two_sat_example.cc

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/*
                          Aleph_w
 
  Data structures & Algorithms
  version 2.0.0b
  https://github.com/lrleon/Aleph-w
 
  This file is part of Aleph-w library
 
  Copyright (c) 2002-2026 Leandro Rabindranath Leon
 
  Permission is hereby granted, free of charge, to any person obtaining a copy
  of this software and associated documentation files (the "Software"), to deal
  in the Software without restriction, including without limitation the rights
  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
  copies of the Software, and to permit persons to whom the Software is
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  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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*/
 
# include <iostream>
# include <string>
 
# include <Two_Sat.H>
# include <tpl_sgraph.H>
# include <tpl_agraph.H>
 
using namespace std;
using namespace Aleph;
 
// =========================================================================
// Example 1: Basic Boolean formula
// =========================================================================
 
void example_basic_formula()
{
  cout << "==========================================================\n";
  cout << "Example 1: Basic Boolean Formula\n";
  cout << "==========================================================\n\n";
 
  // Formula: (x0 OR x1) AND (~x0 OR x2) AND (~x1 OR ~x2) AND (x0 OR x2)
  //
  // We use the literal-index API:
  //   pos_lit(i) = positive literal for variable i
  //   neg_lit(i) = negative literal for variable i
 
  Two_Sat<> sat(3);
 
  sat.add_clause(sat.pos_lit(0), sat.pos_lit(1));   // x0 v x1
  sat.add_clause(sat.neg_lit(0), sat.pos_lit(2));   // ~x0 v x2
  sat.add_clause(sat.neg_lit(1), sat.neg_lit(2));   // ~x1 v ~x2
  sat.add_clause(sat.pos_lit(0), sat.pos_lit(2));   // x0 v x2
 
  cout << "Formula:\n"
       << "  (x0 OR x1) AND (~x0 OR x2) AND (~x1 OR ~x2) AND (x0 OR x2)\n\n";
 
  auto [ok, assignment] = sat.solve();
 
  if (ok)
    {
      cout << "SATISFIABLE.  Assignment:\n";
      for (size_t i = 0; i < assignment.size(); ++i)
        cout << "  x" << i << " = " << (assignment(i) ? "true" : "false")
             << "\n";
    }
  else
    cout << "UNSATISFIABLE\n";
 
  cout << "\n";
}
 
// =========================================================================
// Example 2: Same formula but using the signed 1-based API
// =========================================================================
 
void example_signed_api()
{
  cout << "==========================================================\n";
  cout << "Example 2: Signed 1-based Variable API\n";
  cout << "==========================================================\n\n";
 
  // Same formula: (x0 OR x1) AND (~x0 OR x2) AND (~x1 OR ~x2) AND (x0 OR x2)
  //
  // With signed API: variable k (1-based) is +k, its negation is -k.
  // So x0 -> +1, x1 -> +2, x2 -> +3, ~x0 -> -1, etc.
 
  Two_Sat<> sat(3);
 
  sat.add_clause_signed(1, 2);     // x0 v x1
  sat.add_clause_signed(-1, 3);    // ~x0 v x2
  sat.add_clause_signed(-2, -3);   // ~x1 v ~x2
  sat.add_clause_signed(1, 3);     // x0 v x2
 
  cout << "Formula (same as Example 1, using signed API):\n"
       << "  (+1 OR +2) AND (-1 OR +3) AND (-2 OR -3) AND (+1 OR +3)\n\n";
 
  auto [ok, assignment] = sat.solve();
 
  if (ok)
    {
      cout << "SATISFIABLE.  Assignment:\n";
      for (size_t i = 0; i < assignment.size(); ++i)
        cout << "  x" << i << " = " << (assignment(i) ? "true" : "false")
             << "\n";
    }
  else
    cout << "UNSATISFIABLE\n";
 
  cout << "\n";
}
 
// =========================================================================
// Example 3: Graph 2-coloring
// =========================================================================
 
void example_2_coloring()
{
  cout << "==========================================================\n";
  cout << "Example 3: Graph 2-Coloring via 2-SAT\n";
  cout << "==========================================================\n\n";
 
  // A graph is 2-colorable iff it is bipartite.
  //
  // For each edge (u, v), we require xu XOR xv (different colors).
  // xu XOR xv is encoded as two clauses: (xu OR xv) AND (~xu OR ~xv).
  //
  // A) Path graph: 0-1-2-3 => bipartite => SAT
  {
    cout << "Path graph 0 - 1 - 2 - 3:\n";
    Two_Sat<> sat(4);
    sat.add_xor(sat.pos_lit(0), sat.pos_lit(1));  // edge 0-1
    sat.add_xor(sat.pos_lit(1), sat.pos_lit(2));  // edge 1-2
    sat.add_xor(sat.pos_lit(2), sat.pos_lit(3));  // edge 2-3
 
    auto [ok, assignment] = sat.solve();
    cout << "  2-colorable? " << (ok ? "YES" : "NO") << "\n";
    if (ok)
      {
        cout << "  Coloring:";
        for (size_t i = 0; i < 4; ++i)
          cout << " node" << i << "=" << (assignment(i) ? "A" : "B");
        cout << "\n";
      }
    cout << "\n";
  }
 
