The Blossom Algorithm, introduced by Jack Edmonds in 1961, is a significant combinatorial optimization method. It is commonly employed to address the maximum matching problem in diverse graphs, aiming to identify the largest possible collection of edges without any shared vertices between pairs of edges.
In the realm of graph theory, a matching is a collection of edges where every pair has unique vertices. A maximum matching represents the most extensive matching achievable within a graph. The Hopcroft-Karp algorithm specializes in determining maximum matchings in bipartite graphs. However, its effectiveness diminishes when dealing with general graphs, especially those with odd-length cycles. In such cases, alternative approaches like Edmonds' algorithm are favored. Consequently, the Blossom Algorithm emerges as a prominent choice for investigating matchings.
The Blossom Algorithm proves to be beneficial in real-world scenarios like network configuration, task assignment, scheduling tasks, and distributing resources efficiently.
Matching Concept
In mathematical terms, a matching is characterized as a set of edges within a graph where all edges do not share any vertices. For instance, considering vertices A, B, C, D and edges (A,B), (A,B), (C,D), (C,D), it forms a matching if there is no connection between A and B, or C and D. A matching is considered maximum when it contains the highest possible number of edges while still meeting this criterion.
Blossoms and Augmenting Paths
In a graphical representation, augmenting pathways refer to routes that initiate and conclude at unpaired vertices while switching between paired and unpaired edges. When identified, these pathways allow for an expansion in the match size by identifying paired-unpaired edges throughout the route.
Through patterns such as the odd-length cycle or bloom in graphs, we can identify augmenting routes in a distinct manner compared to bipartite graphs. In a formal sense, these patterns add complexity to the process of finding matches. The Blossom Algorithm operates by condensing these blooms into individual vertices, enabling the algorithm to simplify the graph structure and proceed with its search for augmenting paths.
Steps of the Algorithm Blossom
Here are the high-level explanations to follow during the work of algorithm Blossom:
- Initialization: Start with an empty matching M=∅.
- Find an Augmenting Path: Search for an augmenting path within the current graph. If found, increase the size of the matchings by flipping the edges along the path.
- Detect Blossoms: During the search for an augmenting path, if a blossom is found, shrink this blossom into a vertex and continue the search on the modified, shrunk graph.
- Augmenting Path in the Shrunken Graph: If an augmenting path is found in the shrunk graph, just before proceeding to step 6.1, expand the so-called blossom back to its original state. The path in the expanded graph will still be augmenting and hence the matching gets updated as the need arises.
- Repeat: Repeat the operations of finding augmenting paths and shrinking blossoms until none can be found.
- Termination: The algorithm is terminated when no further augmenting paths can be found, and the current matching is given to be the maximum matching.
- Initial Matching The algorithm starts from an arbitrary matching, empty at first, and iteratively improves it by searching for augmenting paths. Initially, all vertices are considered unmarked.
- Alternating Trees The algorithm constructs alternating trees to explore the graph. An alternating tree is one in which the edges alternate between matching or not. The search for an augmenting path starts with a tree rooted at an unmatched vertex to reach another unmatched vertex at one end of the goal path. If this path is found, then the matching would be augmented. If the path is found, then the matching is augmented.
- In search of blossoms If odd-length cycles are detected during the search for an augmenting path, the algorithm identifies this as a blossom. It is termed when the algorithm has shrunk its blossom by considering the entire cycle a single vertex and continues searching for an augmenting path.
- Expanding Blossoms Whenever a blossom is an augmenting path has been discovered, the blossom is subsequently expanded back to its full size.
How does the Blossom Algorithm Works?
Applications:
The Blossom Algorithm is applied widely in areas where finding optimal pairings or matchings is of prime importance.
- Network Design: It optimises designing communication networks, which requires an efficient way of pairing resources or nodes.
- Conflict-Free Scheduling: The Blossom Algorithm can match tasks to resources in different timing conflicts to maximize efficiency or minimize costs.
- Operations Research: The general optimization in a pairing is stored in the optimization of the operation.
- Graph Theory: It serves as the basis for other results in graph theory and aids in other combinatorial optimiz
- Market Clearing: Here, we see the application of the algorithm in economics, especially in market-clearing problems where optimal matching between the buyers and sellers comes into play.
- Biology: The problems relating to genome sequencing and phylogenetic tree construction are solved in bioinformatics using the Blossom Algorithm.
