mirror of
https://github.com/Z3Prover/z3
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611 lines
22 KiB
C++
611 lines
22 KiB
C++
/*++
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Copyright (c) 2017 Microsoft Corporation
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Module Name:
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<name>
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Abstract:
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<abstract>
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Author:
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Nikolaj Bjorner (nbjorner)
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Lev Nachmanson (levnach)
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Revision History:
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--*/
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#include "math/lp/gomory.h"
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#include "math/lp/int_solver.h"
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#include "math/lp/lar_solver.h"
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#include "math/lp/lp_utils.h"
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#define SMALL_CUTS 1
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namespace lp {
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struct create_cut {
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lar_term & m_t; // the term to return in the cut
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mpq & m_k; // the right side of the cut
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explanation* m_ex; // the conflict explanation
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unsigned m_inf_col; // a basis column which has to be an integer but has a non integral value
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const row_strip<mpq>& m_row;
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int_solver& lia;
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mpq m_f;
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mpq m_one_minus_f;
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mpq m_fj;
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mpq m_one_minus_fj;
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#if SMALL_CUTS
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mpq m_abs_max, m_big_number;
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#endif
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int m_polarity;
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bool m_found_big;
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u_dependency* m_dep;
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const impq & get_value(unsigned j) const { return lia.get_value(j); }
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bool is_int(unsigned j) const { return lia.column_is_int(j) || (lia.is_fixed(j) &&
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lia.lra.column_lower_bound(j).is_int()); }
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bool is_real(unsigned j) const { return !is_int(j); }
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bool at_lower(unsigned j) const { return lia.at_lower(j); }
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bool at_upper(unsigned j) const { return lia.at_upper(j); }
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const impq & lower_bound(unsigned j) const { return lia.lower_bound(j); }
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const impq & upper_bound(unsigned j) const { return lia.upper_bound(j); }
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u_dependency* column_lower_bound_constraint(unsigned j) const { return lia.column_lower_bound_constraint(j); }
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u_dependency* column_upper_bound_constraint(unsigned j) const { return lia.column_upper_bound_constraint(j); }
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bool column_is_fixed(unsigned j) const { return lia.lra.column_is_fixed(j); }
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void push_explanation(u_dependency* d) {
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for (auto ci : lia.lra.flatten(d))
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m_ex->push_back(ci);
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}
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void int_case_in_gomory_cut(unsigned j) {
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lp_assert(is_int(j) && m_fj.is_pos());
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TRACE("gomory_cut_detail",
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tout << " k = " << m_k;
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tout << ", fj: " << m_fj << ", ";
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tout << (at_lower(j)?"at_lower":"at_upper")<< std::endl;
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);
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mpq new_a;
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if (at_lower(j)) {
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// here we have the product of new_a*(xj - lb(j)), so new_a*lb(j) is added to m_k
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new_a = m_fj <= m_one_minus_f ? m_fj / m_one_minus_f : ((1 - m_fj) / m_f);
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lp_assert(new_a.is_pos());
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m_k.addmul(new_a, lower_bound(j).x);
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push_explanation(column_lower_bound_constraint(j));
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}
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else {
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lp_assert(at_upper(j));
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// here we have the expression new_a*(xj - ub), so new_a*ub(j) is added to m_k
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new_a = - (m_fj <= m_f ? m_fj / m_f : ((1 - m_fj) / m_one_minus_f));
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lp_assert(new_a.is_neg());
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m_k.addmul(new_a, upper_bound(j).x);
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push_explanation(column_upper_bound_constraint(j));
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}
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m_t.add_monomial(new_a, j);
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TRACE("gomory_cut_detail", tout << "new_a = " << new_a << ", k = " << m_k << "\n";);
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#if SMALL_CUTS
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// if (numerator(new_a).is_big()) throw found_big();
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if (numerator(new_a) > m_big_number)
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m_found_big = true;
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#endif
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}
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void real_case_in_gomory_cut(const mpq & a, unsigned j) {
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TRACE("gomory_cut_detail_real", tout << "j = " << j << ", a = " << a << ", m_k = " << m_k << "\n";);
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mpq new_a;
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if (at_lower(j)) {
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if (a.is_pos()) {
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// the delta is a (x - f) is positive it has to grow and fight m_one_minus_f
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new_a = a / m_one_minus_f;
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if (m_polarity != 2) {
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if (m_polarity == -1) m_polarity = 2;
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else m_polarity = 1;
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}
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}
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else {
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// the delta is negative and it works again m_f
