mirror of
https://github.com/Z3Prover/z3
synced 2025-04-13 12:28:44 +00:00
752 lines
23 KiB
C++
752 lines
23 KiB
C++
/*
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Copyright (c) 2017 Microsoft Corporation
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Author: Lev Nachmanson
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*/
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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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#include "math/lp/monic.h"
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#include "math/lp/gomory.h"
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#include "math/lp/int_branch.h"
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#include "math/lp/int_cube.h"
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namespace lp {
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int_solver::patcher::patcher(int_solver& lia):
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lia(lia),
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lra(lia.lra),
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lrac(lia.lrac)
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{}
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void int_solver::patcher::remove_fixed_vars_from_base() {
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unsigned num = lra.A_r().column_count();
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for (unsigned v = 0; v < num; v++) {
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if (!lia.is_base(v) || !lia.is_fixed(v))
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continue;
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auto const & r = lra.basic2row(v);
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for (auto const& c : r) {
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if (c.var() != v && !lia.is_fixed(c.var())) {
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lra.pivot(c.var(), v);
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break;
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}
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}
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}
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}
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unsigned int_solver::patcher::count_non_int() {
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unsigned non_int = 0;
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for (auto j : lra.r_basis())
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if (lra.column_is_int(j) && !lra.column_value_is_int(j))
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++non_int;
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return non_int;
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}
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lia_move int_solver::patcher::patch_basic_columns() {
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remove_fixed_vars_from_base();
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lia.settings().stats().m_patches++;
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lp_assert(lia.is_feasible());
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// unsigned non_int_before, non_int_after;
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// non_int_before = count_non_int();
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// unsigned num = lra.A_r().column_count();
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for (unsigned j : lra.r_basis())
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if (!lra.get_value(j).is_int())
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patch_basic_column(j);
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// non_int_after = count_non_int();
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// verbose_stream() << non_int_before << " -> " << non_int_after << "\n";
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if (!lia.has_inf_int()) {
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lia.settings().stats().m_patches_success++;
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return lia_move::sat;
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}
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return lia_move::undef;
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}
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// clang-format on
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/**
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* \brief find integral and minimal, in the absolute values, deltas such that x - alpha*delta is integral too.
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*/
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bool get_patching_deltas(const rational& x, const rational& alpha,
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rational& delta_plus, rational& delta_minus) {
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auto a1 = numerator(alpha);
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auto a2 = denominator(alpha);
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auto x1 = numerator(x);
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auto x2 = denominator(x);
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if (!divides(x2, a2))
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return false;
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// delta has to be integral.
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// We need to find delta such that x1/x2 + (a1/a2)*delta is integral (we are going to flip the delta sign later).
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// Then a2*x1/x2 + a1*delta is integral, but x2 and x1 are coprime:
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// that means that t = a2/x2 is
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// integral. We established that a2 = x2*t Then x1 + a1*delta*(x2/a2) = x1
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// + a1*(delta/t) is integral. Taking into account that t and a1 are
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// coprime we have delta = t*k, where k is an integer.
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rational t = a2 / x2;
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// std::cout << "t = " << t << std::endl;
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// Now we have x1/x2 + (a1/x2)*k is integral, or (x1 + a1*k)/x2 is integral.
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// It is equivalent to x1 + a1*k = x2*m, where m is an integer
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// We know that a2 and a1 are coprime, and x2 divides a2, so x2 and a1 are
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// coprime. We can find u and v such that u*a1 + v*x2 = 1.
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rational u, v;
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gcd(a1, x2, u, v);
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lp_assert(gcd(a1, x2, u, v).is_one());
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// std::cout << "u = " << u << ", v = " << v << std::endl;
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// std::cout << "x= " << (x1 / x2) << std::endl;
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// std::cout << "x + (a1 / a2) * (-u * t) * x1 = "
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// << x + (a1 / a2) * (-u * t) * x1 << std::endl;
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lp_assert((x + (a1 / a2) * (-u * t) * x1).is_int());
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// 1 = (u- l*x2 ) * a1 + (v + l*a1)*x2, for every integer l.
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rational d = u * t * x1;
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// We can prove that x+alpha*d is integral,
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// and any other delta, satisfying x+alpha*delta, is equal to d modulo a2.
