z3/src/math/lp/gomory.cpp

/*++
  Copyright (c) 2017 Microsoft Corporation

  Module Name:

  <name>

  Abstract:

  <abstract>

  Author:
  Nikolaj Bjorner (nbjorner)
  Lev Nachmanson (levnach)

  Revision History:


  --*/
#include "math/lp/gomory.h"
#include "math/lp/int_solver.h"
#include "math/lp/lar_solver.h"
#include "math/lp/lp_utils.h"
namespace lp {

class gomory::imp {
    lar_term   &          m_t; // the term to return in the cut
    mpq        &          m_k; // the right side of the cut
    explanation*          m_ex; // the conflict explanation
    unsigned              m_inf_col; // a basis column which has to be an integer but has a non integral value
    const row_strip<mpq>& m_row;
    const int_solver&     m_int_solver;
    mpq                   m_lcm_den;
    mpq                   m_f;
    mpq                   m_one_minus_f;
    mpq                   m_fj;
    mpq                   m_one_minus_fj;
    
    const impq & get_value(unsigned j) const { return m_int_solver.get_value(j); }
    bool is_real(unsigned j) const { return m_int_solver.is_real(j); }
    bool at_lower(unsigned j) const { return m_int_solver.at_lower(j); }
    bool at_upper(unsigned j) const { return m_int_solver.at_upper(j); }
    const impq & lower_bound(unsigned j) const { return m_int_solver.lower_bound(j); }
    const impq & upper_bound(unsigned j) const  { return m_int_solver.upper_bound(j); }
    constraint_index column_lower_bound_constraint(unsigned j) const { return m_int_solver.column_lower_bound_constraint(j); }
    constraint_index column_upper_bound_constraint(unsigned j) const { return m_int_solver.column_upper_bound_constraint(j); }
    bool column_is_fixed(unsigned j) const { return m_int_solver.m_lar_solver->column_is_fixed(j); }

    void int_case_in_gomory_cut(unsigned j) {
        lp_assert(m_int_solver.column_is_int(j) && m_fj.is_pos());
        TRACE("gomory_cut_detail", 
              tout << " k = " << m_k;
              tout << ", fj: " << m_fj << ", ";
              tout << (at_lower(j)?"at_lower":"at_upper")<< std::endl;
              );
        mpq new_a;
        if (at_lower(j)) {
            new_a = m_fj <= m_one_minus_f ? m_fj / m_one_minus_f : ((1 - m_fj) / m_f);
            lp_assert(new_a.is_pos());
            m_k.addmul(new_a, lower_bound(j).x);
            m_ex->push_justification(column_lower_bound_constraint(j));            
        }
        else {
            lp_assert(at_upper(j));
            // the upper terms are inverted: therefore we have the minus
            new_a = - (m_fj <= m_f ? m_fj / m_f  : ((1 - m_fj) / m_one_minus_f));
            lp_assert(new_a.is_neg());
            m_k.addmul(new_a, upper_bound(j).x);
            m_ex->push_justification(column_upper_bound_constraint(j));
        }
        m_t.add_coeff_var(new_a, j);
        m_lcm_den = lcm(m_lcm_den, denominator(new_a));
        TRACE("gomory_cut_detail", tout << "new_a = " << new_a << ", k = " << m_k << ", lcm_den = " << m_lcm_den << "\n";);
    }

    void real_case_in_gomory_cut(const mpq & a, unsigned j) {
        TRACE("gomory_cut_detail_real", tout << "real\n";);
        mpq new_a;
        if (at_lower(j)) {
            if (a.is_pos()) {
                new_a = a / m_one_minus_f;
            }
            else {
                new_a = - a / m_f;
            }
            m_k.addmul(new_a, lower_bound(j).x); // is it a faster operation than
            // k += lower_bound(j).x * new_a;  
            m_ex->push_justification(column_lower_bound_constraint(j));
        }
        else {
            lp_assert(at_upper(j));
            if (a.is_pos()) {
                new_a =  - a / m_f; 
            }
            else {
                new_a =   a / m_one_minus_f; 
            }
            m_k.addmul(new_a, upper_bound(j).x); //  k += upper_bound(j).x * new_a; 
            m_ex->push_justification(column_upper_bound_constraint(j));
        }
        TRACE("gomory_cut_detail_real", tout << a << "*v" << j << " k: " << m_k << "\n";);
        m_t.add_coeff_var(new_a, j);
    }

