z3-z3-4.13.0.src.math.lp.gomory.cpp Maven / Gradle / Ivy
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/*++
Copyright (c) 2017 Microsoft Corporation
Module Name:
Abstract:
Author:
Nikolaj Bjorner (nbjorner)
Lev Nachmanson (levnach)
Revision History:
--*/
#include "math/lp/gomory.h"
#include "math/lp/int_solver.h"
#include "math/lp/lar_solver.h"
#include "math/lp/lp_utils.h"
namespace lp {
enum class row_polarity { UNDEF, MIN, MAX, MIXED};
struct create_cut {
lar_term & m_t; // the term to return in the cut
mpq & m_k; // the right side of the cut
explanation* m_ex; // the conflict explanation
unsigned m_inf_col; // a basis column which has to be an integer but has a non integral value
const row_strip& m_row;
int_solver& lia;
mpq m_f;
mpq m_one_minus_f;
mpq m_fj;
mpq m_one_minus_fj;
mpq m_abs_max, m_big_number;
row_polarity m_polarity;
bool m_found_big;
u_dependency* m_dep;
const impq & get_value(unsigned j) const { return lia.get_value(j); }
bool is_int(unsigned j) const { return lia.column_is_int(j) || (lia.is_fixed(j) &&
lia.lra.column_lower_bound(j).is_int()); }
bool is_real(unsigned j) const { return !is_int(j); }
bool at_lower(unsigned j) const { return lia.at_lower(j); }
bool at_upper(unsigned j) const { return lia.at_upper(j); }
const impq & lower_bound(unsigned j) const { return lia.lower_bound(j); }
const impq & upper_bound(unsigned j) const { return lia.upper_bound(j); }
u_dependency* column_lower_bound_constraint(unsigned j) const { return lia.column_lower_bound_constraint(j); }
u_dependency* column_upper_bound_constraint(unsigned j) const { return lia.column_upper_bound_constraint(j); }
bool column_is_fixed(unsigned j) const { return lia.lra.column_is_fixed(j); }
void push_explanation(u_dependency* d) {
for (auto ci : lia.lra.flatten(d))
m_ex->push_back(ci);
}
void int_case_in_gomory_cut(unsigned j) {
lp_assert(is_int(j) && m_fj.is_pos());
TRACE("gomory_cut_detail",
tout << " k = " << m_k;
tout << ", fj: " << m_fj << ", ";
tout << (at_lower(j)?"at_lower":"at_upper")<< std::endl;
);
mpq new_a;
if (at_lower(j)) {
// here we have the product of new_a*(xj - lb(j)), so new_a*lb(j) is added to m_k
new_a = m_fj <= m_one_minus_f ? m_fj / m_one_minus_f : ((1 - m_fj) / m_f);
lp_assert(new_a.is_pos());
m_k.addmul(new_a, lower_bound(j).x);
push_explanation(column_lower_bound_constraint(j));
}
else {
lp_assert(at_upper(j));
// here we have the expression new_a*(xj - ub), so new_a*ub(j) is added to m_k
new_a = - (m_fj <= m_f ? m_fj / m_f : ((1 - m_fj) / m_one_minus_f));
lp_assert(new_a.is_neg());
m_k.addmul(new_a, upper_bound(j).x);
push_explanation(column_upper_bound_constraint(j));
}
m_t.add_monomial(new_a, j);
TRACE("gomory_cut_detail", tout << "new_a = " << new_a << ", k = " << m_k << "\n";);
if (numerator(new_a) > m_big_number)
m_found_big = true;
}
void set_polarity(row_polarity p) {
if (m_polarity == row_polarity::MIXED) return;
if (m_polarity == row_polarity::UNDEF) m_polarity = p;
else if (m_polarity != p) m_polarity = row_polarity::MIXED;
}
void real_case_in_gomory_cut(const mpq & a, unsigned j) {
TRACE("gomory_cut_detail_real", tout << "j = " << j << ", a = " << a << ", m_k = " << m_k << "\n";);
mpq new_a;
if (at_lower(j)) {
if (a.is_pos()) {
// the delta is a (x - f) is positive it has to grow and fight m_one_minus_f
new_a = a / m_one_minus_f;
set_polarity(row_polarity::MIN); // reverse the polarity since a = -p.coeff()