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delta_plus = mod(d, a2);
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lp_assert(delta_plus > 0);
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delta_minus = delta_plus - a2;
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lp_assert(delta_minus < 0);
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return true;
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}
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/**
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* \brief try to patch the basic column v
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*/
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bool int_solver::patcher::patch_basic_column_on_row_cell(unsigned v, row_cell<mpq> const& c) {
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if (v == c.var())
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return false;
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if (!lra.column_is_int(c.var())) // could use real to patch integer
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return false;
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if (c.coeff().is_int())
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return false;
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mpq a = fractional_part(c.coeff());
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mpq r = fractional_part(lra.get_value(v));
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lp_assert(0 < r && r < 1);
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lp_assert(0 < a && a < 1);
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mpq delta_plus, delta_minus;
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if (!get_patching_deltas(r, a, delta_plus, delta_minus))
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return false;
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if (lia.random() % 2) {
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return try_patch_column(v, c.var(), delta_plus) ||
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try_patch_column(v, c.var(), delta_minus);
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} else {
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return try_patch_column(v, c.var(), delta_minus) ||
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try_patch_column(v, c.var(), delta_plus);
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}
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}
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bool int_solver::patcher::try_patch_column(unsigned v, unsigned j, mpq const& delta) {
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const auto & A = lra.A_r();
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if (delta < 0) {
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if (lia.has_lower(j) && lia.get_value(j) + impq(delta) < lra.get_lower_bound(j))
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return false;
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}
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else {
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if (lia.has_upper(j) && lia.get_value(j) + impq(delta) > lra.get_upper_bound(j))
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return false;
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}
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for (auto const& c : A.column(j)) {
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unsigned row_index = c.var();
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unsigned bj = lrac.m_r_basis[row_index];
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auto old_val = lia.get_value(bj);
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auto new_val = old_val - impq(c.coeff()*delta);
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if (lia.has_lower(bj) && new_val < lra.get_lower_bound(bj))
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return false;
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if (lia.has_upper(bj) && new_val > lra.get_upper_bound(bj))
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return false;
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if (old_val.is_int() && !new_val.is_int()){
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return false; // do not waste resources on this case
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}
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// if bj == v, then, because we are patching the lra.get_value(v),
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// we just need to assert that the lra.get_value(v) would be integral.
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lp_assert(bj != v || lra.from_model_in_impq_to_mpq(new_val).is_int());
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}
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lra.set_value_for_nbasic_column(j, lia.get_value(j) + impq(delta));
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return true;
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}
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void int_solver::patcher::patch_basic_column(unsigned v) {
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SASSERT(!lia.is_fixed(v));
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for (auto const& c : lra.basic2row(v))
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if (patch_basic_column_on_row_cell(v, c))
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return;
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}
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int_solver::int_solver(lar_solver& lar_slv) :
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lra(lar_slv),
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lrac(lra.m_mpq_lar_core_solver),
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m_gcd(*this),
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m_patcher(*this),
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m_number_of_calls(0),
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m_hnf_cutter(*this),
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m_hnf_cut_period(settings().hnf_cut_period()) {
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lra.set_int_solver(this);
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}
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// this will allow to enable and disable tracking of the pivot rows
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struct check_return_helper {
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lar_solver& lra;
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bool m_track_touched_rows;
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check_return_helper(lar_solver& ls) :
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lra(ls),
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m_track_touched_rows(lra.touched_rows_are_tracked()) {
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lra.track_touched_rows(false);
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}
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~check_return_helper() {
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lra.track_touched_rows(m_track_touched_rows);
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}
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};
