Added a solver for the Maximum Weight Independent Set problem

Mesh/mwis.hpp

+#include "search.hpp"
+
+#include <boost/iterator/counting_iterator.hpp>
+
+#include <boost/graph/random.hpp>
+#include <boost/graph/graph_traits.hpp>
+#include <boost/graph/properties.hpp>
+#include <boost/graph/graph_concepts.hpp>
+#include <boost/graph/adjacency_list.hpp>
+
+#include <random>
+#include <vector>
+#include <functional>
+#include <limits>
+#include <numeric>
+#include <algorithm>
+#include <iostream>
+
+namespace mwis {
+
+/**
+ * A functor to calculate an upper bound to the maximum weight independent set
+ * in a graph.
+ *
+ * A Lagrangian relaxation of the problem is used:
+ *
+ *   UB = min_{λ >= 0} max_{x_i \in {0, 1}}
+ *           (\sum_{i \in V} x_iw_i) -
+ *           \sum_{c \in C} λ_c(\sum_{i \in c} - 1)
+ *
+ * Where C is a set of cliques within the graph, such that if there is an edge
+ * between two nodes in the graph, a clique that contains both nodes is present
+ * in C.
+ *
+ * The set {{u, v} | (u, v) \in E} can always be used as the set of
+ * cliques. However, using larger cliques can lead to better upper bounds.
+ */
+template<typename Graph, typename WeightMap>
+class lagrangian_bound {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+  typedef typename boost::property_traits<WeightMap>::value_type weight;
+
+  typedef typename boost::property_map<
+    Graph, boost::vertex_index_t>::type index_map;
+
+  typedef boost::iterator_property_map<
+    std::vector<bool>::iterator, index_map> selected_map_type;
+  typedef boost::iterator_property_map<
+    std::vector<std::vector<size_t>>::iterator, index_map> clique_map_type;
+  typedef boost::iterator_property_map<
+    typename std::vector<weight>::iterator, index_map> weight_map_type;
+
+  const Graph &_graph;
+  WeightMap _weight;
+  size_t _max_size;
+
+  std::vector<std::vector<vertex>> _clique_contents;
+
+  std::vector<weight> _lambda, _gradient;
+
+  std::vector<std::vector<size_t>> _clique_storage;
+  std::vector<bool> _selected_storage;
+  std::vector<weight> _effective_weight_storage;
+
+  clique_map_type _cliques;
+  selected_map_type _selected;
+  weight_map_type _effective_weight;
+public:
+  typedef weight result_type;
+
+  template<typename CliqueIterator>
+  lagrangian_bound(const Graph &graph, WeightMap weight_map,
+                   CliqueIterator clique_begin,
+                   CliqueIterator clique_end,
+                   size_t max_size = std::numeric_limits<size_t>::max()):
+    _graph(graph), _weight(weight_map), _max_size(max_size) {
+    auto vs = vertices(graph);
+    size_t vertex_count = vs.second - vs.first;
+
+    _clique_storage.resize(vertex_count);
+    _selected_storage.resize(vertex_count);
+    _effective_weight_storage.resize(vertex_count);
+
+    index_map ids = boost::get(boost::vertex_index, graph);
+
+    _cliques  = make_iterator_property_map(_clique_storage.begin(),   ids);
+    _selected = make_iterator_property_map(_selected_storage.begin(), ids);
+    _effective_weight = make_iterator_property_map(
+      _effective_weight_storage.begin(), ids);
+
+    size_t i = 0;
+    for (CliqueIterator it = clique_begin; it != clique_end; it++, i++) {
+      std::vector<vertex> to_add;
+      for (const vertex &v : *it) {
+        to_add.push_back(v);
+        get(_cliques, v).push_back(i);
+      }
+      _clique_contents.push_back(to_add);
+    }
+
+    _lambda.resize(_clique_contents.size());
+    _gradient.resize(_clique_contents.size());
+
+    for (weight &v : _lambda) v = weight(0);
+  }
+
+  /**
+   * Calculates an upper bound. This bound is always admissible, but may not be
+   * optimal.
+   *
+   * Vertices in [solution_begin, solution_end) are all forced to be part of the
+   * solution.
+   *
+   * Vertices in [rest_begin, rest_end) can be selected in any way to maximize
+   * the objective functions.
