boost/graph/maximum_weighted_matching.hpp
//=======================================================================
// Copyright (c) 2018 Yi Ji
//
// Distributed under the Boost Software License, Version 1.0.
// (See accompanying file LICENSE_1_0.txt or copy at
// http://www.boost.org/LICENSE_1_0.txt)
//
//=======================================================================
#ifndef BOOST_GRAPH_MAXIMUM_WEIGHTED_MATCHING_HPP
#define BOOST_GRAPH_MAXIMUM_WEIGHTED_MATCHING_HPP
#include <algorithm> // for std::iter_swap
#include <boost/shared_ptr.hpp>
#include <boost/make_shared.hpp>
#include <boost/graph/max_cardinality_matching.hpp>
namespace boost
{
template <typename Graph, typename MateMap, typename VertexIndexMap>
typename property_traits<typename property_map<Graph, edge_weight_t>::type>::value_type
matching_weight_sum(const Graph& g, MateMap mate, VertexIndexMap vm)
{
typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator_t;
typedef typename graph_traits<Graph>::vertex_descriptor vertex_descriptor_t;
typedef typename property_traits<typename property_map<Graph, edge_weight_t>::type>::value_type edge_property_t;
edge_property_t weight_sum = 0;
vertex_iterator_t vi, vi_end;
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
{
vertex_descriptor_t v = *vi;
if (get(mate, v) != graph_traits<Graph>::null_vertex() && get(vm, v) < get(vm, get(mate,v)))
weight_sum += get(edge_weight, g, edge(v,mate[v],g).first);
}
return weight_sum;
}
template <typename Graph, typename MateMap>
inline typename property_traits<typename property_map<Graph, edge_weight_t>::type>::value_type
matching_weight_sum(const Graph& g, MateMap mate)
{
return matching_weight_sum(g, mate, get(vertex_index,g));
}
template <typename Graph, typename MateMap, typename VertexIndexMap>
class weighted_augmenting_path_finder
{
public:
template <typename T>
struct map_vertex_to_
{
typedef boost::iterator_property_map<typename std::vector<T>::iterator, VertexIndexMap> type;
};
typedef typename graph::detail::VERTEX_STATE vertex_state_t;
typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator_t;
typedef typename graph_traits<Graph>::vertex_descriptor vertex_descriptor_t;
typedef typename std::vector<vertex_descriptor_t>::const_iterator vertex_vec_iter_t;
typedef typename graph_traits<Graph>::out_edge_iterator out_edge_iterator_t;
typedef typename graph_traits<Graph>::edge_descriptor edge_descriptor_t;
typedef typename graph_traits<Graph>::edge_iterator edge_iterator_t;
typedef typename property_traits<typename property_map<Graph, edge_weight_t>::type>::value_type edge_property_t;
typedef std::deque<vertex_descriptor_t> vertex_list_t;
typedef std::vector<edge_descriptor_t> edge_list_t;
typedef typename map_vertex_to_<vertex_descriptor_t>::type vertex_to_vertex_map_t;
typedef typename map_vertex_to_<edge_property_t>::type vertex_to_weight_map_t;
typedef typename map_vertex_to_<bool>::type vertex_to_bool_map_t;
typedef typename map_vertex_to_<std::pair<vertex_descriptor_t, vertex_descriptor_t> >::type vertex_to_pair_map_t;
typedef typename map_vertex_to_<std::pair<edge_descriptor_t, bool> >::type vertex_to_edge_map_t;
typedef typename map_vertex_to_<vertex_to_edge_map_t>::type vertex_pair_to_edge_map_t;
class blossom
{
public:
typedef boost::shared_ptr<blossom> blossom_ptr_t;
std::vector<blossom_ptr_t> sub_blossoms;
edge_property_t dual_var;
blossom_ptr_t father;
blossom() : dual_var(0), father(blossom_ptr_t()) {}
// get the base vertex of a blossom by recursively getting
// its base sub-blossom, which is always the first one in
// sub_blossoms because of how we create and maintain blossoms
virtual vertex_descriptor_t get_base() const
{
const blossom* b = this;
while (!b->sub_blossoms.empty())
b = b->sub_blossoms[0].get();
return b->get_base();
}
// set a sub-blossom as a blossom's base by exchanging it
// with its first sub-blossom
void set_base(const blossom_ptr_t& sub)
{
for (blossom_iterator_t bi = sub_blossoms.begin(); bi != sub_blossoms.end(); ++bi)
{
if (sub.get() == bi->get())
{
std::iter_swap(sub_blossoms.begin(), bi);
break;
}
}
}
// get all vertices inside recursively
virtual std::vector<vertex_descriptor_t> vertices() const
{
std::vector<vertex_descriptor_t> all_vertices;
for (typename std::vector<blossom_ptr_t>::const_iterator bi = sub_blossoms.begin(); bi != sub_blossoms.end(); ++bi)
{
std::vector<vertex_descriptor_t> some_vertices = (*bi)->vertices();