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lia_move int_solver::check(lp::explanation * e) {
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SASSERT(lra.ax_is_correct());
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if (!has_inf_int()) return lia_move::sat;
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m_t.clear();
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m_k.reset();
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m_ex = e;
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m_ex->clear();
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m_upper = false;
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lia_move r = lia_move::undef;
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if (m_gcd.should_apply()) r = m_gcd();
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check_return_helper pc(lra);
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if (settings().get_cancel_flag())
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return lia_move::undef;
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++m_number_of_calls;
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if (r == lia_move::undef && m_patcher.should_apply()) r = m_patcher();
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if (r == lia_move::undef && should_find_cube()) r = int_cube(*this)();
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if (r == lia_move::undef && should_hnf_cut()) r = hnf_cut();
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if (r == lia_move::undef && should_gomory_cut()) r = gomory(*this)();
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if (r == lia_move::undef) r = int_branch(*this)();
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return r;
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}
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std::ostream& int_solver::display_inf_rows(std::ostream& out) const {
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unsigned num = lra.A_r().column_count();
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for (unsigned v = 0; v < num; v++) {
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if (column_is_int(v) && !get_value(v).is_int()) {
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display_column(out, v);
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}
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}
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num = 0;
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for (unsigned i = 0; i < lra.A_r().row_count(); i++) {
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unsigned j = lrac.m_r_basis[i];
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if (column_is_int_inf(j)) {
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num++;
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lra.print_row(lra.A_r().m_rows[i], out);
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out << "\n";
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}
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}
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out << "num of int infeasible: " << num << "\n";
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return out;
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}
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bool int_solver::cut_indices_are_columns() const {
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for (lar_term::ival p : m_t) {
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if (p.column().index() >= lra.A_r().column_count())
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return false;
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}
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return true;
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}
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bool int_solver::current_solution_is_inf_on_cut() const {
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SASSERT(cut_indices_are_columns());
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const auto & x = lrac.m_r_x;
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impq v = m_t.apply(x);
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mpq sign = m_upper ? one_of_type<mpq>() : -one_of_type<mpq>();
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CTRACE("current_solution_is_inf_on_cut", v * sign <= impq(m_k) * sign,
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tout << "m_upper = " << m_upper << std::endl;
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tout << "v = " << v << ", k = " << m_k << std::endl;
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);
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return v * sign > impq(m_k) * sign;
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}
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bool int_solver::has_inf_int() const {
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return lra.has_inf_int();
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}
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constraint_index int_solver::column_upper_bound_constraint(unsigned j) const {
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return lra.get_column_upper_bound_witness(j);
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}
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constraint_index int_solver::column_lower_bound_constraint(unsigned j) const {
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return lra.get_column_lower_bound_witness(j);
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}
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unsigned int_solver::row_of_basic_column(unsigned j) const {
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return lra.row_of_basic_column(j);
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}
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lp_settings& int_solver::settings() {
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return lra.settings();
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}
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const lp_settings& int_solver::settings() const {
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return lra.settings();
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}
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bool int_solver::column_is_int(column_index const& j) const {
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return lra.column_is_int(j);
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}
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bool int_solver::is_real(unsigned j) const {
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return !column_is_int(j);
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}
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bool int_solver::value_is_int(unsigned j) const {
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return lra.column_value_is_int(j);
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}
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unsigned int_solver::random() {
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return settings().random_next();
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}
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const impq& int_solver::upper_bound(unsigned j) const {