+   *
+   * Vertices in neither range are assumed not to be part of the solution.
+   */
+  template<typename SolutionIterator,
+           typename RestIterator>
+  weight operator()(SolutionIterator solution_begin,
+                    SolutionIterator solution_end,
+                    RestIterator rest_begin,
+                    RestIterator rest_end) {
+    weight best(std::numeric_limits<weight>::max());
+
+    for (size_t i = 0; i < _selected_storage.size(); i++)
+      _selected_storage[i] = false;
+
+    for (SolutionIterator it = solution_begin; it != solution_end; it++)
+      put(_selected, *it, true);
+
+    /**
+     * We can get away with very few steps, because the starting point is the
+     * optimum that was found the last time the algorithm was used, and the
+     * input does not usually change much between two calls (therefore, one
+     * would expect the optima to be close to each other).
+     */
+    static const size_t n = 2;
+    for (size_t i = 0; i < n; i++) {
+      compute_weights();
+      if (_max_size == std::numeric_limits<size_t>::max())
+        select(rest_begin, rest_end);
+      else
+        select(rest_begin, rest_end,
+               _max_size - (size_t)std::distance(solution_begin,
+                                                 solution_end));
+      best = std::min(best, evaluate());
+
+      if (i != n - 1) {
+        compute_gradient();
+        follow_gradient(weight(-1e-4 / 1));
+      }
+    }
+
+    return best;
+  }
+
+  /**
+   * For debugging purposes, displays any inconsistencies between the graph and
+   * the sequence of cliques.
+   *
+   * (u, v) \in E <=> there is a clique which contains both u and v.
+   */
+  void check_consistency() {
+    auto es = boost::edges(_graph);
+    for (auto eit = es.first; eit != es.second; eit++) {
+      bool found = false;
+
+      vertex v = boost::source(*eit, _graph);
+      vertex w = boost::target(*eit, _graph);
+
+      const std::vector<size_t> &cliques_v = get(_cliques, v);
+      const std::vector<size_t> &cliques_w = get(_cliques, w);
+
+      for (size_t clique : cliques_v) {
+        if (std::find(cliques_w.begin(), cliques_w.end(), clique) !=
+            cliques_w.end()) {
+          found = true;
+          break;
+        }
+      }
+
+      if (!found) {
+        std::cout << "no shared clique between connected vertices: " <<
+          v << " and " << w << "\n";
+      }
+    }
+
+    for (const std::vector<vertex> &clique : _clique_contents) {
+      for (vertex v : clique) {
+        for (vertex w : clique) {
+          if (v == w) continue;
+
+          if (!edge(v, w, _graph).second) {
+            std::cout << "no edge between " << v << " and " <<
+              w << "\n";
+          }
+        }
+      }
+    }
+  }
+private:
+  /**
+   * Calculates the gradient according to which nodes are currently selected.
+   */
+  void compute_gradient() {
+    std::transform(_clique_contents.begin(), _clique_contents.end(),
+                   _gradient.begin(),
+                   [&](const std::vector<vertex> &vertices) {
+                     weight sum{1};
+                     for (const vertex &v : vertices) {
+                       sum -= weight(get(_selected, v) ? 1 : 0);
+                     }
+
+                     return sum;
+                   });
+  }
+
+  /**
+   * Calculates the effective weight of each node (w_i - \sum_{c, x \in c} λ_c).
+   */
+  void compute_weights() {
+    for (size_t i = 0; i < _effective_weight_storage.size(); i++) {
+      weight result(_weight[i]);
+
+      for (size_t clique : _clique_storage[i])
+        result -= _lambda[clique];
+
+      _effective_weight_storage[i] = result;
+    }
+  }
+
+  /**
+   * Update the Lagrange multipliers in accordance with the last computed
+   * gradient, multiplied by scale.
+   */
+  void follow_gradient(weight scale) {
+    std::transform(_lambda.begin(), _lambda.end(), _gradient.begin(),
+                   _lambda.begin(), [&](const weight &cur, const weight &grad) {
+                     return std::max(weight(0), cur + scale * grad);
+                   });
+  }
+
+  /**
+   * Iterates over the given range, and selects the subset of its nodes which have
+   * a non-negative weight, so as to maximize the objective function.
