might not be always beneficial to
// process the implied bounds even though it seems to be better in average.
//
// TODO(user): Understand & investigate more. In particular, it doesn't seems
// like MIP solvers use implied bounds in the same way. The substitution they
// perform seems to be NewVar = OldVar - (lb + Binary * diff) and not just
// OldVar >= (lb + Binary * diff).
implied_bounds_processor_.ProcessUpperBoundedConstraint(expanded_lp_solution_,
&cut);
// TODO(user): Might be cleaner to only do that in the functions that needs
// this precondition below.
MakeAllVariablesPositive(&cut);
// Fills data for IntegerRoundingCut().
//
// Note(user): we use the current bound here, so the reasonement will only
// produce locally valid cut if we call this at a non-root node. We could
// use the level zero bounds if we wanted to generate a globally valid cut
// at another level, but we will likely not genereate a constraint violating
// the current lp solution in that case.
std::vector<double> lp_values;
std::vector<IntegerValue> var_lbs;
std::vector<IntegerValue> var_ubs;
for (const IntegerVariable var : cut.vars) {
lp_values.push_back(expanded_lp_solution_[var]);
var_lbs.push_back(integer_trail_->LowerBound(var));
var_ubs.push_back(integer_trail_->UpperBound(var));
}
// Add slack.
// definition: integer_lp_[row] + slack_row == bound;
const IntegerVariable first_slack(expanded_lp_solution_.size());
std::vector<RowIndex> slack_rows;
std::vector<IntegerValue> slack_bounds;
for (const auto pair : integer_multipliers) {
const RowIndex row = pair.first;
const IntegerValue coeff = pair.second;
const auto status = simplex_.GetConstraintStatus(row);
if (status == glop::ConstraintStatus::FIXED_VALUE) continue;
lp_values.push_back(0.0);
cut.vars.push_back(first_slack + IntegerVariable(slack_rows.size()));
slack_rows.push_back(row);
cut.coeffs.push_back(coeff);
const IntegerValue diff(
CapSub(integer_lp_[row].ub.value(), integer_lp_[row].lb.value()));
if (coeff > 0) {
slack_bounds.push_back(integer_lp_[row].ub);
var_lbs.push_back(IntegerValue(0));
var_ubs.push_back(diff);
} else {
slack_bounds.push_back(integer_lp_[row].lb);
var_lbs.push_back(-diff);
var_ubs.push_back(IntegerValue(0));
}
}
// Get the cut using some integer rounding heuristic.
RoundingOptions options;
options.max_scaling = sat_parameters_.max_integer_rounding_scaling();
IntegerRoundingCut(options, lp_values, var_lbs, var_ubs, &cut);
// Compute the activity. Warning: the cut no longer have the same size so we
// cannot use lp_values. Note that the substitution below shouldn't change
// the activity by definition.
double activity = 0.0;
for (int i = 0; i < cut.vars.size(); ++i) {
if (cut.vars[i] < first_slack) {
activity += ToDouble(cut.coeffs[i]) * expanded_lp_solution_[cut.vars[i]];
}
}
const double kMinViolation = 1e-4;
const double violation = activity - ToDouble(cut.ub);
if (violation < kMinViolation) {
VLOG(3) << "Bad cut " << activity << " <= " << ToDouble(cut.ub);
return false;
}
// Substitute any slack left.
{
int num_slack = 0;
gtl::ITIVector<ColIndex, IntegerValue> dense_cut(integer_variables_.size(),
IntegerValue(0));
IntegerValue cut_ub = cut.ub;
bool overflow = false;
for (int i = 0; i < cut.vars.size(); ++i) {
// Simple copy for non-slack variables.
if (cut.vars[i] < first_slack) {
CHECK(VariableIsPositive(cut.vars[i]));
const glop::ColIndex col =
gtl::FindOrDie(mirror_lp_variable_, cut.vars[i]);
dense_cut[col] = cut.coeffs[i];
continue;
}
++num_slack;
// Update the constraint.
const int slack_index = cut.vars[i].value() - first_slack.value();
const glop::RowIndex row = slack_rows[slack_index];
const IntegerValue multiplier = -cut.coeffs[i];
if (!AddLinearExpressionMultiple(multiplier, integer_lp_[row].terms,
&dense_cut)) {
overflow = true;
break;
}
// Update rhs.
if (!AddProductTo(multiplier, slack_bounds[slack_index], &cut_ub)) {
overflow = true;
break;
}
}
if (overflow) {
VLOG(1) << "Overflow in slack removal.";
return false;
}
VLOG(3) << " num_slack: " << num_slack;
cut = ConvertToLinearConstraint(dense_cut, cut_ub);
}
const double new_violation =
ComputeActivity(cut, expanded_lp_solution_) - ToDouble(cut.ub);
if (std::abs(violation - new_violation) >= 1e-4) {
VLOG(1) << "Violation discrepancy after slack removal. "
<< " before = " << violation << " after = " << new_violation;
}
DivideByGCD(&cut);
return constraint_manager_.AddCut(cut, name, expanded_lp_solution_);
}
void LinearProgrammingConstraint::AddCGCuts() {
CHECK_EQ(trail_->CurrentDecisionLevel(), 0);
const RowIndex num_rows = lp_data_.num_constraints();
for (RowIndex row(0); row < num_rows; ++row) {
ColIndex basis_col = simplex_.GetBasis(row);
const Fractional lp_value = GetVariableValueAtCpScale(basis_col);
// TODO(user): We could just look at the diff with std::floor() in the hope
// that when we are just under an integer, the exact computation below will
// also be just under it.
if (std::abs(lp_value - std::round(lp_value)) < 0.01) continue;
// If this variable is a slack, we ignore it. This is because the
// corresponding row is not tight under the given lp values.
if (basis_col >= integer_variables_.size()) continue;
const glop::ScatteredRow& lambda = simplex_.GetUnitRowLeftInverse(row);
glop::DenseColumn lp_multipliers(num_rows, 0.0);
double magnitude = 0.0;
int num_non_zeros = 0;
for (RowIndex row(0); row < num_rows; ++row) {
lp_multipliers[row] = lambda.values[glop::RowToColIndex(row)];
if (std::abs(lp_multipliers[row]) < 1e-12) {
lp_multipliers[row] = 0.0;
continue;
}
// There should be no BASIC status, but they could be imprecision
// in the GetUnitRowLeftInverse() code? not sure, so better be safe.
const auto status = simplex_.GetConstraintStatus(row);
if (status == glop::ConstraintStatus::BASIC) {
VLOG(1) << "BASIC row not expected! " << lp_multipliers[row];
lp_multipliers[row] = 0.0;
}
magnitude = std::max(magnitude, std::abs(lp_multipliers[row]));
if (lp_multipliers[row] != 0.0) ++num_non_zeros;
}
if (num_non_zeros == 0) continue;
Fractional scaling;
for (int i = 0; i < 2; ++i) {
if (i == 1) {
// Try other sign.
//
// TODO(user): Maybe add an heuristic to know beforehand which sign to
// use?
for (RowIndex row(0); row < num_rows; ++row) {
lp_multipliers[row] = -lp_multipliers[row];
}
}
// TODO(user): We use a lower value here otherwise we might run into
// overflow while computing the cut. This should be fixable.
const std::vector<std::pair<RowIndex, IntegerValue>> integer_multipliers =
ScaleLpMultiplier(/*take_objective_into_account=*/false,
lp_multipliers, &scaling, /*max_pow=*/52);
AddCutFromConstraints("CG", integer_multipliers);
}
}
}
namespace {
// For each element of a, adds a random one in b and append the pair to output.
void RandomPick(const std::vector<RowIndex>& a, const std::vector<RowIndex>& b,
ModelRandomGenerator* random,
std::vector<std::pair<RowIndex, RowIndex>>* output) {
if (a.empty() || b.empty()) return;
for (const RowIndex row : a) {
const RowIndex other = b[absl::Uniform<int>(*random, 0, b.size() - 1)];
if (other != row) {
output->push_back({row, other});
}
}
}
template <class ListOfTerms>
IntegerValue GetCoeff(ColIndex col, const ListOfTerms& terms) {
for (const auto& term : terms) {
if (term.first == col) return term.second;
}
return IntegerValue(0);
}
} // namespace
void LinearProgrammingConstraint::AddMirCuts() {
CHECK_EQ(trail_->CurrentDecisionLevel(), 0);
// Heuristic to generate MIR_n cuts by combining a small number of rows. This
// works greedily and follow more or less the MIR cut description in the
// literature. We have a current cut, and we add one more row to it while
// eliminating a variable of the current cut whose LP value is far from its
// bound.
//
// A notable difference is that we randomize the variable we eliminate and
// the row we use to do so. We still have weights to indicate our preferred
// choices. This allows to generate different cuts when called again and
// again.
