4.6.4 Maximum Graph Matching (Blossom)

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171

/*

Given an undirected, unweighted graph, determine a maximum matching: a maximum subset of its edges
such that no node is shared between different edges in the resulting subset.

Edmonds' blossom algorithm extends augmenting-path search to general (non-bipartite) graphs, where
odd-length cycles called blossoms can conceal augmenting paths. Each blossom is contracted into a
single node so the search can proceed, and is expanded afterward to recover the matching. For
maximum-weight matching in bipartite graphs, use Hungarian or min-cost max-flow; for maximum-weight
matching in general graphs, use the weighted blossom algorithm in the next section.

- `blossom()` returns a matching for a global, bidirectionally pre-populated adjacency list `adj`
  which must consist of nodes numbered [0, `n`), where `n` is `adj.size()`. The returned vector
  `match` has `match[u] == v` and `match[v] == u` when `u` and `v` are matched, or `match[u] == -1`
  when `u` is unmatched.

Time Complexity:
- O(n^3) per call to `blossom()`, where $n$ is the number of nodes.

Space Complexity:
- O(max(n, m)) for storage of the graph, where $n$ is the number of nodes and $m$ is the number of
  edges.
- O(n) auxiliary heap space for `blossom()`, where $n$ is the number of nodes.

*/

#include <algorithm>
#include <numeric>
#include <queue>
#include <vector>

std::vector<int> blossom(const std::vector<std::vector<int>> &adj) {
  int n = static_cast<int>(adj.size());
  std::vector<int> match(n, -1), label(n), parent(n), base(n), aux(n, -1);
  std::queue<int> q;
  int aux_time = -1;
  auto lca = [&](int u, int v) {
    aux_time++;
    while (true) {
      if (u != -1) {
        u = base[u];
        if (aux[u] == aux_time) {
          return u;
        }
        aux[u] = aux_time;
        u = (match[u] == -1) ? -1 : parent[match[u]];
      }
      std::swap(u, v);
    }
  };
  auto mark_blossom = [&](int u, int v, int b) {
    while (base[u] != b) {
      parent[u] = v;
      v = match[u];
      if (label[v] == 1) {
        label[v] = 0;
        q.push(v);
      }
      base[u] = base[v] = b;
      u = parent[v];
    }
  };
  auto augment = [&](int u) {
    while (u != -1) {
      int p = parent[u];
      int next = match[p];
      match[u] = p;
      match[p] = u;
      u = next;
    }
  };
  auto find_path = [&](int root) {
    std::fill(label.begin(), label.end(), -1);
    std::fill(parent.begin(), parent.end(), -1);
    std::iota(base.begin(), base.end(), 0);
    while (!q.empty()) {
      q.pop();
    }
    label[root] = 0;
    q.push(root);
    while (!q.empty()) {
      int u = q.front();
      q.pop();
      for (int v : adj[u]) {
        if (base[u] == base[v] || match[u] == v) {
          continue;
        }
        if (label[v] == -1) {
          label[v] = 1;
          parent[v] = u;
          if (match[v] == -1) {
            augment(v);
            return true;
          }
          label[match[v]] = 0;
          q.push(match[v]);
        } else if (label[v] == 0) {
          int b = lca(u, v);
          mark_blossom(u, v, b);
          mark_blossom(v, u, b);
        }
      }
    }
    return false;
  };
  for (int i = 0; i < n; i++) {
    if (match[i] == -1) {
      for (int v : adj[i]) {
        if (i != v && match[v] == -1) {
          match[i] = v;
          match[v] = i;
          break;
        }
      }
    }
  }
  for (int i = 0; i < n; i++) {
    if (match[i] == -1) {
      find_path(i);
    }
  }
  return match;
}

/*** Example Usage and Output:

Matched 2 pair(s):
0 1
2 3

***/

#include <cassert>
#include <iostream>
using namespace std;

vector<vector<int>> adj;

void add_edge(int u, int v) {
  adj[u].push_back(v);
  adj[v].push_back(u);
};

int matching_size(const vector<int> &match) {
  return std::count_if(match.begin(), match.end(), [](int v) { return v != -1; }) / 2;
};

int main() {
  int nodes = 4;
  adj.assign(nodes, {});
  add_edge(0, 1);
  add_edge(1, 2);
  add_edge(2, 3);
  add_edge(3, 0);
  vector<int> match = blossom(adj);
  cout << "Matched " << matching_size(match) << " pair(s):" << endl;
  for (int i = 0; i < nodes; i++) {
    if (match[i] != -1 && i < match[i]) {
      cout << i << " " << match[i] << endl;
    }
  }
  nodes = 5;
  adj.assign(nodes, {});
  add_edge(0, 1);
  add_edge(1, 2);
  add_edge(2, 0);
  add_edge(0, 3);
  add_edge(1, 4);
  assert(matching_size(blossom(adj)) == 2);
  return 0;
}