all_vertices.insert(all_vertices.end(), some_vertices.begin(), some_vertices.end());
}
return all_vertices;
}
};
// a trivial_blossom only has one vertex and no sub-blossom;
// for each vertex v, in_blossom[v] is the trivial_blossom that contains it directly
class trivial_blossom : public blossom
{
public:
trivial_blossom(vertex_descriptor_t v) : trivial_vertex(v) {}
virtual vertex_descriptor_t get_base() const
{
return trivial_vertex;
}
virtual std::vector<vertex_descriptor_t> vertices() const
{
std::vector<vertex_descriptor_t> all_vertices;
all_vertices.push_back(trivial_vertex);
return all_vertices;
}
private:
vertex_descriptor_t trivial_vertex;
};
typedef boost::shared_ptr<blossom> blossom_ptr_t;
typedef typename std::vector<blossom_ptr_t>::iterator blossom_iterator_t;
typedef typename map_vertex_to_<blossom_ptr_t>::type vertex_to_blossom_map_t;
weighted_augmenting_path_finder(const Graph& arg_g, MateMap arg_mate, VertexIndexMap arg_vm) :
g(arg_g),
vm(arg_vm),
null_edge(std::pair<edge_descriptor_t, bool>(num_edges(g) == 0 ? edge_descriptor_t() : *edges(g).first, false)),
mate_vector(num_vertices(g)),
label_S_vector(num_vertices(g), graph_traits<Graph>::null_vertex()),
label_T_vector(num_vertices(g), graph_traits<Graph>::null_vertex()),
outlet_vector(num_vertices(g), graph_traits<Graph>::null_vertex()),
tau_idx_vector(num_vertices(g), graph_traits<Graph>::null_vertex()),
dual_var_vector(std::vector<edge_property_t>(num_vertices(g), std::numeric_limits<edge_property_t>::min())),
pi_vector(std::vector<edge_property_t>(num_vertices(g), std::numeric_limits<edge_property_t>::max())),
gamma_vector(std::vector<edge_property_t>(num_vertices(g), std::numeric_limits<edge_property_t>::max())),
tau_vector(std::vector<edge_property_t>(num_vertices(g), std::numeric_limits<edge_property_t>::max())),
in_blossom_vector(num_vertices(g)),
old_label_vector(num_vertices(g)),
critical_edge_vectors(num_vertices(g), std::vector<std::pair<edge_descriptor_t, bool> >(num_vertices(g), null_edge)),
mate(mate_vector.begin(), vm),
label_S(label_S_vector.begin(), vm),
label_T(label_T_vector.begin(), vm),
outlet(outlet_vector.begin(), vm),
tau_idx(tau_idx_vector.begin(), vm),
dual_var(dual_var_vector.begin(), vm),
pi(pi_vector.begin(), vm),
gamma(gamma_vector.begin(), vm),
tau(tau_vector.begin(), vm),
in_blossom(in_blossom_vector.begin(), vm),
old_label(old_label_vector.begin(), vm)
{
vertex_iterator_t vi, vi_end;
edge_iterator_t ei, ei_end;
edge_property_t max_weight = std::numeric_limits<edge_property_t>::min();
for (boost::tie(ei,ei_end) = edges(g); ei != ei_end; ++ei)
max_weight = std::max(max_weight, get(edge_weight, g, *ei));
typename std::vector<std::vector<std::pair<edge_descriptor_t, bool> > >::iterator vei;
for (boost::tie(vi,vi_end) = vertices(g), vei = critical_edge_vectors.begin(); vi != vi_end; ++vi, ++vei)
{
vertex_descriptor_t u = *vi;
mate[u] = get(arg_mate, u);
dual_var[u] = 2*max_weight;
in_blossom[u] = boost::make_shared<trivial_blossom>(u);
outlet[u] = u;
critical_edge_vector.push_back(vertex_to_edge_map_t(vei->begin(), vm));
}
critical_edge = vertex_pair_to_edge_map_t(critical_edge_vector.begin(), vm);
init();
}
// return the top blossom where v is contained inside
blossom_ptr_t in_top_blossom(vertex_descriptor_t v) const
{
blossom_ptr_t b = in_blossom[v];
while (b->father != blossom_ptr_t())
b = b->father;
return b;
}
// check if vertex v is in blossom b
bool is_in_blossom(blossom_ptr_t b, vertex_descriptor_t v) const
{
if (v == graph_traits<Graph>::null_vertex())
return false;
blossom_ptr_t vb = in_blossom[v]->father;
while (vb != blossom_ptr_t())
{
if (vb.get() == b.get())
return true;
vb = vb->father;
}
return false;
}
// return the base vertex of the top blossom that contains v
inline vertex_descriptor_t base_vertex(vertex_descriptor_t v) const
{
return in_top_blossom(v)->get_base();
}
// add an existed top blossom of base vertex v into new top
// blossom b as its sub-blossom
void add_sub_blossom(blossom_ptr_t b, vertex_descriptor_t v)
{
blossom_ptr_t sub = in_top_blossom(v);
sub->father = b;
b->sub_blossoms.push_back(sub);
if (sub->sub_blossoms.empty())
return;
for (blossom_iterator_t bi = top_blossoms.begin(); bi != top_blossoms.end(); ++bi)
{
if (bi->get() == sub.get())
{
top_blossoms.erase(bi);
break;
}
}
}
// when a top blossom is created or its base vertex getting an S-label,
// add all edges incident to this blossom into even_edges
void bloom(blossom_ptr_t b)
{