+   */
+  template<typename Iterator>
+  void select(Iterator begin, Iterator end) {
+    for (Iterator it = begin; it != end; it++) {
+      vertex v = *it;
+      put(_selected, v, get(_effective_weight, v) >= weight(0));
+    }
+  }
+
+  /**
+   * Iterates over the given range, and selects subset of its nodes which have
+   * a non-negative weight, selecting at most n nodes.
+   */
+  template<typename Iterator>
+  void select(Iterator begin, Iterator end, size_t n) {
+    n = std::min((size_t)std::distance(begin, end), n);
+    std::vector<vertex> largest(n);
+
+    std::partial_sort_copy(
+      begin, end,
+      largest.begin(), largest.end(),
+      [&](vertex a ,vertex b) {
+        return get(_effective_weight, a) > get(_effective_weight, b);
+      });
+
+    for (Iterator it = begin; it != end; it++) {
+      vertex v = *it;
+      put(_selected, v, false);
+    }
+
+    for (vertex v : largest) {
+      if (get(_effective_weight, v) <= weight(0))
+        break;
+      put(_selected, v, true);
+    }
+  }
+
+
+  /**
+   * Calculates the upper bound for the current values of the Lagrange
+   * multipliers and the current selection of nodes.
+   */
+  weight evaluate() {
+    weight result(std::accumulate(_lambda.begin(), _lambda.end(),
+                                  weight(0), std::plus<weight>()));
+
+    for (size_t i = 0; i < _selected_storage.size(); i++) {
+      if (_selected_storage[i])
+        result += _effective_weight_storage[i];
+    }
+
+    return result;
+  }
+};
+
+template<typename T>
+struct max_bound {
+  typedef T result_type;
+
+  template<typename SolutionIter, typename RestIter>
+  T operator()(SolutionIter solution_begin, SolutionIter solution_end,
+               RestIter rest_begin, RestIter rest_end) {
+    return std::numeric_limits<T>::max();
+  }
+};
+
+template<typename Graph, typename WeightMap>
+class state_change;
+
+template<typename Graph, typename WeightMap>
+class clique_assignment;
+
+template<typename Graph, typename WeightMap>
+struct state {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+  typedef typename boost::property_traits<WeightMap>::value_type weight;
+
+  typedef typename boost::property_map<
+    Graph, boost::vertex_index_t>::type index_map;
+  typedef boost::iterator_property_map<
+    std::vector<std::vector<size_t>>::iterator, index_map> clique_map_type;
+
+  const Graph &graph;
+  WeightMap weight_map;
+
+  std::vector<std::vector<vertex>> cliques;
+  std::vector<size_t> clique_sizes;
+  std::set<vertex> invalid;
+  std::set<vertex> selectable;
+  std::vector<bool> assigned;
+
+  std::vector<std::vector<size_t>> clique_map_storage;
+  clique_map_type clique_map;
+
+  std::vector<vertex> solution;
+  weight solution_value;
+
+  template<typename CliqueIterator>
+  state(const Graph &graph,
+        WeightMap weight_map,
+        CliqueIterator clique_begin,
+        CliqueIterator clique_end):
+    graph(graph), weight_map(weight_map),
+    cliques(clique_begin, clique_end),
+    clique_sizes(cliques.size()),
+    selectable(vertices(graph).first, vertices(graph).second),
+    assigned(cliques.size(), false),
+    solution_value(0)
+    {
+      clique_map_storage.resize(num_vertices(graph));
+      index_map id_map = get(boost::vertex_index, graph);
+      clique_map = make_iterator_property_map(
+        clique_map_storage.begin(), id_map);
+
+      for (size_t i = 0; i < cliques.size(); i++)
+        clique_sizes[i] = cliques[i].size();
+
+      size_t id = 0;
+      for (std::vector<vertex> &clique : cliques) {
+        for (vertex v : clique)
+          get(clique_map, v).push_back(id);
+
+        std::sort(clique.begin(), clique.end(), [&](vertex a, vertex b) {
+          return get(weight_map, a) > get(weight_map, b);
+        });
+
+        id++;
+      }
+    }
+};
+
+template<typename Graph, typename WeightMap>