//
// TODO(user): We could combine n rows to make sure we eliminate n variables
// far away from their bounds by solving exactly in integer small linear
// system.
gtl::ITIVector<ColIndex, IntegerValue> dense_cut(integer_variables_.size(),
IntegerValue(0));
SparseBitset<ColIndex> non_zeros(ColIndex(integer_variables_.size()));
// We compute all the rows that are tight, these will be used as the base row
// for the MIR_n procedure below.
const RowIndex num_rows = lp_data_.num_constraints();
std::vector<std::pair<RowIndex, IntegerValue>> base_rows;
gtl::ITIVector<RowIndex, double> row_weights(num_rows.value(), 0.0);
for (RowIndex row(0); row < num_rows; ++row) {
const auto status = simplex_.GetConstraintStatus(row);
if (status == glop::ConstraintStatus::BASIC) continue;
if (status == glop::ConstraintStatus::FREE) continue;
if (status == glop::ConstraintStatus::AT_UPPER_BOUND ||
status == glop::ConstraintStatus::FIXED_VALUE) {
base_rows.push_back({row, IntegerValue(1)});
}
if (status == glop::ConstraintStatus::AT_LOWER_BOUND ||
status == glop::ConstraintStatus::FIXED_VALUE) {
base_rows.push_back({row, IntegerValue(-1)});
}
// For now, we use the dual values for the row "weights".
//
// Note that we use the dual at LP scale so that it make more sense when we
// compare different rows since the LP has been scaled.
//
// TODO(user): In Kati Wolter PhD "Implementation of Cutting Plane
// Separators for Mixed Integer Programs" which describe SCIP's MIR cuts
// implementation (or at least an early version of it), a more complex score
// is used.
row_weights[row] = std::abs(simplex_.GetDualValue(row));
}
std::vector<double> weights;
std::vector<std::pair<RowIndex, IntegerValue>> integer_multipliers;
for (const std::pair<RowIndex, IntegerValue>& entry : base_rows) {
// First try to generate a cut directly from this base row (MIR1).
//
// Note(user): We abort on success like it seems to be done in the
// literature. Note that we don't succeed that often in generating an
// efficient cut, so I am not sure aborting will make a big difference
// speedwise. We might generate similar cuts though, but hopefully the cut
// management can deal with that.
integer_multipliers = {entry};
if (AddCutFromConstraints("MIR_1", integer_multipliers)) {
continue;
}
// Cleanup.
for (const ColIndex col : non_zeros.PositionsSetAtLeastOnce()) {
dense_cut[col] = IntegerValue(0);
}
non_zeros.SparseClearAll();
// Copy cut.
const IntegerValue multiplier = entry.second;
for (const std::pair<ColIndex, IntegerValue> term :
integer_lp_[entry.first].terms) {
const ColIndex col = term.first;
const IntegerValue coeff = term.second;
non_zeros.Set(col);
dense_cut[col] += coeff * multiplier;
}
gtl::ITIVector<RowIndex, bool> used_rows(num_rows.value(), false);
used_rows[entry.first] = true;
// We will aggregate at most kMaxAggregation more rows.
//
// TODO(user): optim + tune.
const int kMaxAggregation = 5;
for (int i = 0; i < kMaxAggregation; ++i) {
// First pick a variable to eliminate. We currently pick a random one with
// a weight that depend on how far it is from its closest bound.
IntegerValue max_magnitude(0);
weights.clear();
std::vector<ColIndex> col_candidates;
for (const ColIndex col : non_zeros.PositionsSetAtLeastOnce()) {
if (dense_cut[col] == 0) continue;
max_magnitude = std::max(max_magnitude, IntTypeAbs(dense_cut[col]));
const int col_degree =
lp_data_.GetSparseColumn(col).num_entries().value();
if (col_degree <= 1) continue;
if (simplex_.GetVariableStatus(col) != glop::VariableStatus::BASIC) {
continue;
}
const IntegerVariable var = integer_variables_[col.value()];
const double lp_value = expanded_lp_solution_[var];
const double lb = ToDouble(integer_trail_->LowerBound(var));
const double ub = ToDouble(integer_trail_->UpperBound(var));
const double bound_distance = std::min(ub - lp_value, lp_value - lb);
if (bound_distance > 1e-2) {
weights.push_back(bound_distance);
col_candidates.push_back(col);
}
}
if (col_candidates.empty()) break;
const ColIndex var_to_eliminate =
col_candidates[std::discrete_distribution<>(weights.begin(),
weights.end())(*random_)];
// What rows can we add to eliminate var_to_eliminate?
std::vector<RowIndex> possible_rows;
weights.clear();
for (const auto entry : lp_data_.GetSparseColumn(var_to_eliminate)) {
const RowIndex row = entry.row();
const auto status = simplex_.GetConstraintStatus(row);
if (status == glop::ConstraintStatus::BASIC) continue;
if (status == glop::ConstraintStatus::FREE) continue;
// We dissalow all the rows that contain a variable that we already
// eliminated (or are about to). This mean that we choose rows that
// form a "triangular" matrix on the position we choose to eliminate.
if (used_rows[row]) continue;
used_rows[row] = true;
// TODO(user): Instead of using FIXED_VALUE consider also both direction
// when we almost have an equality? that is if the LP constraints bounds
// are close from each others (<1e-6 ?). Initial experiments shows it
// doesn't change much, so I kept this version for now. Note that it
// might just be better to use the side that constrain the current lp
// optimal solution (that we get from the status).
bool add_row = false;
if (status == glop::ConstraintStatus::FIXED_VALUE ||
status == glop::ConstraintStatus::AT_UPPER_BOUND) {
if (entry.coefficient() > 0.0) {
if (dense_cut[var_to_eliminate] < 0) add_row = true;
} else {
if (dense_cut[var_to_eliminate] > 0) add_row = true;
}
}
if (status == glop::ConstraintStatus::FIXED_VALUE ||
status == glop::ConstraintStatus::AT_LOWER_BOUND) {
if (entry.coefficient() > 0.0) {
if (dense_cut[var_to_eliminate] > 0) add_row = true;
} else {
if (dense_cut[var_to_eliminate] < 0) add_row = true;
}
}
if (add_row) {
possible_rows.push_back(row);
weights.push_back(row_weights[row]);
}
}
if (possible_rows.empty()) break;
const RowIndex row_to_combine =
possible_rows[std::discrete_distribution<>(weights.begin(),
weights.end())(*random_)];
const IntegerValue to_combine_coeff =
GetCoeff(var_to_eliminate, integer_lp_[row_to_combine].terms);
CHECK_NE(to_combine_coeff, 0);
IntegerValue mult1 = -to_combine_coeff;
IntegerValue mult2 = dense_cut[var_to_eliminate];
CHECK_NE(mult2, 0);
if (mult1 < 0) {
mult1 = -mult1;
mult2 = -mult2;
}
const IntegerValue gcd = IntegerValue(
MathUtil::GCD64(std::abs(mult1.value()), std::abs(mult2.value())));
CHECK_NE(gcd, 0);
mult1 /= gcd;
mult2 /= gcd;
// Overflow detection.
//
// TODO(user): do that in the possible_rows selection? only problem is
// that we do not have the integer coefficient there...
if (CapAdd(CapProd(max_magnitude.value(), std::abs(mult1.value())),
CapProd(infinity_norms_[row_to_combine].value(),
std::abs(mult2.value()))) == kint64max) {
break;
}
for (std::pair<RowIndex, IntegerValue>& entry : integer_multipliers) {
entry.second *= mult1;
}
integer_multipliers.push_back({row_to_combine, mult2});
// TODO(user): Not supper efficient to recombine the rows.
if (AddCutFromConstraints(absl::StrCat("MIR_", i + 2),
integer_multipliers)) {
break;
}
// Minor optim: the computation below is only needed if we do one more
// iteration.
if (i + 1 == kMaxAggregation) break;
for (ColIndex col : non_zeros.PositionsSetAtLeastOnce()) {
dense_cut[col] *= mult1;
}
for (const std::pair<ColIndex, IntegerValue> term :
integer_lp_[row_to_combine].terms) {
const ColIndex col = term.first;
const IntegerValue coeff = term.second;
non_zeros.Set(col);
dense_cut[col] += coeff * mult2;
}
}
}
}
void LinearProgrammingConstraint::UpdateSimplexIterationLimit(
const int64 min_iter, const int64 max_iter) {
if (sat_parameters_.linearization_level() < 2) return;
const int64 num_degenerate_columns = CalculateDegeneracy();
const int64 num_cols = simplex_.GetProblemNumCols().value();
if (num_cols <= 0) {
return;
}
CHECK_GT(num_cols, 0);
const int64 decrease_factor = (10 * num_degenerate_columns) / num_cols;
if (simplex_.GetProblemStatus() == glop::ProblemStatus::DUAL_FEASIBLE) {
// We reached here probably because we predicted wrong. We use this as a
// signal to increase the iterations or punish less for degeneracy compare
// to the other part.
if (is_degenerate_) {
next_simplex_iter_ /= std::max(int64{1}, decrease_factor);
} else {
next_simplex_iter_ *= 2;
}
} else if (simplex_.GetProblemStatus() == glop::ProblemStatus::OPTIMAL) {
if (is_degenerate_) {
next_simplex_iter_ /= std::max(int64{1}, 2 * decrease_factor);
} else {
// This is the most common case. We use the size of the problem to
// determine the limit and ignore the previous limit.
next_simplex_iter_ = num_cols / 40;
}
}
next_simplex_iter_ =
std::max(min_iter, std::min(max_iter, next_simplex_iter_));
}
bool LinearProgrammingConstraint::Propagate() {
UpdateBoundsOfLpVariables();
// TODO(user): It seems the time we loose by not stopping early might be worth
// it because we end up with a better explanation at optimality.
glop::GlopParameters parameters = simplex_.GetParameters();
if (/* DISABLES CODE */ (false) && objective_is_defined_) {
// We put a limit on the dual objective since there is no point increasing
// it past our current objective upper-bound (we will already fail as soon
// as we pass it). Note that this limit is properly transformed using the
// objective scaling factor and offset stored in lp_data_.