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new_a = - a / m_f;
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if (m_polarity != 2) {
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if (m_polarity == 1) m_polarity = 2;
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else m_polarity = -1;
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}
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}
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m_k.addmul(new_a, lower_bound(j).x); // is it a faster operation than
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// k += lower_bound(j).x * new_a;
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push_explanation(column_lower_bound_constraint(j));
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}
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else {
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lp_assert(at_upper(j));
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if (a.is_pos()) {
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// the delta is works again m_f
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new_a = - a / m_f;
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if (m_polarity != 2) {
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if (m_polarity == 1) m_polarity = 2;
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else m_polarity = -1;
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}
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}
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else {
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// the delta is positive works again m_one_minus_f
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new_a = a / m_one_minus_f;
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if (m_polarity != 2) {
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if (m_polarity == -1) m_polarity = 2;
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else m_polarity = 1;
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}
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}
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m_k.addmul(new_a, upper_bound(j).x); // k += upper_bound(j).x * new_a;
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push_explanation(column_upper_bound_constraint(j));
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}
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m_t.add_monomial(new_a, j);
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TRACE("gomory_cut_detail_real", tout << "add " << new_a << "*v" << j << ", k: " << m_k << "\n";
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tout << "m_t = "; lia.lra.print_term(m_t, tout) << "\nk: " << m_k << "\n";);
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#if SMALL_CUTS
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// if (numerator(new_a).is_big()) throw found_big();
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if (numerator(new_a) > m_big_number)
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m_found_big = true;
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#endif
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}
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lia_move report_conflict_from_gomory_cut() {
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lp_assert(m_k.is_pos());
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// conflict 0 >= k where k is positive
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return lia_move::conflict;
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}
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std::string var_name(unsigned j) const {
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return std::string("x") + std::to_string(j);
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}
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std::ostream& dump_coeff_val(std::ostream & out, const mpq & a) const {
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if (a.is_int())
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out << a;
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else if (a >= zero_of_type<mpq>())
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out << "(/ " << numerator(a) << " " << denominator(a) << ")";
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else
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out << "(- (/ " << numerator(-a) << " " << denominator(-a) << "))";
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return out;
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}
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template <typename T>
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void dump_coeff(std::ostream & out, const T& c) const {
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dump_coeff_val(out << "(* ", c.coeff()) << " " << var_name(c.column().index()) << ")";
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}
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std::ostream& dump_row_coefficients(std::ostream & out) const {
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mpq lc(1);
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for (const auto& p : m_row)
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lc = lcm(lc, denominator(p.coeff()));
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for (const auto& p : m_row)
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dump_coeff_val(out << " (* ", p.coeff()*lc) << " " << var_name(p.var()) << ")";
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return out;
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}
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void dump_the_row(std::ostream& out) const {
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out << "; the row, excluding fixed vars\n";
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out << "(assert (= (+";
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dump_row_coefficients(out) << ") 0))\n";
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}
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void dump_declaration(std::ostream& out, unsigned v) const {
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out << "(declare-const " << var_name(v) << (is_int(v) ? " Int" : " Real") << ")\n";
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}
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void dump_declarations(std::ostream& out) const {
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// for a column j the var name is vj
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for (const auto & p : m_row)
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dump_declaration(out, p.var());
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for (lar_term::ival p : m_t) {
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auto t = lia.lra.column2tv(p.column());
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if (t.is_term())
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dump_declaration(out, t.id());
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}
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}
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void dump_lower_bound_expl(std::ostream & out, unsigned j) const {
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out << "(assert (>= " << var_name(j) << " " << lower_bound(j).x << "))\n";
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}
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void dump_upper_bound_expl(std::ostream & out, unsigned j) const {
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out << "(assert (<= " << var_name(j) << " " << upper_bound(j).x << "))\n";
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}
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void dump_explanations(std::ostream& out) const {
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for (const auto & p : m_row) {
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unsigned j = p.var();
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if (j == m_inf_col || (!is_real(j) && p.coeff().is_int()))
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continue;
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else if (at_lower(j))