std::vector<vertex_descriptor_t> vertices_of_b = b->vertices();
vertex_vec_iter_t vi;
for (vi = vertices_of_b.begin(); vi != vertices_of_b.end(); ++vi)
{
out_edge_iterator_t oei, oei_end;
for (boost::tie(oei,oei_end) = out_edges(*vi, g); oei != oei_end; ++oei)
{
if (target(*oei,g) != *vi && mate[*vi] != target(*oei,g))
even_edges.push_back(*oei);
}
}
}
// assigning a T-label to a non S-vertex, along with outlet and updating pi value
// if updated pi[v] equals zero, augment the matching from its mate vertex
void put_T_label(vertex_descriptor_t v, vertex_descriptor_t T_label,
vertex_descriptor_t outlet_v, edge_property_t pi_v)
{
if (label_S[v] != graph_traits<Graph>::null_vertex())
return;
label_T[v] = T_label;
outlet[v] = outlet_v;
pi[v] = pi_v;
vertex_descriptor_t v_mate = mate[v];
if (pi[v] == 0)
{
label_T[v_mate] = graph_traits<Graph>::null_vertex();
label_S[v_mate] = v;
bloom(in_top_blossom(v_mate));
}
}
// get the missing T-label for a to-be-expanded base vertex
// the missing T-label is the last vertex of the path from outlet[v] to v
std::pair<vertex_descriptor_t, vertex_descriptor_t> missing_label(vertex_descriptor_t b_base)
{
vertex_descriptor_t missing_outlet = outlet[b_base];
if (outlet[b_base] == b_base)
return std::make_pair(graph_traits<Graph>::null_vertex(), missing_outlet);
vertex_iterator_t vi, vi_end;
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
old_label[*vi] = std::make_pair(label_T[*vi], outlet[*vi]);
std::pair<vertex_descriptor_t, vertex_state_t> child(outlet[b_base], graph::detail::V_EVEN);
blossom_ptr_t b = in_blossom[child.first];
for (; b->father->father != blossom_ptr_t(); b = b->father);
child.first = b->get_base();
if (child.first == b_base)
return std::make_pair(graph_traits<Graph>::null_vertex(), missing_outlet);
while (true)
{
std::pair<vertex_descriptor_t, vertex_state_t> child_parent = parent(child, true);
for (b = in_blossom[child_parent.first]; b->father->father != blossom_ptr_t(); b = b->father);
missing_outlet = child_parent.first;
child_parent.first = b->get_base();
if (child_parent.first == b_base)
break;
else
child = child_parent;
}
return std::make_pair(child.first, missing_outlet);
}
// expand a top blossom, put all its non-trivial sub-blossoms into top_blossoms
blossom_iterator_t expand_blossom(blossom_iterator_t bi, std::vector<blossom_ptr_t>& new_ones)
{
blossom_ptr_t b = *bi;
for (blossom_iterator_t i = b->sub_blossoms.begin(); i != b->sub_blossoms.end(); ++i)
{
blossom_ptr_t sub_blossom = *i;
vertex_descriptor_t sub_base = sub_blossom->get_base();
label_S[sub_base] = label_T[sub_base] = graph_traits<Graph>::null_vertex();
outlet[sub_base] = sub_base;
sub_blossom->father = blossom_ptr_t();
// new top blossoms cannot be pushed back into top_blossoms immediately,
// because push_back() may cause reallocation and then invalid iterators
if (!sub_blossom->sub_blossoms.empty())
new_ones.push_back(sub_blossom);
}
return top_blossoms.erase(bi);
}
// when expanding a T-blossom with base v, it requires more operations:
// supply the missing T-labels for new base vertices by picking the minimum tau from vertices of
// each corresponding new top-blossoms; when label_T[v] is null or we have a smaller tau from
// missing_label(v), replace T-label and outlet of v (but don't bloom v)
blossom_iterator_t expand_T_blossom(blossom_iterator_t bi, std::vector<blossom_ptr_t>& new_ones)
{
blossom_ptr_t b = *bi;
vertex_descriptor_t b_base = b->get_base();
std::pair<vertex_descriptor_t, vertex_descriptor_t> T_and_outlet = missing_label(b_base);
blossom_iterator_t next_bi = expand_blossom(bi, new_ones);
for (blossom_iterator_t i = b->sub_blossoms.begin(); i != b->sub_blossoms.end(); ++i)
{
blossom_ptr_t sub_blossom = *i;
vertex_descriptor_t sub_base = sub_blossom->get_base();
vertex_descriptor_t min_tau_v = graph_traits<Graph>::null_vertex();
edge_property_t min_tau = std::numeric_limits<edge_property_t>::max();
std::vector<vertex_descriptor_t> sub_vertices = sub_blossom->vertices();
for (vertex_vec_iter_t v = sub_vertices.begin(); v != sub_vertices.end(); ++v)
{
if (tau[*v] < min_tau)
{
min_tau = tau[*v];
min_tau_v = *v;
}
}
if (min_tau < std::numeric_limits<edge_property_t>::max())
put_T_label(sub_base, tau_idx[min_tau_v], min_tau_v, tau[min_tau_v]);
}
if (label_T[b_base] == graph_traits<Graph>::null_vertex() || tau[old_label[b_base].second] < pi[b_base])
boost::tie(label_T[b_base], outlet[b_base]) = T_and_outlet;
return next_bi;
}