+class clique_assignment {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+  typedef typename boost::property_traits<WeightMap>::value_type weight;
+
+  size_t _id;
+  vertex _vertex;
+  bool _selected;
+
+public:
+  typedef state_change<Graph, WeightMap> result_type;
+  typedef state<Graph, WeightMap> argument_type;
+
+  clique_assignment(size_t id):
+    _id(id), _selected(false)
+    {}
+
+  clique_assignment(size_t id, vertex vertex_id):
+    _id(id), _vertex(vertex_id), _selected(true)
+    {}
+
+  state_change<Graph, WeightMap> operator()(
+    const state<Graph, WeightMap> &state) const {
+    std::vector<vertex> selected_vertex, removed;
+    std::vector<size_t> assigned;
+
+    weight new_weight(state.solution_value);
+
+    if (_selected) {
+      selected_vertex.push_back(_vertex);
+      new_weight += get(state.weight_map, _vertex);
+
+      auto edges = out_edges(_vertex, state.graph);
+      for (auto eit = edges.first; eit != edges.second; eit++) {
+        const vertex &other = target(*eit, state.graph);
+        if (state.invalid.find(other) == state.invalid.end()) {
+          removed.push_back(other);
+        }
+      }
+
+      for (size_t i : get(state.clique_map, _vertex)) {
+        if (!state.assigned[i]) assigned.push_back(i);
+      }
+    }
+    else {
+      for (vertex v : state.cliques[_id]) {
+        if (state.invalid.find(v) == state.invalid.end())
+          removed.push_back(v);
+      }
+
+      assigned.push_back(_id);
+    }
+
+    std::map<size_t, size_t> size_delta;
+    for (vertex v : removed) {
+      for (size_t clique : get(state.clique_map, v)) {
+        size_delta[clique]++;
+        if (size_delta[clique] == state.clique_sizes[clique]) {
+          if (!state.assigned[clique])
+            assigned.push_back(clique);
+        }
+      }
+    }
+
+    return state_change<Graph, WeightMap>(
+      assigned,
+      size_delta,
+      selected_vertex, removed,
+      state.solution_value, new_weight);
+  }
+};
+
+template<typename Graph, typename WeightMap>
+class state_change {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+  typedef typename boost::property_traits<WeightMap>::value_type weight;
+
+  std::vector<size_t> _assigned_id;
+  std::map<size_t, size_t> _size_delta;
+  std::vector<vertex> _selected;
+  std::vector<vertex> _invalidated;
+  weight _old_weight, _new_weight;
+public:
+  state_change() {}
+
+  state_change(const std::vector<size_t> &assigned_id,
+               const std::map<size_t, size_t> &size_delta,
+               const std::vector<vertex> &selected,
+               const std::vector<vertex> &invalidated,
+               weight old_weight, weight new_weight):
+    _assigned_id(assigned_id),
+    _size_delta(size_delta),
+    _selected(selected), _invalidated(invalidated),
+    _old_weight(old_weight), _new_weight(new_weight)
+    {}
+
+  void apply(state<Graph, WeightMap> &state) const {
+    for (size_t i : _assigned_id)
+      state.assigned[i] = true;
+
+    size_t old_size = state.solution.size();
+    state.solution.resize(state.solution.size() + _selected.size());
+    std::copy(_selected.begin(), _selected.end(),
+              state.solution.begin() + old_size);
+
+    for (vertex v : _selected)
+      state.selectable.erase(v);
+
+    state.solution_value = _new_weight;
+
+    for (vertex v : _invalidated) {
+      state.invalid.insert(v);
+      state.selectable.erase(v);
+    }
+
+    for (auto pair : _size_delta)
+      state.clique_sizes[pair.first] -= pair.second;
+  }
+
+  void reverse(state<Graph, WeightMap> &state) const {
+    for (size_t i : _assigned_id)
+      state.assigned[i] = false;
+
+    state.solution.resize(state.solution.size() - _selected.size());
+
+    for (vertex v : _selected)
+      state.selectable.insert(v);
+
+    state.solution_value = _old_weight;
+