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dump_lower_bound_expl(out, j);
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else {
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lp_assert(at_upper(j));
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dump_upper_bound_expl(out, j);
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}
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}
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}
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std::ostream& dump_term_coefficients(std::ostream & out) const {
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for (lar_term::ival p : m_t)
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dump_coeff(out, p);
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return out;
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}
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std::ostream& dump_term_sum(std::ostream & out) const {
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return dump_term_coefficients(out << "(+ ") << ")";
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}
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std::ostream& dump_term_ge_k(std::ostream & out) const {
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return dump_term_sum(out << "(>= ") << " " << m_k << ")";
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}
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void dump_the_cut_assert(std::ostream & out) const {
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dump_term_ge_k(out << "(assert (not ") << "))\n";
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}
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void dump_cut_and_constraints_as_smt_lemma(std::ostream& out) const {
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dump_declarations(out);
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dump_the_row(out);
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dump_explanations(out);
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dump_the_cut_assert(out);
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out << "(check-sat)\n";
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}
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public:
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void dump(std::ostream& out) {
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out << "applying cut at:\n"; print_linear_combination_indices_only<row_strip<mpq>, mpq>(m_row, out); out << std::endl;
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for (auto & p : m_row)
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lia.lra.print_column_info(p.var(), out);
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out << "inf_col = " << m_inf_col << std::endl;
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}
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lia_move cut() {
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TRACE("gomory_cut", dump(tout););
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m_polarity = 0; // 0: means undefined, 1, -1: the polar case, 2: the mixed case
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// gomory cut will be m_t >= m_k and the current solution has a property m_t < m_k
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m_k = 1;
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m_t.clear();
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m_ex->clear();
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m_found_big = false;
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TRACE("gomory_cut_detail", tout << "m_f: " << m_f << ", ";
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tout << "1 - m_f: " << 1 - m_f << ", get_value(m_inf_col).x - m_f = " << get_value(m_inf_col).x - m_f << "\n";);
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lp_assert(m_f.is_pos() && (get_value(m_inf_col).x - m_f).is_int());
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bool some_int_columns = false;
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#if SMALL_CUTS
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m_abs_max = 0;
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for (const auto & p : m_row) {
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mpq t = abs(ceil(p.coeff()));
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if (t > m_abs_max)
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m_abs_max = t;
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}
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m_big_number = m_abs_max.expt(2);
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#endif
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for (const auto & p : m_row) {
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unsigned j = p.var();
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if (j == m_inf_col) continue;
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// use -p.coeff() to make the format compatible with the format used in: Integrating Simplex with DPLL(T)
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if (lia.is_fixed(j)) {
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push_explanation(column_lower_bound_constraint(j));
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push_explanation(column_upper_bound_constraint(j));
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continue;
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}
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if (is_real(j))
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real_case_in_gomory_cut(- p.coeff(), j);
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else if (!p.coeff().is_int()) {
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some_int_columns = true;
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m_fj = fractional_part(-p.coeff());
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m_one_minus_fj = 1 - m_fj;
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int_case_in_gomory_cut(j);
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if (p.coeff().is_pos()) {
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if (at_lower(j)) {
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if (m_polarity == -1) m_polarity = 2;
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else m_polarity = 1;
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}
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else {
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if (m_polarity == 1) m_polarity = 2;
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else m_polarity = -1;
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}
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}
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else {
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if (at_lower(j)) {
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if (m_polarity == 1) m_polarity = 2;
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else m_polarity = -1;
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}
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else {
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if (m_polarity == -1) m_polarity = 2;
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else m_polarity = 1;
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}
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}
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}
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if (m_found_big) {
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return lia_move::undef;
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}
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}
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if (m_t.is_empty())
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return report_conflict_from_gomory_cut();
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TRACE("gomory_cut", print_linear_combination_of_column_indices_only(m_t.coeffs_as_vector(), tout << "gomory cut: "); tout << " >= " << m_k << std::endl;);
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if (some_int_columns)
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simplify_inequality();