//
// Note that we use a bigger epsilon here to be sure that if we abort
// because of this, we will report a conflict.
parameters.set_objective_upper_limit(
static_cast<double>(integer_trail_->UpperBound(objective_cp_).value() +
100.0 * kCpEpsilon));
}
// Put an iteration limit on the work we do in the simplex for this call. Note
// that because we are "incremental", even if we don't solve it this time we
// will make progress towards a solve in the lower node of the tree search.
if (trail_->CurrentDecisionLevel() == 0) {
// TODO(user): Dynamically change the iteration limit for root node as
// well.
parameters.set_max_number_of_iterations(2000);
} else {
parameters.set_max_number_of_iterations(next_simplex_iter_);
}
if (sat_parameters_.use_exact_lp_reason()) {
parameters.set_change_status_to_imprecise(false);
parameters.set_primal_feasibility_tolerance(1e-7);
parameters.set_dual_feasibility_tolerance(1e-7);
}
simplex_.SetParameters(parameters);
simplex_.NotifyThatMatrixIsUnchangedForNextSolve();
if (!SolveLp()) return true;
// Add new constraints to the LP and resolve?
//
// TODO(user): We might want to do that more than once. Currently we rely on
// this beeing called again on the next IncrementalPropagate() call, but that
// might not always happen at level zero.
if (simplex_.GetProblemStatus() == glop::ProblemStatus::OPTIMAL) {
// We wait for the first batch of problem constraints to be added before we
// begin to generate cuts.
if (!integer_lp_.empty() &&
constraint_manager_.num_cuts() < sat_parameters_.max_num_cuts()) {
// The "generic" cuts are currently part of this class as they are using
// data from the current LP.
//
// TODO(user): Refactor so that they are just normal cut generators?
if (trail_->CurrentDecisionLevel() == 0) {
if (sat_parameters_.add_mir_cuts()) AddMirCuts();
if (sat_parameters_.add_cg_cuts()) AddCGCuts();
}
// Try to add cuts.
if (!cut_generators_.empty() &&
(trail_->CurrentDecisionLevel() == 0 ||
!sat_parameters_.only_add_cuts_at_level_zero())) {
for (const CutGenerator& generator : cut_generators_) {
generator.generate_cuts(expanded_lp_solution_, &constraint_manager_);
}
}
}
glop::BasisState state = simplex_.GetState();
if (constraint_manager_.ChangeLp(expanded_lp_solution_, &state)) {
simplex_.LoadStateForNextSolve(state);
if (!CreateLpFromConstraintManager()) {
return integer_trail_->ReportConflict({});
}
const double old_obj = simplex_.GetObjectiveValue();
if (!SolveLp()) return true;
if (simplex_.GetProblemStatus() == glop::ProblemStatus::OPTIMAL) {
VLOG(1) << "Relaxation improvement " << old_obj << " -> "
<< simplex_.GetObjectiveValue()
<< " diff: " << simplex_.GetObjectiveValue() - old_obj
<< " level: " << trail_->CurrentDecisionLevel();
}
} else {
if (trail_->CurrentDecisionLevel() == 0) {
lp_at_level_zero_is_final_ = true;
}
}
}
// A dual-unbounded problem is infeasible. We use the dual ray reason.
if (simplex_.GetProblemStatus() == glop::ProblemStatus::DUAL_UNBOUNDED) {
if (sat_parameters_.use_exact_lp_reason()) {
if (!FillExactDualRayReason()) return true;
} else {
FillReducedCostReasonIn(simplex_.GetDualRayRowCombination(),
&integer_reason_);
}
return integer_trail_->ReportConflict(integer_reason_);
}
// TODO(user): Update limits for DUAL_UNBOUNDED status as well.
UpdateSimplexIterationLimit(/*min_iter=*/10, /*max_iter=*/1000);
// Optimality deductions if problem has an objective.
if (objective_is_defined_ &&
(simplex_.GetProblemStatus() == glop::ProblemStatus::OPTIMAL ||
simplex_.GetProblemStatus() == glop::ProblemStatus::DUAL_FEASIBLE)) {
// Try to filter optimal objective value. Note that GetObjectiveValue()
// already take care of the scaling so that it returns an objective in the
// CP world.
const double relaxed_optimal_objective = simplex_.GetObjectiveValue();
const IntegerValue approximate_new_lb(
static_cast<int64>(std::ceil(relaxed_optimal_objective - kCpEpsilon)));
// TODO(user): Maybe do a bit less computation when we cannot propagate
// anything.
if (sat_parameters_.use_exact_lp_reason()) {
if (!ExactLpReasonning()) return false;
// Display when the inexact bound would have propagated more.
const IntegerValue propagated_lb =
integer_trail_->LowerBound(objective_cp_);
if (approximate_new_lb > propagated_lb) {
VLOG(2) << "LP objective [ " << ToDouble(propagated_lb) << ", "
<< ToDouble(integer_trail_->UpperBound(objective_cp_))
<< " ] approx_lb += "
<< ToDouble(approximate_new_lb - propagated_lb) << " gap: "
<< integer_trail_->UpperBound(objective_cp_) - propagated_lb;
}
} else {
FillReducedCostReasonIn(simplex_.GetReducedCosts(), &integer_reason_);
const double objective_cp_ub =
ToDouble(integer_trail_->UpperBound(objective_cp_));
ReducedCostStrengtheningDeductions(objective_cp_ub -
relaxed_optimal_objective);
if (!deductions_.empty()) {
deductions_reason_ = integer_reason_;
deductions_reason_.push_back(
integer_trail_->UpperBoundAsLiteral(objective_cp_));
}
// Push new objective lb.
if (approximate_new_lb > integer_trail_->LowerBound(objective_cp_)) {
const IntegerLiteral deduction =
IntegerLiteral::GreaterOrEqual(objective_cp_, approximate_new_lb);
if (!integer_trail_->Enqueue(deduction, {}, integer_reason_)) {
return false;
}
}
// Push reduced cost strengthening bounds.
if (!deductions_.empty()) {
const int trail_index_with_same_reason = integer_trail_->Index();
for (const IntegerLiteral deduction : deductions_) {
if (!integer_trail_->Enqueue(deduction, {}, deductions_reason_,
trail_index_with_same_reason)) {
return false;
}
}
}
}
}
// Copy more info about the current solution.
if (simplex_.GetProblemStatus() == glop::ProblemStatus::OPTIMAL) {
CHECK(lp_solution_is_set_);
lp_objective_ = simplex_.GetObjectiveValue();
lp_solution_is_integer_ = true;
const int num_vars = integer_variables_.size();
for (int i = 0; i < num_vars; i++) {
lp_reduced_cost_[i] = scaler_.UnscaleReducedCost(
glop::ColIndex(i), simplex_.GetReducedCost(glop::ColIndex(i)));
if (std::abs(lp_solution_[i] - std::round(lp_solution_[i])) >
kCpEpsilon) {
lp_solution_is_integer_ = false;
}
}
if (compute_reduced_cost_averages_) {
const int num_vars = integer_variables_.size();
if (sum_cost_down_.size() < num_vars) {
VLOG(1) << " LPReducedCostAverageBranching has #variables: "
<< num_vars;
sum_cost_down_.resize(num_vars, 0.0);
num_cost_down_.resize(num_vars, 0);
sum_cost_up_.resize(num_vars, 0.0);
num_cost_up_.resize(num_vars, 0);
}
// Decay averages.
num_calls_since_reduced_cost_averages_reset_++;
if (num_calls_since_reduced_cost_averages_reset_ == 10000) {
for (int i = 0; i < num_vars; i++) {
sum_cost_up_[i] /= 2;
num_cost_up_[i] /= 2;
sum_cost_down_[i] /= 2;
num_cost_down_[i] /= 2;
}
num_calls_since_reduced_cost_averages_reset_ = 0;
}
// Accumulate pseudo-costs of all unassigned variables.
for (int i = 0; i < num_vars; i++) {
const IntegerVariable var = this->integer_variables_[i];
// Skip ignored and fixed variables.
if (integer_trail_->IsCurrentlyIgnored(var)) continue;
const IntegerValue lb = integer_trail_->LowerBound(var);
const IntegerValue ub = integer_trail_->UpperBound(var);
if (lb == ub) continue;
// Skip reduced costs that are zero or close.
const double rc = this->GetSolutionReducedCost(var);
if (std::abs(rc) < kCpEpsilon) continue;
if (rc < 0.0) {
sum_cost_down_[i] -= rc;
num_cost_down_[i]++;
} else {
sum_cost_up_[i] += rc;
num_cost_up_[i]++;
}
}
}
}
if (sat_parameters_.use_branching_in_lp() && objective_is_defined_ &&
trail_->CurrentDecisionLevel() == 0 && !is_degenerate_ &&
lp_solution_is_set_ && !lp_solution_is_integer_ &&
sat_parameters_.linearization_level() >= 2 &&
compute_reduced_cost_averages_ &&
simplex_.GetProblemStatus() == glop::ProblemStatus::OPTIMAL) {
count_since_last_branching_++;
if (count_since_last_branching_ < branching_frequency_) {
return true;
}
count_since_last_branching_ = 0;
bool branching_successful = false;
// Strong branching on top max_num_branches variable.
const int max_num_branches = 3;
const int num_vars = integer_variables_.size();
std::vector<std::pair<double, IntegerVariable>> branching_vars;
for (int i = 0; i < num_vars; ++i) {
const IntegerVariable var = integer_variables_[i];
const IntegerVariable positive_var = PositiveVariable(var);
// Skip non fractional variables.
const double current_value = GetSolutionValue(positive_var);
if (std::abs(current_value - std::round(current_value)) <= kCpEpsilon) {
continue;
}
// Skip ignored variables.
if (integer_trail_->IsCurrentlyIgnored(var)) continue;
// We can use any metric to select a variable to branch on. Reduced cost
// average is one of the most promissing metric. It captures the history
// of the objective bound improvement in LP due to changes in the given
// variable bounds.