  // B) Triangle graph: 0-1-2-0 => odd cycle => NOT bipartite => UNSAT
  {
    cout << "Triangle 0 - 1 - 2 - 0:\n";
    Two_Sat<> sat(3);
    sat.add_xor(sat.pos_lit(0), sat.pos_lit(1));
    sat.add_xor(sat.pos_lit(1), sat.pos_lit(2));
    sat.add_xor(sat.pos_lit(0), sat.pos_lit(2));
 
    auto [ok, _] = sat.solve();
    cout << "  2-colorable? " << (ok ? "YES" : "NO") << "\n\n";
  }
 
  // C) Cycle of length 4: 0-1-2-3-0 => even cycle => bipartite
  {
    cout << "Square 0 - 1 - 2 - 3 - 0:\n";
    Two_Sat<> sat(4);
    sat.add_xor(sat.pos_lit(0), sat.pos_lit(1));
    sat.add_xor(sat.pos_lit(1), sat.pos_lit(2));
    sat.add_xor(sat.pos_lit(2), sat.pos_lit(3));
    sat.add_xor(sat.pos_lit(3), sat.pos_lit(0));
 
    auto [ok, assignment] = sat.solve();
    cout << "  2-colorable? " << (ok ? "YES" : "NO") << "\n";
    if (ok)
      {
        cout << "  Coloring:";
        for (size_t i = 0; i < 4; ++i)
          cout << " node" << i << "=" << (assignment(i) ? "A" : "B");
        cout << "\n";
      }
    cout << "\n";
  }
}
 
// =========================================================================
// Example 4: Team assignment with constraints
// =========================================================================
 
void example_team_assignment()
{
  cout << "==========================================================\n";
  cout << "Example 4: Team Assignment with Constraints\n";
  cout << "==========================================================\n\n";
 
  // 5 people must be split into two teams (A = true, B = false).
  //
  // Constraints:
  //   - Alice(0) and Bob(1) must be on the same team       => equivalence
  //   - Charlie(2) and Diana(3) cannot both be on team A   => at-most-one
  //   - Eve(4) must be on team A                           => unit
  //   - Eve(4) and Bob(1) must be on different teams       => XOR
 
  const char * names[] = {"Alice", "Bob", "Charlie", "Diana", "Eve"};
 
  Two_Sat<> sat(5);
 
  sat.add_equiv(sat.pos_lit(0), sat.pos_lit(1));             // Alice <-> Bob
  sat.add_clause(sat.neg_lit(2), sat.neg_lit(3));            // ~Charlie v ~Diana
  sat.set_true(sat.pos_lit(4));                              // Eve = true (team A)
  sat.add_xor(sat.pos_lit(4), sat.pos_lit(1));               // Eve XOR Bob
 
  cout << "Constraints:\n"
       << "  Alice and Bob:     same team\n"
       << "  Charlie and Diana: not both on team A\n"
       << "  Eve:               must be on team A\n"
       << "  Eve and Bob:       different teams\n\n";
 
  auto [ok, assignment] = sat.solve();
 
  if (ok)
    {
      cout << "SATISFIABLE.  Assignment:\n";
      for (size_t i = 0; i < 5; ++i)
        cout << "  " << names[i] << " -> team "
             << (assignment(i) ? "A" : "B") << "\n";
    }
  else
    cout << "UNSATISFIABLE: constraints cannot be met\n";
 
  cout << "\n";
}
 
// =========================================================================
// Example 5: Different graph backends
// =========================================================================
 
void example_graph_backends()
{
  cout << "==========================================================\n";
  cout << "Example 5: Different Graph Backends\n";
  cout << "==========================================================\n\n";
 
  // The same formula solved with three graph backends.
  // Formula: (x0 OR x1) AND (~x0 OR ~x1) (exactly one is true)
 
  auto run = [](const string & name, auto & sat) {
    sat.add_xor(sat.pos_lit(0), sat.pos_lit(1));
 
    auto [ok, assignment] = sat.solve();
    cout << "  " << name << ": ";
    if (ok)
      cout << "x0=" << assignment(0) << " x1=" << assignment(1) << "\n";
    else
      cout << "UNSAT\n";
  };
 
  // 1) Default: List_Digraph (tpl_graph.H)
  {
    Two_Sat<> sat(2);
    run("List_Digraph ", sat);
  }
 
  // 2) List_SDigraph (tpl_sgraph.H) — singly-linked lists
  {
    using DG = List_SDigraph<Graph_Snode<Empty_Class>,
                             Graph_Sarc<Empty_Class>>;
    Two_Sat<DG> sat(2);
    run("List_SDigraph", sat);
  }
 
  // 3) Array_Digraph (tpl_agraph.H) — array-based adjacency
  {
    using DG = Array_Digraph<Graph_Anode<Empty_Class>,
                             Graph_Aarc<Empty_Class>>;
    Two_Sat<DG> sat(2);
    run("Array_Digraph", sat);
  }
 
  cout << "\n";
}
 
// =========================================================================
 
int main()
{
  example_basic_formula();
  example_signed_api();
  example_2_coloring();
  example_team_assignment();
  example_graph_backends();
 
  return 0;
}