    lia_move report_conflict_from_gomory_cut() {
        lp_assert(m_k.is_pos());
        // conflict 0 >= k where k is positive
        m_k.neg(); // returning 0 <= -k
        return lia_move::conflict;
    }

    void adjust_term_and_k_for_some_ints_case_gomory() {
        lp_assert(!m_t.is_empty());
        // k = 1 + sum of m_t at bounds
        auto pol = m_t.coeffs_as_vector();
        m_t.clear();
        if (pol.size() == 1) {
            TRACE("gomory_cut_detail", tout << "pol.size() is 1" << std::endl;);
            unsigned v = pol[0].second;
            lp_assert(m_int_solver.column_is_int(v));
            const mpq& a = pol[0].first;
            m_k /= a;
            if (a.is_pos()) { // we have av >= k
                if (!m_k.is_int())
                    m_k = ceil(m_k);
                // switch size
                m_t.add_coeff_var(- mpq(1), v);
                m_k.neg();
            } else {
                if (!m_k.is_int())
                    m_k = floor(m_k);
                m_t.add_coeff_var(mpq(1), v);
            }
        } else {
            m_lcm_den = lcm(m_lcm_den, denominator(m_k));
            lp_assert(m_lcm_den.is_pos());
            TRACE("gomory_cut_detail", tout << "pol.size() > 1 den: " << m_lcm_den << std::endl;);
            if (!m_lcm_den.is_one()) {
                // normalize coefficients of integer parameters to be integers.
                for (auto & pi: pol) {
                    pi.first *= m_lcm_den;
                    SASSERT(!m_int_solver.column_is_int(pi.second) || pi.first.is_int());
                }
                m_k *= m_lcm_den;
            }
            // negate everything to return -pol <= -m_k
            for (const auto & pi: pol)
                m_t.add_coeff_var(-pi.first, pi.second);
            m_k.neg();
        }
        TRACE("gomory_cut_detail", tout << "k = " << m_k << std::endl;);
        lp_assert(m_k.is_int());
    }

    std::string var_name(unsigned j) const {
        return std::string("x") + std::to_string(j);
    }

    std::ostream& dump_coeff_val(std::ostream & out, const mpq & a) const {
        if (a.is_int()) {
            out << a;
        } 
        else if ( a >= zero_of_type<mpq>())
            out << "(/ " << numerator(a) << " " << denominator(a) << ")";
        else {
            out << "(- ( / " <<   numerator(-a) << " " << denominator(-a) << "))";
        }
        return out;
    }
    
    template <typename T>
    void dump_coeff(std::ostream & out, const T& c) const {
        out << "( * ";
        dump_coeff_val(out, c.coeff());
        out << " " << var_name(c.var()) << ")";
    }
    
    std::ostream& dump_row_coefficients(std::ostream & out) const {
        mpq lc(1);
        for (const auto& p : m_row) {
            lc = lcm(lc, denominator(p.coeff())); 
        }        
        for (const auto& p : m_row) {
            dump_coeff_val(out << " (* ", p.coeff()*lc) << " " << var_name(p.var()) << ")";
        }
        return out;
    }
    
    void dump_the_row(std::ostream& out) const {
        out << "; the row, excluding fixed vars\n";
        out << "(assert ( = ( +";
        dump_row_coefficients(out) << ") 0))\n";
    }

    void dump_declaration(std::ostream& out, unsigned v) const {
        out << "(declare-const " << var_name(v) << (m_int_solver.column_is_int(v) ? " Int" : " Real") << ")\n";
    }
    
    void dump_declarations(std::ostream& out) const {
        // for a column j the var name is vj
        for (const auto & p : m_row) {
            dump_declaration(out, p.var());
        }
        for (const auto& p : m_t) {
            unsigned v = p.var();
            if (m_int_solver.m_lar_solver->is_term(v)) {
                dump_declaration(out, v);
            }
        }
    }

    void dump_lower_bound_expl(std::ostream & out, unsigned j) const {
        out << "(assert (>= " << var_name(j) << " " << lower_bound(j).x << "))\n";
    }
    void dump_upper_bound_expl(std::ostream & out, unsigned j) const {
        out << "(assert (<= " << var_name(j) << " " << upper_bound(j).x << "))\n";
    }
    