//
// NOTE: We also experimented using PseudoCosts and most recent reduced
// cost as metrics but it doesn't give better results on benchmarks.
const double cost_i = GetCostFromAverageReducedCosts(i);
std::pair<double, IntegerVariable> branching_var =
std::make_pair(-cost_i, positive_var);
auto iterator = std::lower_bound(branching_vars.begin(),
branching_vars.end(), branching_var);
branching_vars.insert(iterator, branching_var);
if (branching_vars.size() > max_num_branches) {
branching_vars.resize(max_num_branches);
}
}
for (const std::pair<double, IntegerVariable>& branching_var :
branching_vars) {
const IntegerVariable positive_var = branching_var.second;
VLOG(2) << "Branching on: " << positive_var;
if (BranchOnVar(positive_var)) {
VLOG(2) << "Branching successful.";
branching_successful = true;
} else {
break;
}
}
if (!branching_successful) {
branching_frequency_ *= 2;
}
}
return true;
}
// Returns kMinIntegerValue in case of overflow.
//
// TODO(user): Because of PreventOverflow(), this should actually never happen.
IntegerValue LinearProgrammingConstraint::GetImpliedLowerBound(
const LinearConstraint& terms) const {
IntegerValue lower_bound(0);
const int size = terms.vars.size();
for (int i = 0; i < size; ++i) {
const IntegerVariable var = terms.vars[i];
const IntegerValue coeff = terms.coeffs[i];
CHECK_NE(coeff, 0);
const IntegerValue bound = coeff > 0 ? integer_trail_->LowerBound(var)
: integer_trail_->UpperBound(var);
if (!AddProductTo(bound, coeff, &lower_bound)) return kMinIntegerValue;
}
return lower_bound;
}
bool LinearProgrammingConstraint::PossibleOverflow(
const LinearConstraint& constraint) {
IntegerValue lower_bound(0);
const int size = constraint.vars.size();
for (int i = 0; i < size; ++i) {
const IntegerVariable var = constraint.vars[i];
const IntegerValue coeff = constraint.coeffs[i];
CHECK_NE(coeff, 0);
const IntegerValue bound = coeff > 0 ? integer_trail_->LowerBound(var)
: integer_trail_->UpperBound(var);
if (!AddProductTo(bound, coeff, &lower_bound)) {
return true;
}
}
const int64 slack = CapAdd(lower_bound.value(), -constraint.ub.value());
if (slack == kint64min || slack == kint64max) {
return true;
}
return false;
}
namespace {
absl::int128 FloorRatio128(absl::int128 x, IntegerValue positive_div) {
absl::int128 div128(positive_div.value());
absl::int128 result = x / div128;
if (result * div128 > x) return result - 1;
return result;
}
} // namespace
void LinearProgrammingConstraint::PreventOverflow(LinearConstraint* constraint,
int max_pow) {
// Compute the min/max possible partial sum.
//
// Note that since we currently only use this cut locally, it is okay to
// use the current lb/ub here to decide if we have an overflow or not. Below
// however, we do have to use the level zero lower bound.
double sum_min = std::min(0.0, ToDouble(-constraint->ub));
double sum_max = std::max(0.0, ToDouble(-constraint->ub));
const int size = constraint->vars.size();
for (int i = 0; i < size; ++i) {
const IntegerVariable var = constraint->vars[i];
const double coeff = ToDouble(constraint->coeffs[i]);
const double prod1 = coeff * ToDouble(integer_trail_->LowerBound(var));
const double prod2 = coeff * ToDouble(integer_trail_->UpperBound(var));
sum_min += std::min(0.0, std::min(prod1, prod2));
sum_max += std::max(0.0, std::max(prod1, prod2));
}
const double max_value = std::max(sum_max, -sum_min);
const IntegerValue divisor(std::ceil(std::ldexp(max_value, -max_pow)));
if (divisor <= 1) return;
// To be correct, we need to shift all variable so that they are positive.
//
// Important: One might be tempted to think that using the current variable
// bounds is okay here since we only use this to derive cut/constraint that
// only needs to be locally valid. However, in some corner cases (like when
// one term become zero), we might loose the fact that we used one of the
// variable bound to derive the new constraint, so we will miss it in the
// explanation !!
//
// TODO(user): This code is tricky and similar to the one to generate cuts.
// Test and may reduce the duplication? note however that here we use int128
// to deal with potential overflow.
int new_size = 0;
absl::int128 adjust = 0;
for (int i = 0; i < size; ++i) {
const IntegerValue old_coeff = constraint->coeffs[i];
const IntegerValue new_coeff = FloorRatio(old_coeff, divisor);
// Compute the rhs adjustement.
const absl::int128 remainder =
absl::int128(old_coeff.value()) -
absl::int128(new_coeff.value()) * absl::int128(divisor.value());
adjust +=
remainder *
absl::int128(
integer_trail_->LevelZeroLowerBound(constraint->vars[i]).value());
if (new_coeff == 0) continue;
constraint->vars[new_size] = constraint->vars[i];
constraint->coeffs[new_size] = new_coeff;
++new_size;
}
constraint->vars.resize(new_size);
constraint->coeffs.resize(new_size);
constraint->ub = IntegerValue(static_cast<int64>(
FloorRatio128(absl::int128(constraint->ub.value()) - adjust, divisor)));
}
// TODO(user): combine this with RelaxLinearReason() to avoid the extra
// magnitude vector and the weird precondition of RelaxLinearReason().
void LinearProgrammingConstraint::SetImpliedLowerBoundReason(
const LinearConstraint& terms, IntegerValue slack) {
integer_reason_.clear();
std::vector<IntegerValue> magnitudes;
const int size = terms.vars.size();
for (int i = 0; i < size; ++i) {
const IntegerVariable var = terms.vars[i];
const IntegerValue coeff = terms.coeffs[i];
CHECK_NE(coeff, 0);
if (coeff > 0) {
magnitudes.push_back(coeff);
integer_reason_.push_back(integer_trail_->LowerBoundAsLiteral(var));
} else {
magnitudes.push_back(-coeff);
integer_reason_.push_back(integer_trail_->UpperBoundAsLiteral(var));
}
}
CHECK_GE(slack, 0);
if (slack > 0) {
integer_trail_->RelaxLinearReason(slack, magnitudes, &integer_reason_);
}
integer_trail_->RemoveLevelZeroBounds(&integer_reason_);
}
// TODO(user): Provide a sparse interface.
std::vector<std::pair<RowIndex, IntegerValue>>
LinearProgrammingConstraint::ScaleLpMultiplier(
bool take_objective_into_account,
const glop::DenseColumn& dense_lp_multipliers, Fractional* scaling,
int max_pow) const {
double max_sum = 0.0;
std::vector<std::pair<RowIndex, Fractional>> cp_multipliers;
for (RowIndex row(0); row < dense_lp_multipliers.size(); ++row) {
const Fractional lp_multi = dense_lp_multipliers[row];
// We ignore small values since these are likely errors and will not
// contribute much to the new lp constraint anyway.
if (std::abs(lp_multi) < 1e-12) continue;
// Remove trivial bad cases.
//
// TODO(user): It might be better (when possible) to use the OPTIMAL row
// status since in most situation we do want the constraint we add to be
// tight under the current LP solution. Only for infeasible problem we might
// not have access to the status.
if (lp_multi > 0.0 && integer_lp_[row].ub >= kMaxIntegerValue) {
continue;
}
if (lp_multi < 0.0 && integer_lp_[row].lb <= kMinIntegerValue) {
continue;
}
const Fractional cp_multi = scaler_.UnscaleDualValue(row, lp_multi);
cp_multipliers.push_back({row, cp_multi});
max_sum += ToDouble(infinity_norms_[row]) * std::abs(cp_multi);
}
// This behave exactly like if we had another "objective" constraint with
// an lp_multi of 1.0 and a cp_multi of 1.0.
if (take_objective_into_account) {
max_sum += ToDouble(objective_infinity_norm_);
}
std::vector<std::pair<RowIndex, IntegerValue>> integer_multipliers;
if (max_sum == 0.0) {
// Empty linear combinaison.
*scaling = 1;
return integer_multipliers;
}
// We want max_sum * scaling to be <= 2 ^ max_pow and fit on an int64.
// We use a power of 2 as this seems to work better.
const double threshold = std::ldexp(1, max_pow) / max_sum;
*scaling = 1.0;
while (2 * *scaling <= threshold) *scaling *= 2;
// Scale the multipliers by *scaling.
