mirror of
https://github.com/Z3Prover/z3
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464 lines
16 KiB
C++
464 lines
16 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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class 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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const int_solver& lia;
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mpq m_lcm_den;
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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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struct found_big {};
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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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constraint_index column_lower_bound_constraint(unsigned j) const { return lia.column_lower_bound_constraint(j); }
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constraint_index 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 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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m_ex->push_justification(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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m_ex->push_justification(column_upper_bound_constraint(j));
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}
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m_t.add_monomial(new_a, j);
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m_lcm_den = lcm(m_lcm_den, denominator(new_a));
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TRACE("gomory_cut_detail", tout << "new_a = " << new_a << ", k = " << m_k << ", lcm_den = " << m_lcm_den << "\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) throw found_big();
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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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}
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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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}
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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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m_ex->push_justification(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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}
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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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}
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m_k.addmul(new_a, upper_bound(j).x); // k += upper_bound(j).x * new_a;
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m_ex->push_justification(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) throw found_big();
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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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void adjust_term_and_k_for_some_ints_case_gomory() {
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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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auto pol = m_t.coeffs_as_vector();
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m_t.clear();
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if (pol.size() == 1) {
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TRACE("gomory_cut_detail", tout << "pol.size() is 1" << std::endl;);
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unsigned v = pol[0].second;
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lp_assert(is_int(v));
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const mpq& a = pol[0].first;
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if (a.is_pos()) { // we have av >= k
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m_k /= a;
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if (!m_k.is_int())
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m_k = ceil(m_k);
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m_t.add_monomial(mpq(1), v);
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} else {
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m_k /= -a;
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if (!m_k.is_int())
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m_k = ceil(m_k);
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m_t.add_monomial(-mpq(1), v);
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}
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} else {
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m_lcm_den = lcm(m_lcm_den, denominator(m_k));
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lp_assert(m_lcm_den.is_pos());
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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 & pi: pol) {
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pi.first *= m_lcm_den;
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SASSERT(!is_int(pi.second) || pi.first.is_int());
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}
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m_k *= m_lcm_den;
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}
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for (const auto & pi: pol)
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m_t.add_monomial(pi.first, pi.second);
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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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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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}
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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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}
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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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out << "( * ";
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dump_coeff_val(out, c.coeff());
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auto t = lia.lra.column2tv(c.column());
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out << " " << var_name(t.id()) << ")";
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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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}
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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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}
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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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}
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for (const auto& 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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}
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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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}
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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 (const auto& p : m_t) {
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dump_coeff(out, p);
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}
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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_le_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_le_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.m_mpq_lar_core_solver.m_r_solver.print_column_info(p.var(), out);
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}
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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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// gomory will be t >= k and the current solution has a property t < k
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m_k = 1;
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m_t.clear();
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mpq m_lcm_den(1);
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bool some_int_columns = false;
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mpq m_f = fractional_part(get_value(m_inf_col));
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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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#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) 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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mpq one_min_m_f = 1 - m_f;
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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) {
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lp_assert(p.coeff() == one_of_type<mpq>());
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TRACE("gomory_cut_detail", tout << "seeing basic var\n";);
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continue;
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}
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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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try {
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if (lia.is_fixed(j)) {
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m_ex->push_justification(column_lower_bound_constraint(j));
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m_ex->push_justification(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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}
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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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}
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}
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catch (found_big) {
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m_ex->clear();
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m_t.clear();
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m_k = 1;
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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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if (some_int_columns)
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adjust_term_and_k_for_some_ints_case_gomory();
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TRACE("gomory_cut_detail", dump_cut_and_constraints_as_smt_lemma(tout););
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lp_assert(lia.current_solution_is_inf_on_cut());
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// NSB code review: this is also used in nla_core.
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// but it isn't consistent: when theory_lra accesses lar_solver::get_term, the term that is returned uses
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// column indices, not terms.
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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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return lia_move::cut;
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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, const 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_lcm_den(1),
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m_f(fractional_part(get_value(basic_inf_int_j).x)),
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m_one_minus_f(1 - m_f) {}
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};
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lia_move gomory::cut(lar_term & t, mpq & k, explanation* ex, unsigned basic_inf_int_j, const row_strip<mpq>& row) {
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create_cut cc(t, k, ex, basic_inf_int_j, row, lia);
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return cc.cut();
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}
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bool gomory::is_gomory_cut_target(const row_strip<mpq>& row) {
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// All non base variables must be at their bounds and assigned to rationals (that is, infinitesimals are not allowed).
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unsigned j;
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for (const auto & p : row) {
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j = p.var();
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if (!lia.is_base(j) && (!lia.at_bound(j) || !is_zero(lia.get_value(j).y))) {
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TRACE("gomory_cut", tout << "row is not gomory cut target:\n";
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lia.display_column(tout, j);
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tout << "infinitesimal: " << !is_zero(lia.get_value(j).y) << "\n";);
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return false;
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}
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}
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return true;
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}
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int gomory::find_basic_var() {
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int result = -1;
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unsigned n = 0;
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unsigned min_row_size = UINT_MAX;
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#if 0
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bool boxed = false;
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mpq min_range;
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#endif
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// Prefer smaller row size
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// Prefer boxed to non-boxed
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// Prefer smaller ranges
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for (unsigned j : lra.r_basis()) {
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if (!lia.column_is_int_inf(j))
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continue;
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const row_strip<mpq>& row = lra.get_row(lia.row_of_basic_column(j));
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if (!is_gomory_cut_target(row))
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continue;
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#if 0
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if (is_boxed(j) && (min_row_size == UINT_MAX || 4*row.size() < 5*min_row_size)) {
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lar_core_solver & lcs = lra.m_mpq_lar_core_solver;
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auto new_range = lclia.m_r_upper_bounds()[j].x - lclia.m_r_lower_bounds()[j].x;
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if (!boxed) {
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result = j;
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n = 1;
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min_row_size = row.size();
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boxed = true;
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min_range = new_range;
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continue;
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}
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if (min_range > 2*new_range || ((5*min_range > 4*new_range && (random() % (++n)) == 0))) {
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result = j;
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n = 1;
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min_row_size = row.size();
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min_range = std::min(min_range, new_range);
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continue;
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}
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}
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#endif
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if (min_row_size == UINT_MAX ||
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2*row.size() < min_row_size ||
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(4*row.size() < 5*min_row_size && lia.random() % (++n) == 0)) {
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result = j;
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|
n = 1;
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min_row_size = std::min(min_row_size, row.size());
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}
|
|
}
|
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return result;
|
|
}
|
|
|
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lia_move gomory::operator()() {
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if (lra.move_non_basic_columns_to_bounds()) {
|
|
lp_status st = lra.find_feasible_solution();
|
|
(void)st;
|
|
lp_assert(st == lp_status::FEASIBLE || st == lp_status::OPTIMAL);
|
|
}
|
|
|
|
int j = find_basic_var();
|
|
if (j == -1) return lia_move::undef;
|
|
unsigned r = lia.row_of_basic_column(j);
|
|
const row_strip<mpq>& row = lra.get_row(r);
|
|
SASSERT(lra.row_is_correct(r));
|
|
SASSERT(is_gomory_cut_target(row));
|
|
lia.m_upper = false;
|
|
return cut(lia.m_t, lia.m_k, lia.m_ex, j, row);
|
|
}
|
|
|
|
|
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gomory::gomory(int_solver& lia): lia(lia), lra(lia.lra) { }
|
|
|
|
}
|