// when vertices v and w are matched to each other by augmenting,
// we must set v/w as base vertex of any blossom who contains v/w and
// is a sub-blossom of their lowest (smallest) common blossom
void adjust_blossom(vertex_descriptor_t v, vertex_descriptor_t w)
{
blossom_ptr_t vb = in_blossom[v], wb = in_blossom[w], lowest_common_blossom;
std::vector<blossom_ptr_t> v_ancestors, w_ancestors;
while (vb->father != blossom_ptr_t())
{
v_ancestors.push_back(vb->father);
vb = vb->father;
}
while (wb->father != blossom_ptr_t())
{
w_ancestors.push_back(wb->father);
wb = wb->father;
}
typename std::vector<blossom_ptr_t>::reverse_iterator i, j;
i = v_ancestors.rbegin();
j = w_ancestors.rbegin();
while (i != v_ancestors.rend() && j != w_ancestors.rend() && i->get() == j->get())
{
lowest_common_blossom = *i;
++i;++j;
}
vb = in_blossom[v];
wb = in_blossom[w];
while (vb->father != lowest_common_blossom)
{
vb->father->set_base(vb);
vb = vb->father;
}
while (wb->father != lowest_common_blossom)
{
wb->father->set_base(wb);
wb = wb->father;
}
}
// every edge weight is multiplied by 4 to ensure integer weights
// throughout the algorithm if all input weights are integers
inline edge_property_t slack(const edge_descriptor_t& e) const
{
vertex_descriptor_t v, w;
v = source(e, g);
w = target(e, g);
return dual_var[v] + dual_var[w] - 4*get(edge_weight, g, e);
}
// backtrace one step on vertex v along the augmenting path
// by its labels and its vertex state;
// boolean parameter "use_old" means whether we are updating labels,
// if we are, then we use old labels to backtrace and also we
// don't jump to its base vertex when we reach an odd vertex
std::pair<vertex_descriptor_t, vertex_state_t> parent(std::pair<vertex_descriptor_t, vertex_state_t> v,
bool use_old = false) const
{
if (v.second == graph::detail::V_EVEN)
{
// a paranoid check: label_S shoule be the same as mate in backtracing
if (label_S[v.first] == graph_traits<Graph>::null_vertex())
label_S[v.first] = mate[v.first];
return std::make_pair(label_S[v.first], graph::detail::V_ODD);
}
else if (v.second == graph::detail::V_ODD)
{
vertex_descriptor_t w = use_old ? old_label[v.first].first : base_vertex(label_T[v.first]);
return std::make_pair(w, graph::detail::V_EVEN);
}
return std::make_pair(v.first, graph::detail::V_UNREACHED);
}
// backtrace from vertices v and w to their free (unmatched) ancesters,
// return the nearest common ancestor (null_vertex if none) of v and w
vertex_descriptor_t nearest_common_ancestor(vertex_descriptor_t v, vertex_descriptor_t w,
vertex_descriptor_t& v_free_ancestor,
vertex_descriptor_t& w_free_ancestor) const
{
std::pair<vertex_descriptor_t, vertex_state_t> v_up(v, graph::detail::V_EVEN);
std::pair<vertex_descriptor_t, vertex_state_t> w_up(w, graph::detail::V_EVEN);
vertex_descriptor_t nca;
nca = w_free_ancestor = v_free_ancestor = graph_traits<Graph>::null_vertex();
std::vector<bool> ancestor_of_w_vector(num_vertices(g), false);
std::vector<bool> ancestor_of_v_vector(num_vertices(g), false);
vertex_to_bool_map_t ancestor_of_w(ancestor_of_w_vector.begin(), vm);
vertex_to_bool_map_t ancestor_of_v(ancestor_of_v_vector.begin(), vm);
while (nca == graph_traits<Graph>::null_vertex() &&
(v_free_ancestor == graph_traits<Graph>::null_vertex() ||
w_free_ancestor == graph_traits<Graph>::null_vertex()))
{
ancestor_of_w[w_up.first] = true;
ancestor_of_v[v_up.first] = true;
if (w_free_ancestor == graph_traits<Graph>::null_vertex())
w_up = parent(w_up);
if (v_free_ancestor == graph_traits<Graph>::null_vertex())
v_up = parent(v_up);
if (mate[v_up.first] == graph_traits<Graph>::null_vertex())
v_free_ancestor = v_up.first;
if (mate[w_up.first] == graph_traits<Graph>::null_vertex())
w_free_ancestor = w_up.first;
if (ancestor_of_w[v_up.first] == true || v_up.first == w_up.first)
nca = v_up.first;
else if (ancestor_of_v[w_up.first] == true)
nca = w_up.first;
else if (v_free_ancestor == w_free_ancestor &&
v_free_ancestor != graph_traits<Graph>::null_vertex())
nca = v_up.first;
}
return nca;
}
// when a new top blossom b is created by connecting (v, w), we add sub-blossoms into
// b along backtracing from v_prime and w_prime to stop_vertex (the base vertex);
// also, we set labels and outlet for each base vertex we pass by
void make_blossom(blossom_ptr_t b, vertex_descriptor_t w_prime,
vertex_descriptor_t v_prime, vertex_descriptor_t stop_vertex)
{
std::pair<vertex_descriptor_t, vertex_state_t> u(v_prime, graph::detail::V_ODD);