//
// TODO(user): Maybe use int128 to avoid overflow?
for (const auto entry : cp_multipliers) {
const IntegerValue coeff(std::round(entry.second * (*scaling)));
if (coeff != 0) integer_multipliers.push_back({entry.first, coeff});
}
return integer_multipliers;
}
bool LinearProgrammingConstraint::ComputeNewLinearConstraint(
const std::vector<std::pair<RowIndex, IntegerValue>>& integer_multipliers,
gtl::ITIVector<ColIndex, IntegerValue>* dense_terms,
IntegerValue* upper_bound) const {
// Initialize the new constraint.
*upper_bound = 0;
dense_terms->assign(integer_variables_.size(), IntegerValue(0));
// Compute the new constraint by taking the linear combination given by
// integer_multipliers of the integer constraints in integer_lp_.
const ColIndex num_cols(integer_variables_.size());
for (const std::pair<RowIndex, IntegerValue> term : integer_multipliers) {
const RowIndex row = term.first;
const IntegerValue multiplier = term.second;
CHECK_LT(row, integer_lp_.size());
// Update the constraint.
if (!AddLinearExpressionMultiple(multiplier, integer_lp_[row].terms,
dense_terms)) {
return false;
}
// Update the upper bound.
const IntegerValue bound =
multiplier > 0 ? integer_lp_[row].ub : integer_lp_[row].lb;
if (!AddProductTo(multiplier, bound, upper_bound)) return false;
}
return true;
}
// TODO(user): no need to update the multipliers.
void LinearProgrammingConstraint::AdjustNewLinearConstraint(
std::vector<std::pair<glop::RowIndex, IntegerValue>>* integer_multipliers,
gtl::ITIVector<ColIndex, IntegerValue>* dense_terms,
IntegerValue* upper_bound) const {
const IntegerValue kMaxWantedCoeff(1e18);
for (std::pair<RowIndex, IntegerValue>& term : *integer_multipliers) {
const RowIndex row = term.first;
const IntegerValue multiplier = term.second;
if (multiplier == 0) continue;
// We will only allow change of the form "multiplier += to_add" with to_add
// in [-negative_limit, positive_limit].
IntegerValue negative_limit = kMaxWantedCoeff;
IntegerValue positive_limit = kMaxWantedCoeff;
// Make sure we never change the sign of the multiplier, except if the
// row is an equality in which case we don't care.
if (integer_lp_[row].ub != integer_lp_[row].lb) {
if (multiplier > 0) {
negative_limit = std::min(negative_limit, multiplier);
} else {
positive_limit = std::min(positive_limit, -multiplier);
}
}
// Make sure upper_bound + to_add * row_bound never overflow.
const IntegerValue row_bound =
multiplier > 0 ? integer_lp_[row].ub : integer_lp_[row].lb;
if (row_bound != 0) {
const IntegerValue limit1 = FloorRatio(
std::max(IntegerValue(0), kMaxWantedCoeff - IntTypeAbs(*upper_bound)),
IntTypeAbs(row_bound));
const IntegerValue limit2 =
FloorRatio(kMaxWantedCoeff, IntTypeAbs(row_bound));
if (*upper_bound > 0 == row_bound > 0) { // Same sign.
positive_limit = std::min(positive_limit, limit1);
negative_limit = std::min(negative_limit, limit2);
} else {
negative_limit = std::min(negative_limit, limit1);
positive_limit = std::min(positive_limit, limit2);
}
}
// If we add the row to the dense_terms, diff will indicate by how much
// |upper_bound - ImpliedLB(dense_terms)| will change. That correspond to
// increasing the multiplier by 1.
//
// At this stage, we are not sure computing sum coeff * bound will not
// overflow, so we use floating point numbers. It is fine to do so since
// this is not directly involved in the actual exact constraint generation:
// these variables are just used in an heuristic.
double positive_diff = ToDouble(row_bound);
double negative_diff = ToDouble(row_bound);
// TODO(user): we could relax a bit some of the condition and allow a sign
// change. It is just trickier to compute the diff when we allow such
// changes.
for (const auto entry : integer_lp_[row].terms) {
const ColIndex col = entry.first;
const IntegerValue coeff = entry.second;
const IntegerValue abs_coef = IntTypeAbs(coeff);
CHECK_NE(coeff, 0);
const IntegerVariable var = integer_variables_[col.value()];
const IntegerValue lb = integer_trail_->LowerBound(var);
const IntegerValue ub = integer_trail_->UpperBound(var);
// Moving a variable away from zero seems to improve the bound even
// if it reduces the number of non-zero. Note that this is because of
// this that positive_diff and negative_diff are not the same.
const IntegerValue current = (*dense_terms)[col];
if (current == 0) {
const IntegerValue overflow_limit(
FloorRatio(kMaxWantedCoeff, abs_coef));
positive_limit = std::min(positive_limit, overflow_limit);
negative_limit = std::min(negative_limit, overflow_limit);
if (coeff > 0) {
positive_diff -= ToDouble(coeff) * ToDouble(lb);
negative_diff -= ToDouble(coeff) * ToDouble(ub);
} else {
positive_diff -= ToDouble(coeff) * ToDouble(ub);
negative_diff -= ToDouble(coeff) * ToDouble(lb);
}
continue;
}
// We don't want to change the sign of current (except if the variable is
// fixed) or to have an overflow.
IntegerValue before_sign_change = kMaxWantedCoeff;
if (lb != ub) {
before_sign_change = FloorRatio(IntTypeAbs(current), abs_coef);
}
const IntegerValue overflow_limit(
FloorRatio(kMaxWantedCoeff - IntTypeAbs(current), abs_coef));
if (current > 0 == coeff > 0) { // Same sign.
negative_limit = std::min(negative_limit, before_sign_change);
positive_limit = std::min(positive_limit, overflow_limit);
} else {
negative_limit = std::min(negative_limit, overflow_limit);
positive_limit = std::min(positive_limit, before_sign_change);
}
// This is how diff change.
const IntegerValue implied = current > 0 ? lb : ub;
if (implied != 0) {
positive_diff -= ToDouble(coeff) * ToDouble(implied);
negative_diff -= ToDouble(coeff) * ToDouble(implied);
}
}
// Only add a multiple of this row if it tighten the final constraint.
// The positive_diff/negative_diff are supposed to be integer modulo the
// double precision, so we only add a multiple if they seems far away from
// zero.
IntegerValue to_add(0);
if (positive_diff <= -1.0 && positive_limit > 0) {
to_add = positive_limit;
}
if (negative_diff >= 1.0 && negative_limit > 0) {
// Pick this if it is better than the positive sign.
if (to_add == 0 ||
std::abs(ToDouble(negative_limit) * negative_diff) >
std::abs(ToDouble(positive_limit) * positive_diff)) {
to_add = -negative_limit;
}
}
if (to_add != 0) {
term.second += to_add;
*upper_bound += to_add * row_bound;
for (const auto entry : integer_lp_[row].terms) {
const ColIndex col = entry.first;
const IntegerValue coeff = entry.second;
(*dense_terms)[col] += to_add * coeff;
}
}
}
}
// The "exact" computation go as follow:
//
// Given any INTEGER linear combination of the LP constraints, we can create a
// new integer constraint that is valid (its computation must not overflow
// though). Lets call this "linear_combination <= ub". We can then always add to
// it the inequality "objective_terms <= objective_var", so we get:
// ImpliedLB(objective_terms + linear_combination) - ub <= objective_var.
// where ImpliedLB() is computed from the variable current bounds.
//
// Now, if we use for the linear combination and approximation of the optimal
// negated dual LP values (by scaling them and rounding them to integer), we
// will get an EXACT objective lower bound that is more or less the same as the
// inexact bound given by the LP relaxation. This allows to derive exact reasons
// for any propagation done by this constraint.
bool LinearProgrammingConstraint::ExactLpReasonning() {
// Clear old reason and deductions.
integer_reason_.clear();
deductions_.clear();
deductions_reason_.clear();
// The row multipliers will be the negation of the LP duals.
//
// TODO(user): Provide and use a sparse API in Glop to get the duals.
const RowIndex num_rows = simplex_.GetProblemNumRows();
glop::DenseColumn lp_multipliers(num_rows);
for (RowIndex row(0); row < num_rows; ++row) {
lp_multipliers[row] = -simplex_.GetDualValue(row);
}
Fractional scaling;
std::vector<std::pair<RowIndex, IntegerValue>> integer_multipliers =
ScaleLpMultiplier(/*take_objective_into_account=*/true, lp_multipliers,
&scaling);
gtl::ITIVector<ColIndex, IntegerValue> reduced_costs;
IntegerValue rc_ub;
if (!ComputeNewLinearConstraint(integer_multipliers, &reduced_costs,
&rc_ub)) {
VLOG(1) << "Issue while computing the exact LP reason. Aborting.";
return true;
}
// The "objective constraint" behave like if the unscaled cp multiplier was
// 1.0, so we will multiply it by this number and add it to reduced_costs.
const IntegerValue obj_scale(std::round(scaling));
if (obj_scale == 0) {
VLOG(1) << "Overflow during exact LP reasoning. scaling=" << scaling;
return true;
}
CHECK(AddLinearExpressionMultiple(obj_scale, integer_objective_,
&reduced_costs));
AdjustNewLinearConstraint(&integer_multipliers, &reduced_costs, &rc_ub);
// Create the IntegerSumLE that will allow to propagate the objective and more
// generally do the reduced cost fixing.