Example:
Let's consider a program to execute the Blossom Algorithm in C++:
#include <iostream>
#include <vector>
#include <queue>
#include <cstring>
using namespace std;
const int MAXN = 500; // The max number of vertices
vector<int> adjacent[MAXN]; // The adjacency list
int matchValue[MAXN], parentValue[MAXN], baseValue[MAXN];
bool visitedNodes[MAXN], blossomNoded[MAXN];
int findAugmentPath(int u1) {
queue<int> que;
que.push(u1);
visitedNodes[u1] = true;
while (!que.empty()) {
int v1 = que.front();
que.pop();
for (int w : adjacent[v1]) {
if (!visitedNodes[w]) {
if (matchValue[w] == -1) {
// If the match value is found
return w;
}
visitedNodes[w] = true;
que.push(matchValue[w]);
}
}
}
return -1; // If the path isn't found
}
void blossomAlgo(int number) {
memset(matchValue, -1, sizeof(matchValue));
for (int u = 0; u < number; ++u) {
if (matchValue[u] == -1) {
memset(visitedNodes, false, sizeof(visitedNodes));
int v = findAugmentPath(u);
if (v != -1) {
// Updating the matching pair
matchValue[u] = v;
matchValue[v] = u;
}
}
}
}
int main() {
int number, m;
cin >> number >> m; // number: number of vertices, m: edges
for (int i = 0; i < m; ++i) {
int u, v;
cin >> u >> v;
adjacent[u].push_back(v);
adjacent[v].push_back(u);
}
blossomAlgo(number);
// Output the matching
for (int i = 0; i < number; ++i) {
if (matchValue[i] != -1) {
cout << i << " " << matchValue[i] << endl;
}
}
return 0;
}
Output:
2 3
1 5
2 5
3 8
1 5
Explanation:
Variables and Data Structures:
- MAXN: The maximum number of vertices, which receives the value 500 in the context defined in this particular case.
- adjacent[MAXN]: Adjacency list representation of the graph, in which every vertex is assigned a list of its neighbou
- matchValue[MAXN]: An array storing the ones currently matched. If matchValue[i] is j, it means vertex i is matched with vertex j. If matchValue[i] is -1, then vertex i is unmatched.
- visitedNodes[MAXN]: A boolean array used to encompass vertices that have been visited in the course of the BFS search for an augmenting path.
- Goal: To find an augmenting path building upon vertex u1.
- BFS Initialization: A queue is started with the vertex u1, and visitedNodes[u1] is marked true.
- BFS Loop run: For every node v1 found in the queue, examine its neighbours w.
The FindAugmentPath function:
If node w has not been visited and is unmatched (i.e., matchValue[w] equals -1), it indicates the discovery of an augmenting path. In this case, the function should return the node w.
Otherwise, flag w as visited and enqueue its corresponding vertex matchValue[w].
- Outcome: If no augmenting path is discovered, the function returns -1.
blossomAlgo Method:
- Objective: Executes the core algorithm of the Blossom Algorithm to determine the optimal matching.
- Initializing Matching: Assign a value of -1 to all matchValue entries (indicating unmatched status).
Loop Through Nodes: For every node u:
If u is not paired (matchValue[u] == -1), reset the visitedNodes list and try to discover a path that increases the matching starting from u by employing the findAugmentPath(u) function.
In case an augmenting path is detected (v != -1), revise the matching by assigning v to matchValue[u] and setting u as matchValue[v].
Input:
- Read the number of vertices(number) and edges(m).
- Read m edges, where each edge connects two vertices u and v. Add each edge to the adjacency list.
- Run the Blossom Algorithm: Call blossomAlgo(number) to find the maximum matching.
- Output: Print out the matched pairs. For each vertex i, if matched (that is if matchValue[i] != -1), print the pair i and matchValue[i].
Advantages of Blossom Algorithms:
Several advantages of Blossom Algorithms are as follows:
- Blossom Algorithms can handle any graph with the property that there exist odd cycles, unlike simpler algorithms that work only for bipartite graphs.
- With a time complexity of O(V3)O(V3), the algorithm is sufficiently efficient for many practical applications.
- Guaranteed Exact Solution: For any general graph, the algorithm is guaranteed to find maximum matching.
Conclusion:
In summary, the Blossom Algorithm has evolved into a valuable tool for identifying maximum matchings in diverse graphs, going beyond its initial application in bipartite graphs where simpler methods may fall short. By leveraging key concepts like augmenting paths and structures, this algorithm adeptly navigates complex graphs, enabling the determination of maximum matchings even in the presence of odd-length cycles. Striking a balance between time complexity and simplicity, the O(V^3) algorithm offers an optimal solution, albeit at the cost of efficiency. Its wide-ranging uses span network planning, job scheduling, resource distribution, and fundamental graph theory exploration. Ultimately, the Blossom Algorithm bridges the realms of combinatorics theory and practical problem-solving applications.