+    for (vertex v : _invalidated) {
+      state.invalid.erase(v);
+      state.selectable.insert(v);
+    }
+
+    for (auto pair : _size_delta)
+      state.clique_sizes[pair.first] += pair.second;
+  }
+};
+
+template<typename Graph, typename WeightMap>
+struct visit_state {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+  typedef typename boost::property_traits<WeightMap>::value_type weight;
+
+  bool quiet;
+
+  weight best_value;
+  std::vector<vertex> best_solution;
+
+  visit_state(bool quiet = true): quiet(quiet), best_value(0) {}
+
+  void operator()(state<Graph, WeightMap> &state) {
+    if (best_value < state.solution_value) {
+      if (!quiet) {
+        std::cout << "new best: " << state.solution_value <<
+          " (" << state.solution.size() << ")\n";
+      }
+
+      best_value    = state.solution_value;
+      best_solution = state.solution;
+    }
+  }
+};
+
+template<typename Graph, typename WeightMap, typename UpperBound>
+class successor {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+  typedef typename std::vector<vertex>::iterator viterator;
+  typedef typename boost::property_traits<WeightMap>::value_type weight;
+
+  UpperBound &_bound;
+
+  size_t _visited;
+  weight *_best_weight;
+
+  size_t _limit;
+public:
+  typedef clique_assignment<Graph, WeightMap> action_type;
+
+  successor(UpperBound &bound, weight &best_weight,
+            size_t limit = std::numeric_limits<size_t>::max()):
+    _bound(bound), _visited(0), _best_weight(&best_weight), _limit(limit)
+    {}
+
+  template<typename OutputIterator>
+  void operator()(state<Graph, WeightMap> &state, OutputIterator it) {
+    if (_visited > 1000 * 1000 * 10)
+      return;
+    else
+      _visited++;
+
+    if (state.solution.size() == _limit)
+      return;
+
+    auto clique_begin = boost::make_counting_iterator<size_t>(0);
+    auto clique_end   = boost::make_counting_iterator<size_t>(
+      state.cliques.size());
+
+    auto clique_it = std::min_element(
+      clique_begin, clique_end, [&](size_t a, size_t b) {
+        return clique_key(state, a) < clique_key(state, b);
+      });
+
+    if (clique_it == clique_end || state.assigned[*clique_it])
+      return;
+
+    std::vector<vertex> vertices;
+    size_t id = *clique_it;
+    for (vertex v : state.cliques[id]) {
+      if (state.invalid.find(v) == state.invalid.end()) {
+        vertices.push_back(v);
+      }
+    }
+
+    if (vertices.size() > 0) {
+      weight max(_bound(state.solution.begin(), state.solution.end(),
+                        state.selectable.begin(), state.selectable.end()));
+      if (max <= *_best_weight)
+        return;
+    }
+
+    std::sort(vertices.begin(), vertices.end(),
+              [&](vertex a, vertex b) {
+                return vertex_regret(state, a) < vertex_regret(state, b);
+              });
+
+    for (vertex v : vertices)
+      *it++ = clique_assignment<Graph, WeightMap>(id, v);
+
+    *it++ = clique_assignment<Graph, WeightMap>(id);
+  }
+
+  std::tuple<int, size_t, weight>
+  clique_key(state<Graph, WeightMap> &state,
+             size_t id) {
+    if (state.assigned[id]) {
+      return std::make_tuple(1,
+                             std::numeric_limits<size_t>::max(),
+                             weight(0));
+    }
+    else if (state.clique_sizes[id] == 0) {
+      return std::make_tuple(0, 0, weight(0));
+    }
+    else {
+      vertex v = *std::find_if(
+        state.cliques[id].begin(), state.cliques[id].end(), [&](vertex v) {
+          return state.invalid.find(v) == state.invalid.end();
+        });
+
+      return std::make_tuple(id < 1361 ? 0 : 1,
+                             state.clique_sizes[id],
+                             -get(state.weight_map, v));
+    }
+  }
+
+  weight vertex_regret(const state<Graph, WeightMap> &state, vertex v) {
+    weight result(-get(state.weight_map, v));