}
else {
// the delta is negative and it works again m_f
new_a = - a / m_f;
set_polarity(row_polarity::MAX);
}
m_k.addmul(new_a, lower_bound(j).x); // is it a faster operation than
// k += lower_bound(j).x * new_a;
push_explanation(column_lower_bound_constraint(j));
}
else {
lp_assert(at_upper(j));
if (a.is_pos()) {
// the delta is works again m_f
new_a = - a / m_f;
set_polarity(row_polarity::MAX);
}
else {
// the delta is positive works again m_one_minus_f
new_a = a / m_one_minus_f;
set_polarity(row_polarity::MIN);
}
m_k.addmul(new_a, upper_bound(j).x); // k += upper_bound(j).x * new_a;
push_explanation(column_upper_bound_constraint(j));
}
m_t.add_monomial(new_a, j);
TRACE("gomory_cut_detail_real", tout << "add " << new_a << "*v" << j << ", k: " << m_k << "\n";
tout << "m_t = "; lia.lra.print_term(m_t, tout) << "\nk: " << m_k << "\n";);
if (numerator(new_a) > m_big_number)
m_found_big = true;
}
lia_move report_conflict_from_gomory_cut() {
lp_assert(m_k.is_pos());
// conflict 0 >= k where k is positive
return lia_move::conflict;
}
std::string var_name(unsigned j) const {
return std::string("x") + std::to_string(j);
}
std::ostream& dump_coeff_val(std::ostream & out, const mpq & a) const {
if (a.is_int())
out << a;
else if (a >= zero_of_type())
out << "(/ " << numerator(a) << " " << denominator(a) << ")";
else
out << "(- (/ " << numerator(-a) << " " << denominator(-a) << "))";
return out;
}
template
void dump_coeff(std::ostream & out, const T& c) const {
dump_coeff_val(out << "(* ", c.coeff()) << " " << var_name(c.j()) << ")";
}
std::ostream& dump_row_coefficients(std::ostream & out) const {
mpq lc(1);
for (const auto& p : m_row)
lc = lcm(lc, denominator(p.coeff()));
for (const auto& p : m_row)
dump_coeff_val(out << " (* ", p.coeff()*lc) << " " << var_name(p.var()) << ")";
return out;
}
void dump_the_row(std::ostream& out) const {
out << "; the row, excluding fixed vars\n";
out << "(assert (= (+";
dump_row_coefficients(out) << ") 0))\n";
}
void dump_declaration(std::ostream& out, unsigned v) const {
out << "(declare-const " << var_name(v) << (is_int(v) ? " Int" : " Real") << ")\n";
}
void dump_declarations(std::ostream& out) const {
// for a column j the var name is vj
for (const auto & p : m_row)
dump_declaration(out, p.var());
for (lar_term::ival p : m_t) {
if (lia.lra.column_has_term(p.j()))
dump_declaration(out, p.j());
}
}
void dump_lower_bound_expl(std::ostream & out, unsigned j) const {
out << "(assert (>= " << var_name(j) << " " << lower_bound(j).x << "))\n";
}
void dump_upper_bound_expl(std::ostream & out, unsigned j) const {
out << "(assert (<= " << var_name(j) << " " << upper_bound(j).x << "))\n";
}
void dump_explanations(std::ostream& out) const {
for (const auto & p : m_row) {
unsigned j = p.var();
if (j == m_inf_col || (!is_real(j) && p.coeff().is_int()))
continue;
else if (at_lower(j))
dump_lower_bound_expl(out, j);
else {
lp_assert(at_upper(j));
dump_upper_bound_expl(out, j);
}
}
}
std::ostream& dump_term_coefficients(std::ostream & out) const {
for (lar_term::ival p : m_t)
dump_coeff(out, p);
return out;
}
std::ostream& dump_term_sum(std::ostream & out) const {
return dump_term_coefficients(out << "(+ ") << ")";
}
std::ostream& dump_term_ge_k(std::ostream & out) const {
return dump_term_sum(out << "(>= ") << " " << m_k << ")";
}
void dump_the_cut_assert(std::ostream & out) const {