std::pair<vertex_descriptor_t, vertex_state_t> u_up(w_prime, graph::detail::V_EVEN);
for (; u_up.first != stop_vertex; u = u_up, u_up = parent(u))
{
if (u_up.second == graph::detail::V_EVEN)
{
if (!in_top_blossom(u_up.first)->sub_blossoms.empty())
outlet[u_up.first] = label_T[u.first];
label_T[u_up.first] = outlet[u.first];
}
else if (u_up.second == graph::detail::V_ODD)
label_S[u_up.first] = u.first;
add_sub_blossom(b, u_up.first);
}
}
// the design of recursively expanding augmenting path in (reversed_)retrieve_augmenting_path
// functions is inspired by same functions in max_cardinality_matching.hpp;
// except that in weighted matching, we use "outlet" vertices instead of "bridge" vertex pairs:
// if blossom b is the smallest non-trivial blossom that contains its base vertex v, then
// v and outlet[v] are where augmenting path enters and leaves b
void retrieve_augmenting_path(vertex_descriptor_t v, vertex_descriptor_t w, vertex_state_t v_state)
{
if (v == w)
aug_path.push_back(v);
else if (v_state == graph::detail::V_EVEN)
{
aug_path.push_back(v);
retrieve_augmenting_path(label_S[v], w, graph::detail::V_ODD);
}
else if (v_state == graph::detail::V_ODD)
{
if (outlet[v] == v)
aug_path.push_back(v);
else
reversed_retrieve_augmenting_path(outlet[v], v, graph::detail::V_EVEN);
retrieve_augmenting_path(label_T[v], w, graph::detail::V_EVEN);
}
}
void reversed_retrieve_augmenting_path(vertex_descriptor_t v, vertex_descriptor_t w, vertex_state_t v_state)
{
if (v == w)
aug_path.push_back(v);
else if (v_state == graph::detail::V_EVEN)
{
reversed_retrieve_augmenting_path(label_S[v], w, graph::detail::V_ODD);
aug_path.push_back(v);
}
else if (v_state == graph::detail::V_ODD)
{
reversed_retrieve_augmenting_path(label_T[v], w, graph::detail::V_EVEN);
if (outlet[v] != v)
retrieve_augmenting_path(outlet[v], v, graph::detail::V_EVEN);
else
aug_path.push_back(v);
}
}
// correct labels for vertices in the augmenting path
void relabel(vertex_descriptor_t v)
{
blossom_ptr_t b = in_blossom[v]->father;
if (!is_in_blossom(b, mate[v]))
{ // if v is a new base vertex
std::pair<vertex_descriptor_t, vertex_state_t> u(v, graph::detail::V_EVEN);
while (label_S[u.first] != u.first && is_in_blossom(b, label_S[u.first]))
u = parent(u, true);
vertex_descriptor_t old_base = u.first;
if (label_S[old_base] != old_base)
{ // if old base is not exposed
label_T[v] = label_S[old_base];
outlet[v] = old_base;
}
else
{ // if old base is exposed then new label_T[v] is not in b,
// we must (i) make b2 the smallest blossom containing v but not as base vertex
// (ii) backtrace from b2's new base vertex to b
label_T[v] = graph_traits<Graph>::null_vertex();
for (b = b->father; b != blossom_ptr_t() && b->get_base() == v; b = b->father);
if (b != blossom_ptr_t())
{
u = std::make_pair(b->get_base(), graph::detail::V_ODD);
while (!is_in_blossom(in_blossom[v]->father, old_label[u.first].first))
u = parent(u, true);
label_T[v] = u.first;
outlet[v] = old_label[u.first].first;
}
}
}
else if (label_S[v] == v || !is_in_blossom(b, label_S[v]))
{ // if v is an old base vertex
// let u be the new base vertex; backtrace from u's old T-label
std::pair<vertex_descriptor_t, vertex_state_t> u(b->get_base(), graph::detail::V_ODD);
while (old_label[u.first].first != graph_traits<Graph>::null_vertex() && old_label[u.first].first != v)
u = parent(u, true);
label_T[v] = old_label[u.first].second;
outlet[v] = v;
}
else // if v is neither a new nor an old base vertex
label_T[v] = label_S[v];
}
void augmenting(vertex_descriptor_t v, vertex_descriptor_t v_free_ancestor,
vertex_descriptor_t w, vertex_descriptor_t w_free_ancestor)
{
vertex_iterator_t vi, vi_end;
// retrieve the augmenting path and put it in aug_path
reversed_retrieve_augmenting_path(v, v_free_ancestor, graph::detail::V_EVEN);
retrieve_augmenting_path(w, w_free_ancestor, graph::detail::V_EVEN);
// augment the matching along aug_path
vertex_descriptor_t a, b;
vertex_list_t reversed_aug_path;
while (!aug_path.empty())
{
a = aug_path.front();
aug_path.pop_front();
reversed_aug_path.push_back(a);
b = aug_path.front();
aug_path.pop_front();
reversed_aug_path.push_back(b);
mate[a] = b;
mate[b] = a;
// reset base vertex for every blossom in augment path
adjust_blossom(a, b);
}
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
old_label[*vi] = std::make_pair(label_T[*vi], outlet[*vi]);
// correct labels for in-blossom vertices along aug_path
while (!reversed_aug_path.empty())
{