    void dump_explanations(std::ostream& out) const {
        for (const auto & p : m_row) {            
            unsigned j = p.var();
            if (j == m_inf_col || (!is_real(j) && p.coeff().is_int())) {
                continue;
            }
            else if (at_lower(j)) {
                dump_lower_bound_expl(out, j);
            } else {
                lp_assert(at_upper(j));
                dump_upper_bound_expl(out, j);
            }
        }
    }

    std::ostream& dump_term_coefficients(std::ostream & out) const {
        for (const auto& p : m_t) {
            dump_coeff(out, p);
        }
        return out;
    }
    
    std::ostream& dump_term_sum(std::ostream & out) const {
        return dump_term_coefficients(out << "(+ ") << ")";
    }
    
    std::ostream& dump_term_le_k(std::ostream & out) const {
        return dump_term_sum(out << "(<= ") << " " << m_k << ")";
    }

    void dump_the_cut_assert(std::ostream & out) const {
        dump_term_le_k(out << "(assert (not ") << "))\n";
    }

    void dump_cut_and_constraints_as_smt_lemma(std::ostream& out) const {
        dump_declarations(out);
        dump_the_row(out);
        dump_explanations(out);
        dump_the_cut_assert(out);
        out << "(check-sat)\n";
    }

public:

    lia_move create_cut() {
        TRACE("gomory_cut",
              print_linear_combination_of_column_indices_only(m_row, tout << "applying cut at:\n"); tout << std::endl;
              for (auto & p : m_row) {
                  m_int_solver.m_lar_solver->m_mpq_lar_core_solver.m_r_solver.print_column_info(p.var(), tout);
              }
              tout << "inf_col = " << m_inf_col << std::endl;
              );
        
        // gomory will be   t <= k and the current solution has a property t > k
        m_k = 1;
        m_t.clear();
        mpq m_lcm_den(1);
        bool some_int_columns = false;
        mpq m_f  = fractional_part(get_value(m_inf_col));
        TRACE("gomory_cut_detail", tout << "m_f: " << m_f << ", ";
              tout << "1 - m_f: " << 1 - m_f << ", get_value(m_inf_col).x - m_f = " << get_value(m_inf_col).x - m_f << "\n";);
        lp_assert(m_f.is_pos() && (get_value(m_inf_col).x - m_f).is_int());  

        mpq one_min_m_f = 1 - m_f;
        for (const auto & p : m_row) {
            unsigned j = p.var();
            if (j == m_inf_col) {
                lp_assert(p.coeff() == one_of_type<mpq>());
                TRACE("gomory_cut_detail", tout << "seeing basic var\n";);
                continue;
            }

             // use -p.coeff() to make the format compatible with the format used in: Integrating Simplex with DPLL(T)
            if (is_real(j)) {  
                real_case_in_gomory_cut(- p.coeff(), j);
            } else {
                if (p.coeff().is_int()) {
                    // m_fj will be zero and no monomial will be added
                    continue;
                }
                some_int_columns = true;
                m_fj = fractional_part(-p.coeff());
                m_one_minus_fj = 1 - m_fj;
                int_case_in_gomory_cut(j);
            }
        }

        if (m_t.is_empty())
            return report_conflict_from_gomory_cut();
        if (some_int_columns)
            adjust_term_and_k_for_some_ints_case_gomory();
        lp_assert(m_int_solver.current_solution_is_inf_on_cut());
        TRACE("gomory_cut_detail", dump_cut_and_constraints_as_smt_lemma(tout););
        m_int_solver.m_lar_solver->subs_term_columns(m_t);
        TRACE("gomory_cut", print_linear_combination_of_column_indices_only(m_t, tout << "gomory cut:"); tout << " <= " << m_k << std::endl;);
        return lia_move::cut;
    }

    imp(lar_term & t, mpq & k, explanation* ex, unsigned basic_inf_int_j, const row_strip<mpq>& row, const int_solver& int_slv ) :
        m_t(t),
        m_k(k),
        m_ex(ex),
        m_inf_col(basic_inf_int_j),
        m_row(row),
        m_int_solver(int_slv),
        m_lcm_den(1),
        m_f(fractional_part(get_value(basic_inf_int_j).x)),
        m_one_minus_f(1 - m_f) {}
    
};

lia_move gomory::create_cut() {
    return m_imp->create_cut();
}

gomory::gomory(lar_term & t, mpq & k, explanation* ex, unsigned basic_inf_int_j, const row_strip<mpq>& row, const int_solver& s) {
    m_imp = alloc(imp, t, k, ex, basic_inf_int_j, row, s);
}

gomory::~gomory() { dealloc(m_imp); }

}