dump_term_ge_k(out << "(assert (not ") << "))\n";
}
void dump_cut_and_constraints_as_smt_lemma(std::ostream& out) const {
dump_declarations(out);
dump_the_row(out);
dump_explanations(out);
dump_the_cut_assert(out);
out << "(check-sat)\n";
}
public:
void dump(std::ostream& out) {
out << "applying cut at:\n"; print_linear_combination_indices_only, mpq>(m_row, out); out << std::endl;
for (auto & p : m_row)
lia.lra.print_column_info(p.var(), out);
out << "inf_col = " << m_inf_col << std::endl;
}
lia_move cut() {
TRACE("gomory_cut", dump(tout););
// If m_polarity is MAX, then
// the row constraints the base variable to be at the maximum,
// MIN - at the minimum,
// MIXED : the row does not constraint the base variable to be at an extremum
// UNDEF is the initial state
m_polarity = row_polarity::UNDEF;
// gomory cut will be m_t >= m_k and the current solution has a property m_t < m_k
m_k = 1;
m_t.clear();
m_ex->clear();
m_found_big = false;
TRACE("gomory_cut_detail", tout << "m_f: " << m_f << ", ";
tout << "1 - m_f: " << 1 - m_f << ", get_value(m_inf_col).x - m_f = " << get_value(m_inf_col).x - m_f << "\n";);
lp_assert(m_f.is_pos() && (get_value(m_inf_col).x - m_f).is_int());
auto set_polarity_for_int = [&](const mpq & a, lpvar j) {
if (a.is_pos()) {
if (at_lower(j))
set_polarity(row_polarity::MAX);
else if (at_upper(j))
set_polarity(row_polarity::MIN);
else
set_polarity(row_polarity::MIXED);
}
else {
if (at_lower(j))
set_polarity(row_polarity::MIN);
else if (at_upper(j))
set_polarity(row_polarity::MAX);
else
set_polarity(row_polarity::MIXED);
}
};
m_abs_max = 0;
for (const auto & p : m_row) {
mpq t = abs(ceil(p.coeff()));
if (t > m_abs_max)
m_abs_max = t;
}
m_big_number = m_abs_max.expt(2);
for (const auto & p : m_row) {
unsigned j = p.var();
if (j == m_inf_col) continue;
// use -p.coeff() to make the format compatible with the format used in: Integrating Simplex with DPLL(T)
if (lia.is_fixed(j)) {
push_explanation(column_lower_bound_constraint(j));
push_explanation(column_upper_bound_constraint(j));
continue;
}
if (is_real(j))
real_case_in_gomory_cut(- p.coeff(), j);
else {
if (!p.coeff().is_int()) {
m_fj = fractional_part(-p.coeff());
m_one_minus_fj = 1 - m_fj;
int_case_in_gomory_cut(j);
}
if (m_polarity != row_polarity::MIXED)
set_polarity_for_int(p.coeff(), j);
}
if (m_found_big) {
return lia_move::undef;
}
}
if (m_t.is_empty()) {
return report_conflict_from_gomory_cut();
}
TRACE("gomory_cut", print_linear_combination_of_column_indices_only(m_t.coeffs_as_vector(), tout << "gomory cut: "); tout << " >= " << m_k << std::endl;);
m_dep = nullptr;
for (auto c : *m_ex)
m_dep = lia.lra.join_deps(lia.lra.dep_manager().mk_leaf(c.ci()), m_dep);
TRACE("gomory_cut_detail", dump_cut_and_constraints_as_smt_lemma(tout);
lia.lra.display(tout));
SASSERT(lia.current_solution_is_inf_on_cut());
lia.settings().stats().m_gomory_cuts++;
return lia_move::cut;
}
create_cut(lar_term & t, mpq & k, explanation* ex, unsigned basic_inf_int_j, const row_strip& row, int_solver& lia) :
m_t(t),
m_k(k),
m_ex(ex),
m_inf_col(basic_inf_int_j),
m_row(row),
lia(lia),
m_f(fractional_part(get_value(basic_inf_int_j).x)),
m_one_minus_f(1 - m_f) {}
};
bool gomory::is_gomory_cut_target(lpvar k) {
SASSERT(lia.is_base(k));
const row_strip& row = lra.get_row(lia.row_of_basic_column(k));
// Consider monomial c*x from the row, where x is non-basic.