a = reversed_aug_path.front();
reversed_aug_path.pop_front();
if (in_blossom[a]->father != blossom_ptr_t())
relabel(a);
}
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
{
vertex_descriptor_t u = *vi;
if (mate[u] != graph_traits<Graph>::null_vertex())
label_S[u] = mate[u];
}
// expand blossoms with zero dual variables
std::vector<blossom_ptr_t> new_top_blossoms;
for (blossom_iterator_t bi = top_blossoms.begin(); bi != top_blossoms.end();)
{
if ((*bi)->dual_var <= 0)
bi = expand_blossom(bi, new_top_blossoms);
else
++bi;
}
top_blossoms.insert(top_blossoms.end(), new_top_blossoms.begin(), new_top_blossoms.end());
init();
}
// create a new blossom and set labels for vertices inside
void blossoming(vertex_descriptor_t v, vertex_descriptor_t v_prime,
vertex_descriptor_t w, vertex_descriptor_t w_prime,
vertex_descriptor_t nca)
{
vertex_iterator_t vi, vi_end;
std::vector<bool> is_old_base_vector(num_vertices(g));
vertex_to_bool_map_t is_old_base(is_old_base_vector.begin(), vm);
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
{
if (*vi == base_vertex(*vi))
is_old_base[*vi] = true;
}
blossom_ptr_t b = boost::make_shared<blossom>();
add_sub_blossom(b, nca);
label_T[w_prime] = v;
label_T[v_prime] = w;
outlet[w_prime] = w;
outlet[v_prime] = v;
make_blossom(b, w_prime, v_prime, nca);
make_blossom(b, v_prime, w_prime, nca);
label_T[nca] = graph_traits<Graph>::null_vertex();
outlet[nca] = nca;
top_blossoms.push_back(b);
bloom(b);
// set gamma[b_base] = min_slack{critical_edge(b_base, other_base)} where each critical edge
// is updated before, by argmin{slack(old_bases_in_b, other_base)};
vertex_vec_iter_t i, j;
std::vector<vertex_descriptor_t> b_vertices = b->vertices(), old_base_in_b, other_base;
vertex_descriptor_t b_base = b->get_base();
for (i = b_vertices.begin(); i != b_vertices.end(); ++i)
{
if (is_old_base[*i])
old_base_in_b.push_back(*i);
}
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
{
if (*vi != b_base && *vi == base_vertex(*vi))
other_base.push_back(*vi);
}
for (i = other_base.begin(); i != other_base.end(); ++i)
{
edge_property_t min_slack = std::numeric_limits<edge_property_t>::max();
std::pair<edge_descriptor_t, bool> b_vi = null_edge;
for (j = old_base_in_b.begin(); j != old_base_in_b.end(); ++j)
{
if (critical_edge[*j][*i] != null_edge && min_slack > slack(critical_edge[*j][*i].first))
{
min_slack = slack(critical_edge[*j][*i].first);
b_vi = critical_edge[*j][*i];
}
}
critical_edge[b_base][*i] = critical_edge[*i][b_base] = b_vi;
}
gamma[b_base] = std::numeric_limits<edge_property_t>::max();
for (i = other_base.begin(); i != other_base.end(); ++i)
{
if (critical_edge[b_base][*i] != null_edge)
gamma[b_base] = std::min(gamma[b_base], slack(critical_edge[b_base][*i].first));
}
}
void init()
{
even_edges.clear();
vertex_iterator_t vi, vi_end;
typename std::vector<std::vector<std::pair<edge_descriptor_t, bool> > >::iterator vei;
for (boost::tie(vi,vi_end) = vertices(g), vei = critical_edge_vectors.begin(); vi != vi_end; ++vi, ++vei)
{
vertex_descriptor_t u = *vi;
out_edge_iterator_t ei, ei_end;
gamma[u] = tau[u] = pi[u] = std::numeric_limits<edge_property_t>::max();
std::fill(vei->begin(), vei->end(), null_edge);
if (base_vertex(u) != u)
continue;
label_S[u] = label_T[u] = graph_traits<Graph>::null_vertex();
outlet[u] = u;
if (mate[u] == graph_traits<Graph>::null_vertex())
{
label_S[u] = u;
bloom(in_top_blossom(u));
}
}
}
bool augment_matching()
{
vertex_descriptor_t v, w, w_free_ancestor, v_free_ancestor;
v = w = w_free_ancestor = v_free_ancestor = graph_traits<Graph>::null_vertex();
bool found_alternating_path = false;
// note that we only use edges of zero slack value for augmenting
while (!even_edges.empty() && !found_alternating_path)
{
// search for augmenting paths depth-first
edge_descriptor_t current_edge = even_edges.back();
even_edges.pop_back();
v = source(current_edge, g);
w = target(current_edge, g);
vertex_descriptor_t v_prime = base_vertex(v);
vertex_descriptor_t w_prime = base_vertex(w);
// w_prime == v_prime implies that we get an edge that has been shrunk into a blossom
if (v_prime == w_prime)
continue;
// a paranoid check
if (label_S[v_prime] == graph_traits<Graph>::null_vertex())
{
std::swap(v_prime, w_prime);
std::swap(v, w);
}
// w_prime may be unlabeled or have a T-label; replace the existed T-label if the edge slack
// is smaller than current pi[w_prime] and update it. Note that a T-label is "deserved" only when pi equals zero.
