4.6.4 Maximum Graph Matching (Blossom)

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/*

Given an undirected, unweighted simple 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.

The graph must be simple: self-loops and parallel edges are not supported.

- `max_matching(adj)` returns a matching for a bidirectionally pre-populated adjacency list `adj`,
  whose indices represent the nodes. The returned vector `match` has `match[u] == v` and
  `match[v] == u` when `u` and `v` are matched, or `match[u]` is $-1$ when `u` is unmatched.

Time Complexity:
- O(n^3) per call, 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.

*/

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

std::vector<int> max_matching(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) {
    label.assign(n, -1);
    parent.assign(n, -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 ***/

#include <cassert>
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 count_if(match.begin(), match.end(), [](int v) { return v != -1; }) / 2;
}

int main() {
  {
    // 0---1
    // |   |
    // 3---2
    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 = max_matching(adj);
    assert(matching_size(match) == 2);
    assert((match == vector<int>{1, 0, 3, 2}));
  }
  {
    // 3---0---1---4
    //      \ /
    //       2
    int 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(max_matching(adj)) == 2);
  }
  return 0;
}