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return lra.column_upper_bound(j);
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}
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const impq& int_solver::lower_bound(unsigned j) const {
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return lra.column_lower_bound(j);
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}
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bool int_solver::is_term(unsigned j) const {
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return lra.column_corresponds_to_term(j);
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}
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unsigned int_solver::column_count() const {
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return lra.column_count();
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}
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bool int_solver::should_find_cube() {
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return m_number_of_calls % settings().m_int_find_cube_period == 0;
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}
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bool int_solver::should_gomory_cut() {
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return m_number_of_calls % settings().m_int_gomory_cut_period == 0;
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}
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bool int_solver::should_hnf_cut() {
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return settings().enable_hnf() && m_number_of_calls % m_hnf_cut_period == 0;
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}
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lia_move int_solver::hnf_cut() {
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lia_move r = m_hnf_cutter.make_hnf_cut();
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if (r == lia_move::undef) {
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m_hnf_cut_period *= 2;
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}
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else {
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m_hnf_cut_period = settings().hnf_cut_period();
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}
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return r;
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}
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bool int_solver::has_lower(unsigned j) const {
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switch (lrac.m_column_types()[j]) {
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case column_type::fixed:
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case column_type::boxed:
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case column_type::lower_bound:
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return true;
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default:
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return false;
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}
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}
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bool int_solver::has_upper(unsigned j) const {
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switch (lrac.m_column_types()[j]) {
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case column_type::fixed:
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case column_type::boxed:
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case column_type::upper_bound:
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return true;
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default:
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return false;
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}
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}
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static void set_lower(impq & l, bool & inf_l, impq const & v ) {
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if (inf_l || v > l) {
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l = v;
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inf_l = false;
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}
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}
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static void set_upper(impq & u, bool & inf_u, impq const & v) {
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if (inf_u || v < u) {
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u = v;
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inf_u = false;
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}
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}
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// this function assumes that all basic columns dependend on j are feasible
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bool int_solver::get_freedom_interval_for_column(unsigned j, bool & inf_l, impq & l, bool & inf_u, impq & u, mpq & m) {
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if (lrac.m_r_heading[j] >= 0 || is_fixed(j)) // basic or fixed var
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return false;
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TRACE("random_update", display_column(tout, j) << ", is_int = " << column_is_int(j) << "\n";);
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impq const & xj = get_value(j);
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inf_l = true;
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inf_u = true;
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l = u = zero_of_type<impq>();
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m = mpq(1);
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if (has_lower(j))
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set_lower(l, inf_l, lower_bound(j) - xj);
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if (has_upper(j))
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set_upper(u, inf_u, upper_bound(j) - xj);
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const auto & A = lra.A_r();
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TRACE("random_update", tout << "m = " << m << "\n";);
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auto delta = [](mpq const& x, impq const& y, impq const& z) {
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if (x.is_one())
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return y - z;
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if (x.is_minus_one())
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return z - y;
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return (y - z) / x;
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};
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for (auto c : A.column(j)) {
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unsigned row_index = c.var();
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const mpq & a = c.coeff();
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unsigned i = lrac.m_r_basis[row_index];
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impq const & xi = get_value(i);
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lp_assert(lrac.m_r_solver.column_is_feasible(i));
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if (column_is_int(i) && !a.is_int() && xi.is_int())
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m = lcm(m, denominator(a));