// also update tau and tau_idx so that tau_idx becomes T-label when a T-blossom is expanded
if (label_S[w_prime] == graph_traits<Graph>::null_vertex())
{
if (slack(current_edge) < pi[w_prime])
put_T_label(w_prime, v, w, slack(current_edge));
if (slack(current_edge) < tau[w])
{
if (in_blossom[w]->father == blossom_ptr_t() || label_T[w_prime] == v ||
label_T[w_prime] == graph_traits<Graph>::null_vertex() ||
nearest_common_ancestor(v_prime, label_T[w_prime],
v_free_ancestor, w_free_ancestor) == graph_traits<Graph>::null_vertex())
{
tau[w] = slack(current_edge);
tau_idx[w] = v;
}
}
}
else
{
if (slack(current_edge) > 0)
{
// update gamma and critical_edges when we have a smaller edge slack
gamma[v_prime] = std::min(gamma[v_prime], slack(current_edge));
gamma[w_prime] = std::min(gamma[w_prime], slack(current_edge));
if (critical_edge[v_prime][w_prime] == null_edge ||
slack(critical_edge[v_prime][w_prime].first) > slack(current_edge))
{
critical_edge[v_prime][w_prime] = std::pair<edge_descriptor_t, bool>(current_edge, true);
critical_edge[w_prime][v_prime] = std::pair<edge_descriptor_t, bool>(current_edge, true);
}
continue;
}
else if (slack(current_edge) == 0)
{
// if nca is null_vertex then we have an augmenting path; otherwise we have
// a new top blossom with nca as its base vertex
vertex_descriptor_t nca = nearest_common_ancestor(v_prime, w_prime, v_free_ancestor, w_free_ancestor);
if (nca == graph_traits<Graph>::null_vertex())
found_alternating_path = true; //to break out of the loop
else
blossoming(v, v_prime, w, w_prime, nca);
}
}
}
if (!found_alternating_path)
return false;
augmenting(v, v_free_ancestor, w, w_free_ancestor);
return true;
}
// slack the vertex and blossom dual variables when there is no augmenting path found
// according to the primal-dual method
bool adjust_dual()
{
edge_property_t delta1, delta2, delta3, delta4, delta;
delta1 = delta2 = delta3 = delta4 = std::numeric_limits<edge_property_t>::max();
vertex_iterator_t vi, vi_end;
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
{
delta1 = std::min(delta1, dual_var[*vi]);
delta4 = pi[*vi] > 0 ? std::min(delta4, pi[*vi]) : delta4;
if (*vi == base_vertex(*vi))
delta3 = std::min(delta3, gamma[*vi]/2);
}
for (blossom_iterator_t bi = top_blossoms.begin(); bi != top_blossoms.end(); ++bi)
{
vertex_descriptor_t b_base = (*bi)->get_base();
if (label_T[b_base] != graph_traits<Graph>::null_vertex() && pi[b_base] == 0)
delta2 = std::min(delta2, (*bi)->dual_var/2);
}
delta = std::min(std::min(delta1, delta2), std::min(delta3, delta4));
// start updating dual variables, note that the order is important
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
{
vertex_descriptor_t v = *vi, v_prime = base_vertex(v);
if (label_S[v_prime] != graph_traits<Graph>::null_vertex())
dual_var[v] -= delta;
else if (label_T[v_prime] != graph_traits<Graph>::null_vertex() && pi[v_prime] == 0)
dual_var[v] += delta;
if (v == v_prime)
gamma[v] -= 2*delta;
}
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
{
vertex_descriptor_t v_prime = base_vertex(*vi);
if (pi[v_prime] > 0)
tau[*vi] -= delta;
}
for (blossom_iterator_t bi = top_blossoms.begin(); bi != top_blossoms.end(); ++bi)
{
vertex_descriptor_t b_base = (*bi)->get_base();
if (label_T[b_base] != graph_traits<Graph>::null_vertex() && pi[b_base] == 0)
(*bi)->dual_var -= 2*delta;
if (label_S[b_base] != graph_traits<Graph>::null_vertex())
(*bi)->dual_var += 2*delta;
}
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
{
vertex_descriptor_t v = *vi;
if (pi[v] > 0)
pi[v] -= delta;
// when some T-vertices have zero pi value, bloom their mates so that matching can be further augmented
if (label_T[v] != graph_traits<Graph>::null_vertex() && pi[v] == 0)
put_T_label(v, label_T[v], outlet[v], pi[v]);
}
// optimal solution reached, halt
if (delta == delta1)
return false;
// expand odd blossoms with zero dual variables and zero pi value of their base vertices
if (delta == delta2 && delta != delta3)
{
std::vector<blossom_ptr_t> new_top_blossoms;
for (blossom_iterator_t bi = top_blossoms.begin(); bi != top_blossoms.end();)
{
const blossom_ptr_t b = *bi;
vertex_descriptor_t b_base = b->get_base();
if (b->dual_var == 0 && label_T[b_base] != graph_traits<Graph>::null_vertex() && pi[b_base] == 0)
bi = expand_T_blossom(bi, new_top_blossoms);
else
++bi;
}
top_blossoms.insert(top_blossoms.end(), new_top_blossoms.begin(), new_top_blossoms.end());
}
while (true)
{
// find a zero-slack critical edge (v, w) of zero gamma values
std::pair<edge_descriptor_t, bool> best_edge = null_edge;
std::vector<vertex_descriptor_t> base_nodes;
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
{
if (*vi == base_vertex(*vi))
base_nodes.push_back(*vi);
}
for (vertex_vec_iter_t i = base_nodes.begin(); i != base_nodes.end(); ++i)
{
if (gamma[*i] == 0)
{
for (vertex_vec_iter_t j = base_nodes.begin(); j != base_nodes.end(); ++j)
{
if (critical_edge[*i][*j] != null_edge && slack(critical_edge[*i][*j].first) == 0)
best_edge = critical_edge[*i][*j];
}
}
}
// if not found, continue finding other augment matching
if (best_edge == null_edge)
{
bool augmented = augment_matching();
return augmented || delta != delta1;
}
// if found, determine either augmenting or blossoming
vertex_descriptor_t v = source(best_edge.first, g), w = target(best_edge.first, g);
vertex_descriptor_t v_prime = base_vertex(v), w_prime = base_vertex(w), v_free_ancestor, w_free_ancestor;