LinearConstraint new_constraint =
ConvertToLinearConstraint(reduced_costs, rc_ub);
new_constraint.vars.push_back(objective_cp_);
new_constraint.coeffs.push_back(-obj_scale);
DivideByGCD(&new_constraint);
PreventOverflow(&new_constraint);
DCHECK(!PossibleOverflow(new_constraint));
DCHECK(constraint_manager_.DebugCheckConstraint(new_constraint));
IntegerSumLE* cp_constraint =
new IntegerSumLE({}, new_constraint.vars, new_constraint.coeffs,
new_constraint.ub, model_);
if (trail_->CurrentDecisionLevel() == 0) {
// Since we will never ask the reason for a constraint at level 0, we just
// keep the last one.
optimal_constraints_.clear();
}
optimal_constraints_.emplace_back(cp_constraint);
rev_optimal_constraints_size_ = optimal_constraints_.size();
return cp_constraint->Propagate();
}
bool LinearProgrammingConstraint::FillExactDualRayReason() {
Fractional scaling;
std::vector<std::pair<RowIndex, IntegerValue>> integer_multipliers =
ScaleLpMultiplier(/*take_objective_into_account=*/false,
simplex_.GetDualRay(), &scaling);
gtl::ITIVector<ColIndex, IntegerValue> dense_new_constraint;
IntegerValue new_constraint_ub;
if (!ComputeNewLinearConstraint(integer_multipliers, &dense_new_constraint,
&new_constraint_ub)) {
VLOG(1) << "Isse while computing the exact dual ray reason. Aborting.";
return false;
}
AdjustNewLinearConstraint(&integer_multipliers, &dense_new_constraint,
&new_constraint_ub);
LinearConstraint new_constraint =
ConvertToLinearConstraint(dense_new_constraint, new_constraint_ub);
DivideByGCD(&new_constraint);
PreventOverflow(&new_constraint);
DCHECK(!PossibleOverflow(new_constraint));
DCHECK(constraint_manager_.DebugCheckConstraint(new_constraint));
const IntegerValue implied_lb = GetImpliedLowerBound(new_constraint);
if (implied_lb <= new_constraint.ub) {
VLOG(1) << "LP exact dual ray not infeasible,"
<< " implied_lb: " << implied_lb.value() / scaling
<< " ub: " << new_constraint.ub.value() / scaling;
return false;
}
const IntegerValue slack = (implied_lb - new_constraint.ub) - 1;
SetImpliedLowerBoundReason(new_constraint, slack);
return true;
}
int64 LinearProgrammingConstraint::CalculateDegeneracy() {
const glop::ColIndex num_vars = simplex_.GetProblemNumCols();
int num_non_basic_with_zero_rc = 0;
for (glop::ColIndex i(0); i < num_vars; ++i) {
const double rc = simplex_.GetReducedCost(i);
if (rc != 0.0) continue;
if (simplex_.GetVariableStatus(i) == glop::VariableStatus::BASIC) {
continue;
}
num_non_basic_with_zero_rc++;
}
const int64 num_cols = simplex_.GetProblemNumCols().value();
is_degenerate_ = num_non_basic_with_zero_rc >= 0.3 * num_cols;
return num_non_basic_with_zero_rc;
}
void LinearProgrammingConstraint::ReducedCostStrengtheningDeductions(
double cp_objective_delta) {
deductions_.clear();
// TRICKY: while simplex_.GetObjectiveValue() use the objective scaling factor
// stored in the lp_data_, all the other functions like GetReducedCost() or
// GetVariableValue() do not.
const double lp_objective_delta =
cp_objective_delta / lp_data_.objective_scaling_factor();
const int num_vars = integer_variables_.size();
for (int i = 0; i < num_vars; i++) {
const IntegerVariable cp_var = integer_variables_[i];
const glop::ColIndex lp_var = glop::ColIndex(i);
const double rc = simplex_.GetReducedCost(lp_var);
const double value = simplex_.GetVariableValue(lp_var);
if (rc == 0.0) continue;
const double lp_other_bound = value + lp_objective_delta / rc;
const double cp_other_bound =
scaler_.UnscaleVariableValue(lp_var, lp_other_bound);
if (rc > kLpEpsilon) {
const double ub = ToDouble(integer_trail_->UpperBound(cp_var));
const double new_ub = std::floor(cp_other_bound + kCpEpsilon);
if (new_ub < ub) {
// TODO(user): Because rc > kLpEpsilon, the lower_bound of cp_var
// will be part of the reason returned by FillReducedCostsReason(), but
// we actually do not need it here. Same below.
const IntegerValue new_ub_int(static_cast<IntegerValue>(new_ub));
deductions_.push_back(IntegerLiteral::LowerOrEqual(cp_var, new_ub_int));
}
} else if (rc < -kLpEpsilon) {
const double lb = ToDouble(integer_trail_->LowerBound(cp_var));
const double new_lb = std::ceil(cp_other_bound - kCpEpsilon);
if (new_lb > lb) {
const IntegerValue new_lb_int(static_cast<IntegerValue>(new_lb));
deductions_.push_back(
IntegerLiteral::GreaterOrEqual(cp_var, new_lb_int));
}
}
}
}
namespace {
// Add a cut of the form Sum_{outgoing arcs from S} lp >= rhs_lower_bound.
//
// Note that we used to also add the same cut for the incoming arcs, but because
// of flow conservation on these problems, the outgoing flow is always the same
// as the incoming flow, so adding this extra cut doesn't seem relevant.
void AddOutgoingCut(int num_nodes, int subset_size,
const std::vector<bool>& in_subset,
const std::vector<int>& tails,
const std::vector<int>& heads,
const std::vector<Literal>& literals,
const std::vector<double>& literal_lp_values,
int64 rhs_lower_bound,
const gtl::ITIVector<IntegerVariable, double>& lp_values,
LinearConstraintManager* manager, Model* model) {
LinearConstraintBuilder outgoing(model, IntegerValue(rhs_lower_bound),
kMaxIntegerValue);
double sum_outgoing = 0.0;
// Add outgoing arcs, compute outgoing flow.
for (int i = 0; i < tails.size(); ++i) {
if (in_subset[tails[i]] && !in_subset[heads[i]]) {
sum_outgoing += literal_lp_values[i];
CHECK(outgoing.AddLiteralTerm(literals[i], IntegerValue(1)));
}
}
// A node is said to be optional if it can be excluded from the subcircuit,
// in which case there is a self-loop on that node.
// If there are optional nodes, use extended formula:
// sum(cut) >= 1 - optional_loop_in - optional_loop_out
// where optional_loop_in's node is in subset, optional_loop_out's is out.
// TODO(user): Favor optional loops fixed to zero at root.
int num_optional_nodes_in = 0;
int num_optional_nodes_out = 0;
int optional_loop_in = -1;
int optional_loop_out = -1;
for (int i = 0; i < tails.size(); ++i) {
if (tails[i] != heads[i]) continue;
if (in_subset[tails[i]]) {
num_optional_nodes_in++;
if (optional_loop_in == -1 ||
literal_lp_values[i] < literal_lp_values[optional_loop_in]) {
optional_loop_in = i;
}
} else {
num_optional_nodes_out++;
if (optional_loop_out == -1 ||
literal_lp_values[i] < literal_lp_values[optional_loop_out]) {
optional_loop_out = i;
}
}
}
if (num_optional_nodes_in + num_optional_nodes_out > 0) {
CHECK_EQ(rhs_lower_bound, 1);
// When all optionals of one side are excluded in lp solution, no cut.
if (num_optional_nodes_in == subset_size &&
(optional_loop_in == -1 ||
literal_lp_values[optional_loop_in] > 1.0 - 1e-6)) {
return;
}
if (num_optional_nodes_out == num_nodes - subset_size &&
(optional_loop_out == -1 ||
literal_lp_values[optional_loop_out] > 1.0 - 1e-6)) {
return;
}
// There is no mandatory node in subset, add optional_loop_in.
if (num_optional_nodes_in == subset_size) {
CHECK(
outgoing.AddLiteralTerm(literals[optional_loop_in], IntegerValue(1)));
sum_outgoing += literal_lp_values[optional_loop_in];
}
// There is no mandatory node out of subset, add optional_loop_out.
if (num_optional_nodes_out == num_nodes - subset_size) {
CHECK(outgoing.AddLiteralTerm(literals[optional_loop_out],
IntegerValue(1)));
sum_outgoing += literal_lp_values[optional_loop_out];
}
}
if (sum_outgoing < rhs_lower_bound - 1e-6) {
manager->AddCut(outgoing.Build(), "Circuit", lp_values);
}
}
} // namespace
// We roughly follow the algorithm described in section 6 of "The Traveling
// Salesman Problem, A computational Study", David L. Applegate, Robert E.
// Bixby, Vasek Chvatal, William J. Cook.