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if (!inf_l && !inf_u) {
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if (l == u)
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continue;
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}
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if (a.is_neg()) {
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if (has_lower(i))
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set_lower(l, inf_l, delta(a, xi, lra.get_lower_bound(i)));
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if (has_upper(i))
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set_upper(u, inf_u, delta(a, xi, lra.get_upper_bound(i)));
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}
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else {
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if (has_upper(i))
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set_lower(l, inf_l, delta(a, xi, lra.get_upper_bound(i)));
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if (has_lower(i))
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set_upper(u, inf_u, delta(a, xi, lra.get_lower_bound(i)));
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}
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}
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l += xj;
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u += xj;
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TRACE("freedom_interval",
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tout << "freedom variable for:\n";
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tout << lra.get_variable_name(j);
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tout << "[";
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if (inf_l) tout << "-oo"; else tout << l;
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tout << "; ";
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if (inf_u) tout << "oo"; else tout << u;
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tout << "]\n";
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tout << "val = " << get_value(j) << "\n";
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tout << "return " << (inf_l || inf_u || l <= u);
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);
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return (inf_l || inf_u || l <= u);
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}
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bool int_solver::is_feasible() const {
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lp_assert(
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lrac.m_r_solver.calc_current_x_is_feasible_include_non_basis() ==
|
|
lrac.m_r_solver.current_x_is_feasible());
|
|
return lrac.m_r_solver.current_x_is_feasible();
|
|
}
|
|
|
|
const impq & int_solver::get_value(unsigned j) const {
|
|
return lrac.m_r_x[j];
|
|
}
|
|
|
|
std::ostream& int_solver::display_column(std::ostream & out, unsigned j) const {
|
|
return lrac.m_r_solver.print_column_info(j, out);
|
|
}
|
|
|
|
bool int_solver::column_is_int_inf(unsigned j) const {
|
|
return column_is_int(j) && (!value_is_int(j));
|
|
}
|
|
|
|
bool int_solver::is_base(unsigned j) const {
|
|
return lrac.m_r_heading[j] >= 0;
|
|
}
|
|
|
|
bool int_solver::is_boxed(unsigned j) const {
|
|
return lrac.m_column_types[j] == column_type::boxed;
|
|
}
|
|
|
|
bool int_solver::is_fixed(unsigned j) const {
|
|
return lrac.m_column_types[j] == column_type::fixed;
|
|
}
|
|
|
|
bool int_solver::is_free(unsigned j) const {
|
|
return lrac.m_column_types[j] == column_type::free_column;
|
|
}
|
|
|
|
bool int_solver::at_bound(unsigned j) const {
|
|
auto & mpq_solver = lrac.m_r_solver;
|
|
switch (mpq_solver.m_column_types[j] ) {
|
|
case column_type::fixed:
|
|
case column_type::boxed:
|
|
return
|
|
mpq_solver.m_lower_bounds[j] == get_value(j) ||
|
|
mpq_solver.m_upper_bounds[j] == get_value(j);
|
|
case column_type::lower_bound:
|
|
return mpq_solver.m_lower_bounds[j] == get_value(j);
|
|
case column_type::upper_bound:
|
|
return mpq_solver.m_upper_bounds[j] == get_value(j);
|
|
default:
|
|
return false;
|
|
}
|
|
}
|
|
|
|
bool int_solver::at_lower(unsigned j) const {
|
|
auto & mpq_solver = lrac.m_r_solver;
|
|
switch (mpq_solver.m_column_types[j] ) {
|
|
case column_type::fixed:
|
|
case column_type::boxed:
|
|
case column_type::lower_bound:
|
|
return mpq_solver.m_lower_bounds[j] == get_value(j);
|
|
default:
|
|
return false;
|
|
}
|
|
}
|
|
|
|
bool int_solver::at_upper(unsigned j) const {
|
|
auto & mpq_solver = lrac.m_r_solver;
|
|
switch (mpq_solver.m_column_types[j] ) {
|
|
case column_type::fixed:
|
|
case column_type::boxed:
|
|
case column_type::upper_bound:
|
|
return mpq_solver.m_upper_bounds[j] == get_value(j);
|
|
default:
|
|
return false;
|
|
}
|
|
}
|
|
|
|
std::ostream & int_solver::display_row(std::ostream & out, lp::row_strip<rational> const & row) const {
|
|
bool first = true;
|
|
auto & rslv = lrac.m_r_solver;
|
|
for (const auto &c : row) {
|
|
if (is_fixed(c.var())) {
|
|
if (!get_value(c.var()).is_zero()) {
|
|
impq val = get_value(c.var()) * c.coeff();
|
|
if (!first && val.is_pos())
|
|
out << "+";
|
|
if (val.y.is_zero())
|
|
out << val.x << " ";
|
|
else
|
|
out << val << " ";
|
|
}
|
|
first = false;
|
|
continue;
|
|
}
|
|
if (c.coeff().is_one())
|
|
{
|
|
if (!first)
|
|
out << "+";
|
|
}
|
|
else if (c.coeff().is_minus_one())
|
|
out << "-";
|
|
else {
|
|
if (c.coeff().is_pos() && !first)
|
|
out << "+";
|
|
if (c.coeff().is_big())
|
|
out << " b*";
|
|
else
|
|
out << c.coeff();
|
|
}
|
|
out << rslv.column_name(c.var()) << " ";
|
|
first = false;
|
|
}
|
|
out << "\n";
|
|
for (const auto &c : row) {
|
|
if (is_fixed(c.var()))
|
|
continue;
|
|
rslv.print_column_info(c.var(), out);
|
|
if (is_base(c.var()))
|
|
out << "j" << c.var() << " base\n";
|
|
}
|
|
return out;
|
|
}
|
|
|
|
std::ostream& int_solver::display_row_info(std::ostream & out, unsigned row_index) const {
|
|
auto & rslv = lrac.m_r_solver;
|
|
auto const& row = rslv.m_A.m_rows[row_index];
|
|
return display_row(out, row);
|
|
}
|
|
|
|
bool int_solver::shift_var(unsigned j, unsigned range) {
|
|
if (is_fixed(j) || is_base(j))
|
|
return false;
|
|
if (settings().get_cancel_flag())
|
|
return false;
|
|
bool inf_l = false, inf_u = false;
|
|
impq l, u;
|
|
mpq m;
|
|
if (!get_freedom_interval_for_column(j, inf_l, l, inf_u, u, m))
|
|
return false;
|
|
if (settings().get_cancel_flag())
|
|
return false;
|
|
const impq & x = get_value(j);
|
|
// x, the value of j column, might be shifted on a multiple of m
|
|
|
|
if (inf_l && inf_u) {
|
|
impq new_val = m * impq(random() % (range + 1)) + x;
|
|
lra.set_value_for_nbasic_column(j, new_val);
|
|
return true;
|
|
}
|
|
if (column_is_int(j)) {
|
|
if (!inf_l) {
|
|
l = impq(ceil(l));
|
|
}
|
|
if (!inf_u) {
|
|
u = impq(floor(u));
|
|
}
|
|
}
|
|
if (!inf_l && !inf_u && l >= u)
|
|
return false;
|
|
|
|
|
|
if (inf_u) {
|
|
SASSERT(!inf_l);
|
|
impq new_val = x + m * impq(random() % (range + 1));
|
|
lra.set_value_for_nbasic_column(j, new_val);
|
|
return true;
|
|
}
|
|
|
|
if (inf_l) {
|
|
SASSERT(!inf_u);
|
|
impq new_val = x - m * impq(random() % (range + 1));
|
|
lra.set_value_for_nbasic_column(j, new_val);
|
|
return true;
|
|
}
|
|
|
|
SASSERT(!inf_l && !inf_u);
|
|
// The shift has to be a multiple of m: let us look for s, such that the shift is m*s.