vertex_descriptor_t nca = nearest_common_ancestor(v_prime, w_prime, v_free_ancestor, w_free_ancestor);
if (nca == graph_traits<Graph>::null_vertex())
{
augmenting(v, v_free_ancestor, w, w_free_ancestor);
return true;
}
else
blossoming(v, v_prime, w, w_prime, nca);
}
return false;
}
template <typename PropertyMap>
void get_current_matching(PropertyMap pm)
{
vertex_iterator_t vi, vi_end;
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
put(pm, *vi, mate[*vi]);
}
private:
const Graph& g;
VertexIndexMap vm;
const std::pair<edge_descriptor_t, bool> null_edge;
// storage for the property maps below
std::vector<vertex_descriptor_t> mate_vector;
std::vector<vertex_descriptor_t> label_S_vector, label_T_vector;
std::vector<vertex_descriptor_t> outlet_vector;
std::vector<vertex_descriptor_t> tau_idx_vector;
std::vector<edge_property_t> dual_var_vector;
std::vector<edge_property_t> pi_vector, gamma_vector, tau_vector;
std::vector<blossom_ptr_t> in_blossom_vector;
std::vector<std::pair<vertex_descriptor_t, vertex_descriptor_t> > old_label_vector;
std::vector<vertex_to_edge_map_t> critical_edge_vector;
std::vector<std::vector<std::pair<edge_descriptor_t, bool> > > critical_edge_vectors;
// iterator property maps
vertex_to_vertex_map_t mate;
vertex_to_vertex_map_t label_S; // v has an S-label -> v can be an even vertex, label_S[v] is its mate
vertex_to_vertex_map_t label_T; // v has a T-label -> v can be an odd vertex, label_T[v] is its predecessor in aug_path
vertex_to_vertex_map_t outlet;
vertex_to_vertex_map_t tau_idx;
vertex_to_weight_map_t dual_var;
vertex_to_weight_map_t pi, gamma, tau;
vertex_to_blossom_map_t in_blossom; // map any vertex v to the trivial blossom containing v
vertex_to_pair_map_t old_label; // <old T-label, old outlet> before relabeling or expanding T-blossoms
vertex_pair_to_edge_map_t critical_edge; // an not matched edge (v, w) is critical if v and w belongs to different S-blossoms
vertex_list_t aug_path;
edge_list_t even_edges;
std::vector<blossom_ptr_t> top_blossoms;
};
template <typename Graph, typename MateMap, typename VertexIndexMap>
void maximum_weighted_matching(const Graph& g, MateMap mate, VertexIndexMap vm)
{
empty_matching<Graph, MateMap>::find_matching(g, mate);
weighted_augmenting_path_finder<Graph, MateMap, VertexIndexMap> augmentor(g, mate, vm);
// can have |V| times augmenting at most
for (std::size_t t = 0; t < num_vertices(g); ++t)
{
bool augmented = false;
while (!augmented)
{
augmented = augmentor.augment_matching();
if (!augmented)
{
// halt if adjusting dual variables can't bring potential augment
if (!augmentor.adjust_dual())
break;
}
}
if (!augmented)
break;
}
augmentor.get_current_matching(mate);
}
template <typename Graph, typename MateMap>
inline void maximum_weighted_matching(const Graph& g, MateMap mate)
{
maximum_weighted_matching(g, mate, get(vertex_index,g));
}
// brute-force matcher searches all possible combinations of matched edges to get the maximum weighted matching
// which can be used for testing on small graphs (within dozens vertices)
template <typename Graph, typename MateMap, typename VertexIndexMap>
class brute_force_matching
{
public:
typedef typename graph_traits<Graph>::vertex_descriptor vertex_descriptor_t;
typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator_t;
typedef typename std::vector<vertex_descriptor_t>::iterator vertex_vec_iter_t;
typedef typename graph_traits<Graph>::edge_iterator edge_iterator_t;
typedef boost::iterator_property_map<vertex_vec_iter_t, VertexIndexMap> vertex_to_vertex_map_t;
brute_force_matching(const Graph& arg_g, MateMap arg_mate, VertexIndexMap arg_vm) :
g(arg_g),
vm(arg_vm),
mate_vector(num_vertices(g)),
best_mate_vector(num_vertices(g)),
mate(mate_vector.begin(), vm),
best_mate(best_mate_vector.begin(), vm)
{
vertex_iterator_t vi,vi_end;
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
best_mate[*vi] = mate[*vi] = get(arg_mate, *vi);
}
template <typename PropertyMap>
void find_matching(PropertyMap pm)
{
edge_iterator_t ei;
boost::tie(ei, ei_end) = edges(g);
select_edge(ei);
vertex_iterator_t vi,vi_end;
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
put(pm, *vi, best_mate[*vi]);
}
private:
const Graph& g;
VertexIndexMap vm;
std::vector<vertex_descriptor_t> mate_vector, best_mate_vector;
vertex_to_vertex_map_t mate, best_mate;
edge_iterator_t ei_end;
void select_edge(edge_iterator_t ei)
{
if (ei == ei_end)
{
if (matching_weight_sum(g, mate) > matching_weight_sum(g, best_mate))
{
vertex_iterator_t vi, vi_end;
for (boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
best_mate[*vi] = mate[*vi];
}
return;
}
vertex_descriptor_t v, w;
v = source(*ei, g);
w = target(*ei, g);
select_edge(++ei);
if (mate[v] == graph_traits<Graph>::null_vertex() &&
mate[w] == graph_traits<Graph>::null_vertex())
{
mate[v] = w;
mate[w] = v;
select_edge(ei);
mate[v] = mate[w] = graph_traits<Graph>::null_vertex();
}
}
};
template <typename Graph, typename MateMap, typename VertexIndexMap>
void brute_force_maximum_weighted_matching(const Graph& g, MateMap mate, VertexIndexMap vm)
{
empty_matching<Graph, MateMap>::find_matching(g, mate);
brute_force_matching<Graph, MateMap, VertexIndexMap> brute_force_matcher(g, mate, vm);
brute_force_matcher.find_matching(mate);
}
template <typename Graph, typename MateMap>
inline void brute_force_maximum_weighted_matching(const Graph& g, MateMap mate)
{
brute_force_maximum_weighted_matching(g, mate, get(vertex_index, g));
}
}
#endif