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m_dep = nullptr;
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for (auto c : *m_ex)
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m_dep = lia.lra.join_deps(lia.lra.dep_manager().mk_leaf(c.ci()), m_dep);
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TRACE("gomory_cut", print_linear_combination_of_column_indices_only(m_t.coeffs_as_vector(), tout << "gomory cut: "); tout << " >= " << m_k << std::endl;);
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TRACE("gomory_cut_detail", dump_cut_and_constraints_as_smt_lemma(tout);
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lia.lra.display(tout));
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SASSERT(lia.current_solution_is_inf_on_cut()); // checks that indices are columns
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lia.settings().stats().m_gomory_cuts++;
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return lia_move::cut;
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}
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// TODO: use this also for HNF cuts?
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mpq m_lcm_den = { mpq(1) };
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void simplify_inequality() {
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auto divd = [](mpq& r, mpq const& d) {
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r /= d;
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if (!r.is_int())
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r = ceil(r);
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};
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SASSERT(!lia.m_upper);
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lp_assert(!m_t.is_empty());
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// k = 1 + sum of m_t at bounds
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lar_term t = lia.lra.unfold_nested_subterms(m_t);
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auto pol = t.coeffs_as_vector();
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m_t.clear();
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if (pol.size() == 1 && is_int(pol[0].second)) {
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TRACE("gomory_cut_detail", tout << "pol.size() is 1" << std::endl;);
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auto const& [a, v] = pol[0];
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lp_assert(is_int(v));
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if (a.is_pos()) { // we have av >= k
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divd(m_k, a);
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m_t.add_monomial(mpq(1), v);
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}
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else {
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// av >= k
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// a/-a*v >= k / - a
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// -v >= k / - a
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// -v >= ceil(k / -a)
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divd(m_k, -a);
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m_t.add_monomial(-mpq(1), v);
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}
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}
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else {
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m_lcm_den = denominator(m_k);
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for (auto const& [c, v] : pol)
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m_lcm_den = lcm(m_lcm_den, denominator(c));
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lp_assert(m_lcm_den.is_pos());
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bool int_row = all_of(pol, [&](auto const& kv) { return is_int(kv.second); });
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TRACE("gomory_cut_detail", tout << "pol.size() > 1 den: " << m_lcm_den << std::endl;);
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if (!m_lcm_den.is_one()) {
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// normalize coefficients of integer parameters to be integers.
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for (auto & [c,v]: pol) {
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c *= m_lcm_den;
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SASSERT(!is_int(v) || c.is_int());
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}
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m_k *= m_lcm_den;
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}
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// ax + by >= k
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// b > 0, c1 <= y <= c2
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// ax + b*c2 >= ax + by >= k
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// =>
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// ax >= k - by >= k - b*c1
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// b < 0
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// ax + b*c1 >= ax + by >= k
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//
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unsigned j = 0, i = 0;
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for (auto & [c, v] : pol) {
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if (lia.is_fixed(v)) {
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push_explanation(column_lower_bound_constraint(v));
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push_explanation(column_upper_bound_constraint(v));
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m_k -= c * lower_bound(v).x;
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}
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else
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pol[j++] = pol[i];
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++i;
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}
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pol.shrink(j);
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// gcd reduction is loss-less:
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mpq g(1);
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for (const auto & [c, v] : pol)
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g = gcd(g, c);
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if (!int_row)
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g = gcd(g, m_k);
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if (g != 1) {
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for (auto & [c, v] : pol)
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c /= g;
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divd(m_k, g);
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}
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for (const auto & [c, v]: pol)
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m_t.add_monomial(c, v);
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VERIFY(m_t.size() > 0);
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}
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TRACE("gomory_cut_detail", tout << "k = " << m_k << std::endl;);
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lp_assert(m_k.is_int());
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}
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create_cut(lar_term & t, mpq & k, explanation* ex, unsigned basic_inf_int_j, const row_strip<mpq>& row, int_solver& lia) :
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m_t(t),
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m_k(k),
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m_ex(ex),
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m_inf_col(basic_inf_int_j),
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m_row(row),
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lia(lia),
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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));
|
|
// All non base variables must be at their bounds and assigned to rationals (that is, infinitesimals are not allowed).