|
|
// We have new_val = x+m*s <= u, so m*s <= u-x and, finally, s <= floor((u- x)/m) = a
|
|
// The symmetric reasoning gives us s >= ceil((l-x)/m) = b
|
|
// We randomly pick s in the segment [b, a]
|
|
mpq a = floor((u - x) / m);
|
|
mpq b = ceil((l - x) / m);
|
|
mpq r = a - b;
|
|
if (!r.is_pos())
|
|
return false;
|
|
TRACE("int_solver", tout << "a = " << a << ", b = " << b << ", r = " << r<< ", m = " << m << "\n";);
|
|
if (r < mpq(range))
|
|
range = static_cast<unsigned>(r.get_uint64());
|
|
|
|
mpq s = b + mpq(random() % (range + 1));
|
|
impq new_val = x + m * impq(s);
|
|
TRACE("int_solver", tout << "new_val = " << new_val << "\n";);
|
|
SASSERT(l <= new_val && new_val <= u);
|
|
lra.set_value_for_nbasic_column(j, new_val);
|
|
return true;
|
|
}
|
|
|
|
// not used:
|
|
bool int_solver::non_basic_columns_are_at_bounds() const {
|
|
for (unsigned j : lrac.m_r_nbasis) {
|
|
auto & val = lrac.m_r_x[j];
|
|
switch (lrac.m_column_types()[j]) {
|
|
case column_type::boxed:
|
|
if (val != lrac.m_r_lower_bounds()[j] && val != lrac.m_r_upper_bounds()[j])
|
|
return false;
|
|
break;
|
|
case column_type::lower_bound:
|
|
if (val != lrac.m_r_lower_bounds()[j])
|
|
return false;
|
|
break;
|
|
case column_type::upper_bound:
|
|
if (val != lrac.m_r_upper_bounds()[j])
|
|
return false;
|
|
break;
|
|
default:
|
|
if (column_is_int(j) && !val.is_int()) {
|
|
return false;
|
|
}
|
|
}
|
|
}
|
|
return true;
|
|
}
|
|
|
|
int int_solver::select_int_infeasible_var() {
|
|
int result = -1;
|
|
mpq range;
|
|
mpq new_range;
|
|
mpq small_value(1024);
|
|
unsigned n = 0;
|
|
lar_core_solver & lcs = lra.m_mpq_lar_core_solver;
|
|
unsigned prev_usage = 0; // to quiet down the compile
|
|
|
|
enum state { small_box, is_small_value, any_value, not_found };
|
|
state st = not_found;
|
|
|
|
for (unsigned j : lra.r_basis()) {
|
|
if (!column_is_int_inf(j))
|
|
continue;
|
|
unsigned usage = lra.usage_in_terms(j);
|
|
if (is_boxed(j) && (new_range = lcs.m_r_upper_bounds()[j].x - lcs.m_r_lower_bounds()[j].x - rational(2*usage)) <= small_value) {
|
|
SASSERT(!is_fixed(j));
|
|
if (st != small_box) {
|
|
n = 0;
|
|
st = small_box;
|
|
}
|
|
if (n == 0 || new_range < range) {
|
|
result = j;
|
|
range = new_range;
|
|
n = 1;
|
|
}
|
|
else if (new_range == range && (random() % (++n) == 0)) {
|
|
result = j;
|
|
}
|
|
continue;
|
|
}
|
|
if (st == small_box)
|
|
continue;
|
|
impq const& value = get_value(j);
|
|
if (abs(value.x) < small_value ||
|
|
(has_upper(j) && small_value > upper_bound(j).x - value.x) ||
|
|
(has_lower(j) && small_value > value.x - lower_bound(j).x)) {
|
|
if (st != is_small_value) {
|
|
n = 0;
|
|
st = is_small_value;
|
|
}
|
|
if (random() % (++n) == 0)
|
|
result = j;
|
|
}
|
|
if (st == is_small_value)
|
|
continue;
|
|
SASSERT(st == not_found || st == any_value);
|
|
st = any_value;
|
|
if (n == 0 || usage > prev_usage) {
|
|
result = j;
|
|
prev_usage = usage;
|
|
n = 1;
|
|
}
|
|
else if (usage > 0 && usage == prev_usage && (random() % (++n) == 0))
|
|
result = j;
|
|
}
|
|
|
|
return result;
|
|
}
|
|
|
|
|
|
}
|