// Then, for each such monomial, one of following conditions
// has to hold for the row to be eligible for Gomory cut:
// 1) c is integral and x integral varible with an integral value
// 2) the value of x is at a bound and has no infinitesimals.
unsigned j;
for (const auto & p : row) {
j = p.var();
if (k == j) continue;
if (p.coeff().is_int() && lia.column_is_int(j) && lia.get_value(j).is_int()) continue;
if ( !lia.at_bound(j) || lia.get_value(j).y != 0) {
TRACE("gomory_cut", tout << "row is not gomory cut target:\n";
lia.display_column(tout, j);
tout << "infinitesimal: " << !(lia.get_value(j).y ==0) << "\n";);
return false;
}
}
return true;
// Condition 1) above can be relaxed even more, allowing any value for x, but it will change the calculation for m_f.
}
// return the minimal distance from the variable value to an integer
mpq get_gomory_score(const int_solver& lia, lpvar j) {
const mpq& val = lia.get_value(j).x;
auto l = val - floor(val);
if (l <= mpq(1, 2))
return l;
return mpq(1) - l;
}
unsigned_vector gomory::gomory_select_int_infeasible_vars(unsigned num_cuts) {
std::list sorted_vars;
std::unordered_map score;
for (lpvar j : lra.r_basis()) {
if (!lia.column_is_int_inf(j) || !is_gomory_cut_target(j))
continue;
SASSERT(!lia.is_fixed(j));
sorted_vars.push_back(j);
score[j] = get_gomory_score(lia, j);
}
// prefer the variables with the values close to integers
sorted_vars.sort([&](lpvar j, lpvar k) {
auto diff = score[j] - score[k];
if (diff.is_neg())
return true;
if (diff.is_pos())
return false;
return lra.usage_in_terms(j) > lra.usage_in_terms(k);
});
unsigned_vector ret;
unsigned n = static_cast(sorted_vars.size());
while (num_cuts-- && n > 0) {
unsigned k = lia.random() % n;
double k_ratio = k / (double) n;
k_ratio *= k_ratio*k_ratio; // square k_ratio to make it smaller
k = static_cast(std::floor(k_ratio * n));
// these operations move k to the beginning of the indices range
SASSERT(0 <= k && k < n);
auto it = sorted_vars.begin();
while(k--) it++;
ret.push_back(*it);
sorted_vars.erase(it);
n--;
}
return ret;
}
row_polarity test_row_polarity(const int_solver& lia, const row_strip& row, lpvar basic_j) {
row_polarity ret = row_polarity::UNDEF;
for (const auto& p : row) {
lpvar j = p.var();
if (j == basic_j)
continue;
if (lia.is_fixed(j))
continue;
row_polarity rp;
if (p.coeff().is_pos()) {
if (lia.at_lower(j))
rp = row_polarity::MAX;
else if (lia.at_upper(j))
rp = row_polarity::MIN;
else
rp = row_polarity::MIXED;
}
else {
if (lia.at_lower(j))
rp = row_polarity::MIN;
else if (lia.at_upper(j))
rp = row_polarity::MAX;
else
rp = row_polarity::MIXED;
}
if (ret == row_polarity::UNDEF)
ret = rp;
if (ret != rp)
return row_polarity::MIXED;
}
return ret;
}
u_dependency* gomory::add_deps(u_dependency* dep, const row_strip& row, lpvar basic_var) {