+
+    auto es = out_edges(v, state.graph);
+    for (auto eit = es.first; eit != es.second; eit++) {
+      vertex u = target(*eit, state.graph);
+      if (state.invalid.find(u) != state.invalid.end())
+        result += get(state.weight_map, u);
+    }
+
+    return result;
+  }
+};
+
+template<typename Graph, typename WeightMap>
+class evaluator {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+  typedef typename boost::property_traits<WeightMap>::value_type weight;
+
+  std::vector<std::vector<vertex>> _cliques;
+  size_t _limit;
+public:
+  typedef weight result_type;
+  typedef state<Graph, WeightMap> argument_type;
+
+  template<typename Iterator>
+  evaluator(Iterator begin, Iterator end, size_t limit):
+    _cliques(begin, end), _limit(limit)
+    {}
+
+  weight operator()(state<Graph, WeightMap> &cur) const {
+    visit_state<Graph, WeightMap> visitor;
+
+    // max_bound<weight> bound;
+    lagrangian_bound<Graph, WeightMap>
+      bound(cur.graph, cur.weight_map, _cliques.begin(), _cliques.end(),
+            cur.solution.size() + _limit);
+    successor<Graph, WeightMap, decltype(bound)>
+      successor(bound, visitor.best_value, cur.solution.size() + _limit);
+
+    visitor(cur);
+    search::depth_limited_search(cur, visitor, successor, _limit);
+
+    return visitor.best_value;
+  }
+};
+
+template<typename Graph, typename WeightMap>
+struct lns_state {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+
+  const Graph &graph;
+  WeightMap weight_map;
+
+  std::vector<vertex> solution;
+
+  lns_state(const Graph &graph, WeightMap weight_map,
+            const std::vector<vertex> &solution):
+    graph(graph), weight_map(weight_map), solution(solution)
+    {}
+};
+
+template<typename Graph>
+struct lns_fragment {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+
+  std::set<vertex> vertices;
+
+  lns_fragment(const std::set<vertex> &vertices):
+    vertices(vertices)
+    {}
+};
+
+/**
+ * Behaves like a queue, but elements can only be inserted into the queue a
+ * limited amount of times.
+ */
+template<typename T>
+class limited_queue {
+  std::queue<T> _queue;
+  size_t _size, _max_size;
+public:
+  typedef T value_type;
+  typedef typename std::queue<T>::size_type size_type;
+
+  limited_queue(size_t max_size):
+    _size(0), _max_size(max_size) {}
+
+  void push(const T &t) {
+    if (_size != _max_size) {
+      _size++;
+      _queue.push(t);
+    }
+  }
+
+  void pop() { _queue.pop(); }
+
+  T &top() { return _queue.front(); }
+  const T &top() const { return _queue.front(); }
+
+  size_type size() const { return _queue.size(); }
+
+  bool empty() { return _queue.empty(); }
+};
+
+/**
+ * A visitor for graph traversal algoritms which records the target of every
+ * visited edge into a set.
+ */
+template<typename Graph, typename Tag>
+class set_recorder : public boost::base_visitor<set_recorder<Graph, Tag>> {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+  typedef typename boost::graph_traits<Graph>::edge_descriptor edge;
+
+  std::set<vertex> &_set;
+public:
+  typedef Tag event_filter;
+
+  set_recorder(std::set<vertex> &set): _set(set) {}
+
+  void operator()(edge e, const Graph &g) {
+    _set.insert(boost::target(e, g));
+  }
+};
+
+template<typename Graph, typename WeightMap>
+lns_fragment<Graph> fragment_selector(const lns_state<Graph, WeightMap> &state) {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+
+  std::random_device dev;
+  std::mt19937 gen(dev());
+
+  vertex v = boost::random_vertex(state.graph, gen);
+
+  size_t neighborhood_size = std::uniform_int_distribution<size_t>(50, 80)(gen);
+
+  std::set<vertex> set;
+  set.insert(v);
+  set_recorder<Graph, boost::on_black_target> recorder(set);
+
+  limited_queue<vertex> queue(neighborhood_size);
+
+  boost::breadth_first_search(
+    state.graph, v,
+    boost::buffer(queue).