//
// Note that this is mainly a "symmetric" case algo, but it does still work for
// the asymmetric case.
void SeparateSubtourInequalities(
int num_nodes, const std::vector<int>& tails, const std::vector<int>& heads,
const std::vector<Literal>& literals,
const gtl::ITIVector<IntegerVariable, double>& lp_values,
absl::Span<const int64> demands, int64 capacity,
LinearConstraintManager* manager, Model* model) {
if (num_nodes <= 2) return;
// We will collect only the arcs with a positive lp_values to speed up some
// computation below.
struct Arc {
int tail;
int head;
double lp_value;
};
std::vector<Arc> relevant_arcs;
// Sort the arcs by non-increasing lp_values.
std::vector<double> literal_lp_values(literals.size());
std::vector<std::pair<double, int>> arc_by_decreasing_lp_values;
auto* encoder = model->GetOrCreate<IntegerEncoder>();
for (int i = 0; i < literals.size(); ++i) {
double lp_value;
const IntegerVariable direct_view = encoder->GetLiteralView(literals[i]);
if (direct_view != kNoIntegerVariable) {
lp_value = lp_values[direct_view];
} else {
lp_value =
1.0 - lp_values[encoder->GetLiteralView(literals[i].Negated())];
}
literal_lp_values[i] = lp_value;
if (lp_value < 1e-6) continue;
relevant_arcs.push_back({tails[i], heads[i], lp_value});
arc_by_decreasing_lp_values.push_back({lp_value, i});
}
std::sort(arc_by_decreasing_lp_values.begin(),
arc_by_decreasing_lp_values.end(),
std::greater<std::pair<double, int>>());
// We will do a union-find by adding one by one the arc of the lp solution
// in the order above. Every intermediate set during this construction will
// be a candidate for a cut.
//
// In parallel to the union-find, to efficiently reconstruct these sets (at
// most num_nodes), we construct a "decomposition forest" of the different
// connected components. Note that we don't exploit any asymmetric nature of
// the graph here. This is exactly the algo 6.3 in the book above.
int num_components = num_nodes;
std::vector<int> parent(num_nodes);
std::vector<int> root(num_nodes);
for (int i = 0; i < num_nodes; ++i) {
parent[i] = i;
root[i] = i;
}
auto get_root_and_compress_path = [&root](int node) {
int r = node;
while (root[r] != r) r = root[r];
while (root[node] != r) {
const int next = root[node];
root[node] = r;
node = next;
}
return r;
};
for (const auto pair : arc_by_decreasing_lp_values) {
if (num_components == 2) break;
const int tail = get_root_and_compress_path(tails[pair.second]);
const int head = get_root_and_compress_path(heads[pair.second]);
if (tail != head) {
// Update the decomposition forest, note that the number of nodes is
// growing.
const int new_node = parent.size();
parent.push_back(new_node);
parent[head] = new_node;
parent[tail] = new_node;
--num_components;
// It is important that the union-find representative is the same node.
root.push_back(new_node);
root[head] = new_node;
root[tail] = new_node;
}
}
// For each node in the decomposition forest, try to add a cut for the set
// formed by the nodes and its children. To do that efficiently, we first
// order the nodes so that for each node in a tree, the set of children forms
// a consecutive span in the pre_order vector. This vector just lists the
// nodes in the "pre-order" graph traversal order. The Spans will point inside
// the pre_order vector, it is why we initialize it once and for all.
int new_size = 0;
std::vector<int> pre_order(num_nodes);
std::vector<absl::Span<const int>> subsets;
{
std::vector<absl::InlinedVector<int, 2>> graph(parent.size());
for (int i = 0; i < parent.size(); ++i) {
if (parent[i] != i) graph[parent[i]].push_back(i);
}
std::vector<int> queue;
std::vector<bool> seen(graph.size(), false);
std::vector<int> start_index(parent.size());
for (int i = num_nodes; i < parent.size(); ++i) {
// Note that because of the way we constructed 'parent', the graph is a
// binary tree. This is not required for the correctness of the algorithm
// here though.
CHECK(graph[i].empty() || graph[i].size() == 2);
if (parent[i] != i) continue;
// Explore the subtree rooted at node i.
CHECK(!seen[i]);
queue.push_back(i);
while (!queue.empty()) {
const int node = queue.back();
if (seen[node]) {
queue.pop_back();
// All the children of node are in the span [start, end) of the
// pre_order vector.
const int start = start_index[node];
if (new_size - start > 1) {
subsets.emplace_back(&pre_order[start], new_size - start);
}
continue;
}
seen[node] = true;
start_index[node] = new_size;
if (node < num_nodes) pre_order[new_size++] = node;
for (const int child : graph[node]) {
if (!seen[child]) queue.push_back(child);
}
}
}
}
// Compute the total demands in order to know the minimum incoming/outgoing
// flow.
int64 total_demands = 0;
if (!demands.empty()) {
for (const int64 demand : demands) total_demands += demand;
}
// Process each subsets and add any violated cut.
CHECK_EQ(pre_order.size(), num_nodes);
std::vector<bool> in_subset(num_nodes, false);
for (const absl::Span<const int> subset : subsets) {
CHECK_GT(subset.size(), 1);
CHECK_LT(subset.size(), num_nodes);
// These fields will be left untouched if demands.empty().
bool contain_depot = false;
int64 subset_demand = 0;
// Initialize "in_subset" and the subset demands.
for (const int n : subset) {
in_subset[n] = true;
if (!demands.empty()) {
if (n == 0) contain_depot = true;
subset_demand += demands[n];
}
}
// Compute a lower bound on the outgoing flow.
//
// TODO(user): This lower bound assume all nodes in subset must be served,
// which is not the case. For TSP we do the correct thing in
// AddOutgoingCut() but not for CVRP... Fix!!
//
// TODO(user): It could be very interesting to see if this "min outgoing
// flow" cannot be automatically infered from the constraint in the
// precedence graph. This might work if we assume that any kind of path
// cumul constraint is encoded with constraints:
// [edge => value_head >= value_tail + edge_weight].
// We could take the minimum incoming edge weight per node in the set, and
// use the cumul variable domain to infer some capacity.
int64 min_outgoing_flow = 1;
if (!demands.empty()) {
min_outgoing_flow =
contain_depot
? (total_demands - subset_demand + capacity - 1) / capacity
: (subset_demand + capacity - 1) / capacity;
}
// We still need to serve nodes with a demand of zero, and in the corner
// case where all node in subset have a zero demand, the formula above
// result in a min_outgoing_flow of zero.
min_outgoing_flow = std::max(min_outgoing_flow, int64{1});
// Compute the current outgoing flow out of the subset.
//
// This can take a significant portion of the running time, it is why it is
// faster to do it only on arcs with non-zero lp values which should be in
// linear number rather than the total number of arc which can be quadratic.
//
// TODO(user): For the symmetric case there is an even faster algo. See if
// it can be generalized to the asymmetric one if become needed.
// Reference is algo 6.4 of the "The Traveling Salesman Problem" book
// mentionned above.
double outgoing_flow = 0.0;
for (const auto arc : relevant_arcs) {
if (in_subset[arc.tail] && !in_subset[arc.head]) {
outgoing_flow += arc.lp_value;
}
}
// Add a cut if the current outgoing flow is not enough.
if (outgoing_flow < min_outgoing_flow - 1e-6) {
AddOutgoingCut(num_nodes, subset.size(), in_subset, tails, heads,
literals, literal_lp_values,
/*rhs_lower_bound=*/min_outgoing_flow, lp_values, manager,
model);
}
// Sparse clean up.
for (const int n : subset) in_subset[n] = false;
}
}
namespace {
// Returns for each literal its integer view, or the view of its negation.
std::vector<IntegerVariable> GetAssociatedVariables(
const std::vector<Literal>& literals, Model* model) {
auto* encoder = model->GetOrCreate<IntegerEncoder>();
std::vector<IntegerVariable> result;
for (const Literal l : literals) {
const IntegerVariable direct_view = encoder->GetLiteralView(l);
if (direct_view != kNoIntegerVariable) {
result.push_back(direct_view);
} else {
result.push_back(encoder->GetLiteralView(l.Negated()));
DCHECK_NE(result.back(), kNoIntegerVariable);
}
}
return result;
}
} // namespace
// We use a basic algorithm to detect components that are not connected to the
// rest of the graph in the LP solution, and add cuts to force some arcs to
// enter and leave this component from outside.
CutGenerator CreateStronglyConnectedGraphCutGenerator(
int num_nodes, const std::vector<int>& tails, const std::vector<int>& heads,
const std::vector<Literal>& literals, Model* model) {
CutGenerator result;
result.vars = GetAssociatedVariables(literals, model);
result.generate_cuts =
[num_nodes, tails, heads, literals, model](
const gtl::ITIVector<IntegerVariable, double>& lp_values,
LinearConstraintManager* manager) {
SeparateSubtourInequalities(
num_nodes, tails, heads, literals, lp_values,
/*demands=*/{}, /*capacity=*/0, manager, model);
};
return result;
}
CutGenerator CreateCVRPCutGenerator(int num_nodes,
const std::vector<int>& tails,
const std::vector<int>& heads,
const std::vector<Literal>& literals,
const std::vector<int64>& demands,
int64 capacity, Model* model) {
CutGenerator result;
result.vars = GetAssociatedVariables(literals, model);
result.generate_cuts =
[num_nodes, tails, heads, demands, capacity, literals, model](
const gtl::ITIVector<IntegerVariable, double>& lp_values,
LinearConstraintManager* manager) {
SeparateSubtourInequalities(num_nodes, tails, heads, literals,
lp_values, demands, capacity, manager,
model);
};
return result;
}
std::function<LiteralIndex()>
LinearProgrammingConstraint::HeuristicLPMostInfeasibleBinary(Model* model) {
IntegerTrail* integer_trail = integer_trail_;
IntegerEncoder* integer_encoder = model->GetOrCreate<IntegerEncoder>();
// Gather all 0-1 variables that appear in some LP.
std::vector<IntegerVariable> variables;
for (IntegerVariable var : integer_variables_) {
if (integer_trail_->LowerBound(var) == 0 &&
integer_trail_->UpperBound(var) == 1) {
variables.push_back(var);
}
}
VLOG(1) << "HeuristicLPMostInfeasibleBinary has " << variables.size()
<< " variables.";
return [this, variables, integer_trail, integer_encoder]() {
const double kEpsilon = 1e-6;
// Find most fractional value.