|
|
const row_strip<mpq>& row = lra.get_row(lia.row_of_basic_column(k));
|
|
unsigned j;
|
|
for (const auto & p : row) {
|
|
j = p.var();
|
|
if ( k != j && (!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;
|
|
}
|
|
|
|
// 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<lpvar> sorted_vars;
|
|
std::unordered_map<lpvar, mpq> 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<unsigned>(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<unsigned>(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;
|
|
}
|
|
|
|
|
|
lia_move gomory::get_gomory_cuts(unsigned num_cuts) {
|
|
struct cut_result {u_dependency *dep; lar_term t; mpq k; int polarity; lpvar j;};
|
|
vector<cut_result> big_cuts;
|
|
struct polar_pair {int polarity; u_dependency *dep;};
|
|
std::unordered_map<lpvar, polar_pair> polar_vars;
|
|
unsigned_vector columns_for_cuts = gomory_select_int_infeasible_vars(num_cuts);
|
|
|
|
// define inline helper functions
|
|
bool has_small_cut = false;
|
|
auto is_small_cut = [&](lar_term const& t) {
|
|
return all_of(t, [&](auto ci) { return ci.coeff().is_small(); });
|
|
};
|
|
auto add_cut = [&](cut_result const& cr) {
|
|
u_dependency* dep = cr.dep;
|
|
lp::lpvar term_index = lra.add_term(cr.t.coeffs_as_vector(), UINT_MAX);
|
|
term_index = lra.map_term_index_to_column_index(term_index);
|
|
lra.update_column_type_and_bound(term_index,
|
|
lp::lconstraint_kind::GE,
|
|
lia.m_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) {
|
|
unsigned row_index = lia.row_of_basic_column(j);
|
|
const row_strip<mpq>& 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)
|
|
continue;
|
|
cut_result cr = {cc.m_dep, lia.m_t, lia.m_k, cc.m_polarity, j};
|
|
if (abs(cr.polarity) == 1) // need to delay the update of the bounds for j in a polar case, because simplify_inequality relies on the old bounds
|
|
polar_vars[j] = {cr.polarity, cc.m_dep};
|
|
|
|
if (!is_small_cut(lia.m_t)) {
|
|
big_cuts.push_back(cr);
|
|
continue;
|
|
}
|
|
has_small_cut = true;
|
|
add_cut(cr);
|
|
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);
|
|
bool feas = _check_feasible();
|
|
lra.pop(1);
|
|
|
|
if (lia.settings().get_cancel_flag())
|
|
return lia_move::undef;
|
|
|
|
if (!feas)
|
|
return lia_move::conflict;
|
|
}
|
|
|
|
// this way we create bounds for the variables in polar cases even where the terms had big numbers
|
|
for (auto const& p : polar_vars) {
|
|
if (p.second.polarity == 1) {
|
|
lra.update_column_type_and_bound(p.first, lp::lconstraint_kind::LE, floor(lra.get_column_value(p.first).x), p.second.dep);
|
|
}
|
|
else {
|
|
SASSERT(p.second.polarity == -1);
|
|
lra.update_column_type_and_bound(p.first, lp::lconstraint_kind::GE, ceil(lra.get_column_value(p.first).x), p.second.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) { }
|
|
}
|