+    visitor(boost::make_bfs_visitor(recorder)));
+
+  return lns_fragment<Graph>(set);
+}
+
+template<typename Graph>
+class lns_assignment {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+  std::vector<vertex> _vertices;
+public:
+  lns_assignment(const std::vector<vertex> &vertices): _vertices(vertices) {}
+
+  template<typename WeightMap>
+  void operator()(lns_state<Graph, WeightMap> &state,
+                  const lns_fragment<Graph> &fragment) const {
+    static int i = 0;
+    std::cout << "\rsize: " << state.solution.size() << " " << i++;
+    std::cout.flush();
+
+    state.solution.erase(
+      std::remove_if(
+        state.solution.begin(), state.solution.end(),
+        [&](vertex v) {
+          return fragment.vertices.find(v) != fragment.vertices.end();
+        }),
+      state.solution.end());
+
+    std::copy(_vertices.begin(), _vertices.end(),
+              std::back_inserter(state.solution));
+  }
+};
+
+template<typename Graph, typename WeightMap>
+class lns_search {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+  typedef typename boost::property_traits<WeightMap>::value_type weight;
+
+  std::vector<std::vector<vertex>> _cliques;
+public:
+  typedef lns_assignment<Graph> result_type;
+  typedef lns_state<Graph, WeightMap> first_argument_type;
+  typedef lns_fragment<Graph> second_argument_type;
+
+  template<typename Iterator>
+  lns_search(Iterator begin, Iterator end): _cliques(begin, end) {}
+
+  lns_assignment<Graph> operator()(const lns_state<Graph, WeightMap> &l_state,
+                                   const lns_fragment<Graph> &fragment) const {
+    state<Graph, WeightMap> search_state(l_state.graph, l_state.weight_map,
+                                         _cliques.begin(), _cliques.end());
+
+    std::vector<vertex> selected_vertex;
+    std::set<vertex> removed;
+    std::set<size_t> assigned;
+
+    weight new_weight(0);
+
+    for (vertex v : l_state.solution) {
+      if (fragment.vertices.find(v) != fragment.vertices.end())
+        continue;
+
+      selected_vertex.push_back(v);
+
+      new_weight += get(l_state.weight_map, v);
+
+      auto edges = out_edges(v, l_state.graph);
+      for (auto eit = edges.first; eit != edges.second; eit++) {
+        const vertex &other = target(*eit, l_state.graph);
+        removed.insert(other);
+      }
+
+      for (size_t i : get(search_state.clique_map, v))
+        assigned.insert(i);
+    }
+
+    auto vs = boost::vertices(l_state.graph);
+    for (auto it = vs.first; it != vs.second; it++) {
+      vertex v = *it;
+      if (fragment.vertices.find(v) == fragment.vertices.end())
+        removed.insert(v);
+    }
+
+    std::map<size_t, size_t> size_delta;
+    for (vertex v : removed) {
+      for (size_t clique : get(search_state.clique_map, v)) {
+        size_delta[clique]++;
+        if (size_delta[clique] == search_state.clique_sizes[clique]) {
+          assigned.insert(clique);
+        }
+      }
+    }
+
+    std::vector<size_t> assigned_v(assigned.begin(), assigned.end());
+    std::vector<vertex> removed_v(removed.begin(), removed.end());
+
+    state_change<Graph, WeightMap> delta(
+      assigned_v, size_delta,
+      selected_vertex, removed_v,
+      weight(0), new_weight);
+
+    delta.apply(search_state);
+
+    lagrangian_bound<Graph, WeightMap>
+      bound(l_state.graph, l_state.weight_map,
+            _cliques.begin(), _cliques.end());
+
+    visit_state<Graph, WeightMap> visitor;
+    visitor.best_solution = l_state.solution;
+    visitor.best_value = weight(0);
+    for (vertex v : visitor.best_solution)
+      visitor.best_value += get(l_state.weight_map, v);
+
+    successor<Graph, WeightMap, decltype(bound)> successor(
+      bound, visitor.best_value);
+
+    search::depth_first_search(search_state, visitor, successor);
+
+    std::vector<vertex> newly_selected;
+    for (vertex v : visitor.best_solution) {
+      if (fragment.vertices.find(v) != fragment.vertices.end())
+        newly_selected.push_back(v);
+    }
+
+    return lns_assignment<Graph>(newly_selected);
+  }
+};
+
+template<typename Graph, typename WeightMap>
+class direct_evaluator {