u_dependency* ret = dep;
for (const auto& p : row) {
lpvar j = p.var();
if (j == basic_var)
continue;
if (lia.is_fixed(j))
continue;
if (lia.is_real(j)) continue;
if (!p.coeff().is_int()) continue;
// the explanation for all above have been already added
if (lia.at_lower(j))
ret = lia.lra.dep_manager().mk_join(lia.column_lower_bound_constraint(j), ret);
else {
SASSERT(lia.at_upper(j));
ret = lia.lra.dep_manager().mk_join(lia.column_upper_bound_constraint(j), ret);
}
}
return ret;
}
lia_move gomory::get_gomory_cuts(unsigned num_cuts) {
struct cut_result {lar_term t; mpq k; u_dependency *dep;};
vector big_cuts;
unsigned_vector columns_for_cuts = gomory_select_int_infeasible_vars(num_cuts);
bool has_small_cut = false;
// define inline helper functions
auto is_small_cut = [&](lar_term const& t) {
return all_of(t, [&](auto ci) { return ci.coeff().is_small(); });
};
auto add_cut = [&](const lar_term& t, const mpq& k, u_dependency * dep) {
lp::lpvar j = lra.add_term(t.coeffs_as_vector(), UINT_MAX);
lra.update_column_type_and_bound(j, lp::lconstraint_kind::GE, k, dep);
};
auto _check_feasible = [&](void) {
lra.find_feasible_solution();
if (!lra.is_feasible() && !lia.settings().get_cancel_flag()) {
lra.get_infeasibility_explanation(*lia.m_ex);
return false;
}
return true;
};
// start creating cuts
for (unsigned j : columns_for_cuts) {
SASSERT(is_gomory_cut_target(j));
unsigned row_index = lia.row_of_basic_column(j);
const row_strip& row = lra.get_row(row_index);
create_cut cc(lia.m_t, lia.m_k, lia.m_ex, j, row, lia);
auto r = cc.cut();
if (r != lia_move::cut) {
if (r == lia_move::conflict)
return lia_move::conflict;
continue;
}
SASSERT(test_row_polarity(lia, row, j) == cc.m_polarity);
if (cc.m_polarity == row_polarity::MAX)
lra.update_column_type_and_bound(j, lp::lconstraint_kind::LE, floor(lra.get_column_value(j).x), add_deps(cc.m_dep, row, j));
else if (cc.m_polarity == row_polarity::MIN)
lra.update_column_type_and_bound(j, lp::lconstraint_kind::GE, ceil(lra.get_column_value(j).x), add_deps(cc.m_dep, row, j));
if (!is_small_cut(lia.m_t)) {
big_cuts.push_back({cc.m_t, cc.m_k, cc.m_dep});
continue;
}
has_small_cut = true;
add_cut(cc.m_t, cc.m_k, cc.m_dep);
if (lia.settings().get_cancel_flag())
return lia_move::undef;
}
if (big_cuts.size()) {
lra.push();
for (auto const& cut : big_cuts)
add_cut(cut.t, cut.k, cut.dep);
bool feas = _check_feasible();
lra.pop(1);
if (!feas)
for (auto const& cut : big_cuts)
add_cut(cut.t, cut.k, cut.dep);
}
if (!_check_feasible())
return lia_move::conflict;
if (!lia.has_inf_int())
return lia_move::sat;
if (has_small_cut || big_cuts.size())
return lia_move::continue_with_check;
lra.move_non_basic_columns_to_bounds();
return lia_move::undef;
}
gomory::gomory(int_solver& lia): lia(lia), lra(lia.lra) { }
}