IntegerVariable fractional_var = kNoIntegerVariable;
double fractional_distance_best = -1.0;
for (const IntegerVariable var : variables) {
// Skip ignored and fixed variables.
if (integer_trail_->IsCurrentlyIgnored(var)) continue;
const IntegerValue lb = integer_trail_->LowerBound(var);
const IntegerValue ub = integer_trail_->UpperBound(var);
if (lb == ub) continue;
// Check variable's support is fractional.
const double lp_value = this->GetSolutionValue(var);
const double fractional_distance =
std::min(std::ceil(lp_value - kEpsilon) - lp_value,
lp_value - std::floor(lp_value + kEpsilon));
if (fractional_distance < kEpsilon) continue;
// Keep variable if it is farther from integrality than the previous.
if (fractional_distance > fractional_distance_best) {
fractional_var = var;
fractional_distance_best = fractional_distance;
}
}
if (fractional_var != kNoIntegerVariable) {
return integer_encoder
->GetOrCreateAssociatedLiteral(
IntegerLiteral::GreaterOrEqual(fractional_var, IntegerValue(1)))
.Index();
}
return kNoLiteralIndex;
};
}
std::function<LiteralIndex()>
LinearProgrammingConstraint::HeuristicLPPseudoCostBinary(Model* model) {
// Gather all 0-1 variables that appear in this LP.
std::vector<IntegerVariable> variables;
for (IntegerVariable var : integer_variables_) {
if (integer_trail_->LowerBound(var) == 0 &&
integer_trail_->UpperBound(var) == 1) {
variables.push_back(var);
}
}
VLOG(1) << "HeuristicLPPseudoCostBinary has " << variables.size()
<< " variables.";
// Store average of reduced cost from 1 to 0. The best heuristic only sets
// variables to one and cares about cost to zero, even though classic
// pseudocost will use max_var min(cost_to_one[var], cost_to_zero[var]).
const int num_vars = variables.size();
std::vector<double> cost_to_zero(num_vars, 0.0);
std::vector<int> num_cost_to_zero(num_vars);
int num_calls = 0;
IntegerEncoder* integer_encoder = model->GetOrCreate<IntegerEncoder>();
return [=]() mutable {
const double kEpsilon = 1e-6;
// Every 10000 calls, decay pseudocosts.
num_calls++;
if (num_calls == 10000) {
for (int i = 0; i < num_vars; i++) {
cost_to_zero[i] /= 2;
num_cost_to_zero[i] /= 2;
}
num_calls = 0;
}
// Accumulate pseudo-costs of all unassigned variables.
for (int i = 0; i < num_vars; i++) {
const IntegerVariable var = variables[i];
// Skip ignored and fixed variables.
if (integer_trail_->IsCurrentlyIgnored(var)) continue;
const IntegerValue lb = integer_trail_->LowerBound(var);
const IntegerValue ub = integer_trail_->UpperBound(var);
if (lb == ub) continue;
const double rc = this->GetSolutionReducedCost(var);
// Skip reduced costs that are nonzero because of numerical issues.
if (std::abs(rc) < kEpsilon) continue;
const double value = std::round(this->GetSolutionValue(var));
if (value == 1.0 && rc < 0.0) {
cost_to_zero[i] -= rc;
num_cost_to_zero[i]++;
}
}
// Select noninstantiated variable with highest pseudo-cost.
int selected_index = -1;
double best_cost = 0.0;
for (int i = 0; i < num_vars; i++) {
const IntegerVariable var = variables[i];
// Skip ignored and fixed variables.
if (integer_trail_->IsCurrentlyIgnored(var)) continue;
const IntegerValue lb = integer_trail_->LowerBound(var);
const IntegerValue ub = integer_trail_->UpperBound(var);
if (lb == ub) continue;
if (num_cost_to_zero[i] > 0 &&
best_cost < cost_to_zero[i] / num_cost_to_zero[i]) {
best_cost = cost_to_zero[i] / num_cost_to_zero[i];
selected_index = i;
}
}
if (selected_index >= 0) {
const Literal decision = integer_encoder->GetOrCreateAssociatedLiteral(
IntegerLiteral::GreaterOrEqual(variables[selected_index],
IntegerValue(1)));
return decision.Index();
}
return kNoLiteralIndex;
};
}
std::function<LiteralIndex()>
LinearProgrammingConstraint::LPReducedCostAverageBranching() {
const int num_vars = integer_variables_.size();
if (sum_cost_down_.size() < num_vars) {
VLOG(1) << " LPReducedCostAverageBranching has #variables: " << num_vars;
sum_cost_down_.resize(num_vars, 0.0);
num_cost_down_.resize(num_vars, 0);
sum_cost_up_.resize(num_vars, 0.0);
num_cost_up_.resize(num_vars, 0);
}
return [this]() { return this->LPReducedCostAverageDecision(); };
}
double LinearProgrammingConstraint::GetCostFromAverageReducedCosts(
int position) {
// If only one direction exist, we takes its value divided by 2, so that
// such variable should have a smaller cost than the min of the two side
// except if one direction have a really high reduced costs.
if (num_cost_down_[position] > 0 && num_cost_up_[position] > 0) {
return std::min(sum_cost_down_[position] / num_cost_down_[position],
sum_cost_up_[position] / num_cost_up_[position]);
} else {
const double divisor = num_cost_down_[position] + num_cost_up_[position];
if (divisor != 0) {
return 0.5 * (sum_cost_down_[position] + sum_cost_up_[position]) /
divisor;
}
}
return 0.0;
}
LiteralIndex LinearProgrammingConstraint::LPReducedCostAverageDecision() {
const int num_vars = integer_variables_.size();
// Select noninstantiated variable with highest pseudo-cost.
int selected_index = -1;
double best_cost = 0.0;
for (int i = 0; i < num_vars; i++) {
const IntegerVariable var = this->integer_variables_[i];
// Skip ignored and fixed variables.
if (integer_trail_->IsCurrentlyIgnored(var)) continue;
const IntegerValue lb = integer_trail_->LowerBound(var);
const IntegerValue ub = integer_trail_->UpperBound(var);
if (lb == ub) continue;
const double cost_i = GetCostFromAverageReducedCosts(i);
if (selected_index == -1 || cost_i > best_cost) {
best_cost = cost_i;
selected_index = i;
}
}
if (selected_index == -1) return kNoLiteralIndex;
const IntegerVariable var = integer_variables_[selected_index];
// If ceil(value) is current upper bound, try var == upper bound first.
// Guarding with >= prevents numerical problems.
// With 0/1 variables, this will tend to try setting to 1 first,
// which produces more shallow trees.
const IntegerValue ub = integer_trail_->UpperBound(var);
const IntegerValue value_ceil(
std::ceil(this->GetSolutionValue(var) - kCpEpsilon));
if (value_ceil >= ub) {
const Literal result = integer_encoder_->GetOrCreateAssociatedLiteral(
IntegerLiteral::GreaterOrEqual(var, ub));
CHECK(!trail_->Assignment().LiteralIsAssigned(result));
return result.Index();
}
// If floor(value) is current lower bound, try var == lower bound first.
// Guarding with <= prevents numerical problems.
const IntegerValue lb = integer_trail_->LowerBound(var);
const IntegerValue value_floor(
std::floor(this->GetSolutionValue(var) + kCpEpsilon));
if (value_floor <= lb) {
const Literal result = integer_encoder_->GetOrCreateAssociatedLiteral(
IntegerLiteral::LowerOrEqual(var, lb));
CHECK(!trail_->Assignment().LiteralIsAssigned(result))
<< " " << lb << " " << ub;
return result.Index();
}
// Here lb < value_floor <= value_ceil < ub.
// Try the most promising split between var <= floor or var >= ceil.
if (sum_cost_down_[selected_index] / num_cost_down_[selected_index] <
sum_cost_up_[selected_index] / num_cost_up_[selected_index]) {
const Literal result = integer_encoder_->GetOrCreateAssociatedLiteral(
IntegerLiteral::LowerOrEqual(var, value_floor));
CHECK(!trail_->Assignment().LiteralIsAssigned(result));
return result.Index();
} else {
const Literal result = integer_encoder_->GetOrCreateAssociatedLiteral(
IntegerLiteral::GreaterOrEqual(var, value_ceil));
CHECK(!trail_->Assignment().LiteralIsAssigned(result));
return result.Index();
}
}
} // namespace sat
} // namespace operations_research
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