+public:
+  typedef typename boost::property_traits<WeightMap>::value_type result_type;
+  typedef state<Graph, WeightMap> argument_type;
+
+private:
+  result_type _bound;
+
+public:
+  direct_evaluator(result_type bound): _bound(bound) {}
+
+  result_type operator()(const argument_type &state) const {
+    return state.solution_value / _bound;
+  }
+};
+
+template<typename Graph, typename OutputIterator>
+void find_cliques(const Graph &graph, OutputIterator out) {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+
+  std::set<std::pair<vertex, vertex>> visited_edges;
+
+  auto es = edges(graph);
+  for (auto eit = es.first; eit != es.second; eit++) {
+    auto u = source(*eit, graph);
+    auto v = target(*eit, graph);
+
+    if (visited_edges.find(std::make_pair(u, v)) == visited_edges.end()) {
+      std::vector<vertex> contents, candidates;
+
+      auto connected_es = out_edges(u, graph);
+      for (auto eit = connected_es.first; eit != connected_es.second; eit++) {
+        auto new_v = target(*eit, graph);
+        if (new_v != v && visited_edges.find(std::make_pair(u, new_v)) ==
+            visited_edges.end()) {
+          candidates.push_back(new_v);
+        }
+      }
+
+      candidates.push_back(v);
+
+      contents.push_back(u);
+
+      while (!candidates.empty()) {
+        vertex v = candidates.back();
+        candidates.pop_back();
+
+        contents.push_back(v);
+
+        candidates.erase(
+          std::remove_if(candidates.begin(), candidates.end(),
+                         [&](vertex w) {
+                           return visited_edges.find(std::make_pair(v, w)) ==
+                             visited_edges.end();
+                         }),
+          candidates.end());
+      }
+
+      for (size_t i = 0; i < contents.size(); i++) {
+        for (size_t j = 0; j < i; j++) {
+          visited_edges.insert(std::make_pair(contents[i], contents[j]));
+          visited_edges.insert(std::make_pair(contents[j], contents[i]));
+        }
+      }
+
+      *out++ = contents;
+    }
+  }
+}
+
+}
+
+template<typename Graph, typename WeightMap, typename OutputIterator>
+void maximum_weight_independent_set(const Graph &graph, WeightMap weight_map,
+                                    OutputIterator out) {
+  typedef typename boost::graph_traits<Graph>::vertex_descriptor vertex;
+  typedef typename boost::property_traits<WeightMap>::value_type weight;
+
+  std::vector<std::vector<vertex>> cliques;
+  mwis::find_cliques(graph, std::back_inserter(cliques));
+
+  mwis::lagrangian_bound<Graph, WeightMap> bound(
+    graph, weight_map,
+    cliques.begin(), cliques.end());
+
+  bound.check_consistency();
+
+  mwis::visit_state<Graph, WeightMap> visitor;
+  mwis::successor<Graph, WeightMap, decltype(bound)> successor(
+    bound, visitor.best_value);
+
+  mwis::state<Graph, WeightMap> state(
+    graph, weight_map,
+    cliques.begin(), cliques.end());
+
+  // search::depth_first_search(state, visitor, successor);
+  // for (vertex v : visitor.best_solution)
+  //   *out++ = v;
+
+  auto vs = vertices(graph);
+  weight root_bound(bound(vs.second, vs.second, vs.first, vs.second));
+
+  mwis::direct_evaluator<Graph, WeightMap> direct_eval(root_bound);
+
+  mwis::max_bound<weight> max_bound;
+  mwis::successor<Graph, WeightMap, decltype(max_bound)>
+    max_successor(max_bound, visitor.best_value);
+
+  search::monte_carlo_tree_search(state, max_successor, direct_eval);
+  std::cout << "\n";
+
+  for (vertex v : state.solution)
+    *out++ = v;
+
+  // mwis::evaluator<Graph, WeightMap> eval(cliques.begin(), cliques.end(), 5);
+
+  // visitor(state);
+  // search::greedy_search(state, successor, eval);
+
+  // mwis::lns_state<Graph, WeightMap> l_state(graph, weight_map, state.solution);
+  // mwis::lns_fragment<Graph> (*l_selector)(const decltype(l_state)&) =
+  //   mwis::fragment_selector<Graph, WeightMap>;
+  // mwis::lns_search<Graph, WeightMap> l_search(cliques.begin(), cliques.end());
+
+  // search::large_neighborhood_search(l_state, l_selector, l_search, 300);
+  // std::cout << "\n";
+
+  // for (vertex v : l_state.solution)
+  //   *out++ = v;
+}