Overview of Edmons’ Blossom Algorithm and Examples of Algorithms and Implementations

Machine Learning Natural Language Processing Artificial Intelligence Digital Transformation Semantic Web Knowledge Information Processing Graph Data Algorithm Relational Data Learning Recommend Technology Python Time Series Data Analysis Navigation of this blog

Overview of Edmonds’ Blossom Algorithm

Edmonds’ Blossom Algorithm is designed to find a maximum matching in general graphs (including non-bipartite graphs). Unlike algorithms limited to bipartite graphs, this method is applicable to a wide range of graph structures, making it useful in fields such as social networks, scheduling, and structural analysis.

The term “blossom” refers to an odd-length cycle, which is a structural obstacle when searching for an augmenting path. Such cycles can block the extension of a current matching. To overcome this, the algorithm contracts (temporarily merges) these odd cycles into a single pseudo-vertex, allowing the search to continue.

This process of contracting and later expanding blossoms is the core idea of the algorithm and the origin of its name. Thanks to this mechanism, it becomes possible to detect augmenting paths in general graphs and derive a correct maximum matching.

The algorithm incrementally increases the size of the matching by repeatedly searching for augmenting paths—paths in which unmatched and matched edges alternate, and whose endpoints are free (unmatched) vertices.

In general graphs, such paths may be blocked by the presence of odd-length cycles (blossoms). The algorithm deals with this by contracting the blossom, continuing the search as if the cycle were a single vertex, and then, after finding and augmenting the path, expanding the blossom to restore the original structure and matching.

Step-by-Step Procedure

Initialization:
Start with an empty or partial matching.
Augmenting Path Search:
From a free vertex, traverse edges alternating between unmatched and matched, aiming to find an augmenting path.
Blossom Detection and Contraction:
If an odd-length cycle is encountered during the search, treat it as a pseudo-vertex by contracting it.
Matching Augmentation:
If an augmenting path is found, reverse the matched/unmatched status along the path to increase the matching size by one.
Search Continuation or Termination:
If no more augmenting paths can be found, the current matching is maximum.
Blossom Expansion and Matching Restoration:
Expand the previously contracted blossoms and correctly restore internal matching status.

Through this loop of contract → search → augment → expand, the algorithm guarantees a correct maximum matching even in general graphs.

Time Complexity

The classical version of the algorithm, as introduced by Edmonds in 1965, has a time complexity of O(V³). This results from the repeated operations of searching, contracting, and expanding, which can involve a large number of vertex manipulations.

However, later improvements by researchers like Gabow and Micali–Vazirani led to faster algorithms that approach a time complexity of O(√V × E), enabling practical performance even for large-scale graphs.

Related Algorithm

1. Maximum Matching Algorithms (Directly Related)

Maximum matching algorithms aim to find the largest possible set of edges in a graph where no two edges share a vertex. These vary depending on the graph type (bipartite or general) and the optimization goal (maximum size vs. minimum cost).

Algorithm Name	Graph Type	Features	Time Complexity
Edmonds’ Blossom Algorithm	General Graph	Contracts odd-length cycles (blossoms)	O(V³) (original)
Gabow’s Algorithm	General Graph	Improved version of Edmonds (more efficient)	O(V³), faster in practice
Micali–Vazirani Algorithm	General Graph	Fast search for augmenting paths	O(√V × E)
Hopcroft–Karp Algorithm	Bipartite Graph	Batch search of augmenting paths	O(√V × E)
Kuhn’s Algorithm (DFS)	Bipartite Graph	Basic DFS-based matching	O(V × E)
Hungarian Algorithm	Bipartite (Weighted)	Minimum-cost perfect matching	O(V³)

2. Max-Flow Related Algorithms

Maximum matching in bipartite graphs can be reduced to a maximum flow problem. This relationship allows flow-based algorithms to solve matching problems efficiently.

Algorithm Name	Main Purpose	Time Complexity	Notes
Ford–Fulkerson Algorithm	Max Flow	O(max_flow × E)	Simple but slow
Edmonds–Karp Algorithm	Max Flow	O(V × E²)	Based on BFS
Dinic’s Algorithm	Max Flow	O(E × √V) (unit capacity)	Combines BFS and DFS
Push–Relabel Algorithm	Max Flow	O(V³)	More complex implementation

3. Related Optimization Algorithms

Matching-related optimization algorithms deal with weighted matchings, stable pairings, and assignment problems. They are designed to satisfy constraints like cost, stability, or preferences.

Algorithm Name	Problem Type	Features
Assignment Problem Solver	Minimum-cost matching	Closely related to the Hungarian method
Stable Marriage Algorithm	Stable matching	Solved using the Gale–Shapley algorithm
Maximum Weighted Matching	Weighted general graph	Extension of Edmonds’ method (uses labels)
Vertex Cover via Matching	Bipartite graphs	Equivalent to matching via König’s Theorem

4. Applications and Related Research Areas

Maximum matching has widespread applications across disciplines. Graph matching techniques are used to assign resources, optimize pairings, and analyze structures.

Domain	Related Techniques
Compiler Optimization	Register allocation using interference graphs
Chemical Structure Analysis	Graph-based modeling of molecular bonds and optimization
Computer Vision	Feature point matching between image pairs
Operations Research	Task allocation, resource optimization, and scheduling

Applied Implementation Example: Matching People to Projects (General Graph Case)

Scenario:

Matching individuals to projects, where:

Each person can apply to multiple projects.
Some projects conflict with each other (e.g., Project A conflicts with B, but B doesn’t conflict with C).
These inter-project constraints form a non-bipartite structure.

Objective:

Select the maximum number of non-conflicting person–project pairs.

Solution:

Model the scenario as a general graph, where:

Nodes represent both people and projects.
Edges represent applications or constraints.
Use Edmonds’ Blossom Algorithm to compute the maximum matching.

Python Implementation (Using NetworkX)

Python’s networkx library includes an implementation of Edmonds’ algorithm for general graph matching.

import networkx as nx

# Create a general graph
G = nx.Graph()

# Nodes (people and projects)
nodes = ['Alice', 'Bob', 'ProjA', 'ProjB', 'ProjC']
G.add_nodes_from(nodes)

# Edges (application or conflict relationship)
edges = [
    ('Alice', 'ProjA'),
    ('Alice', 'ProjB'),
    ('Bob', 'ProjA'),
    ('Bob', 'ProjC'),
    ('ProjA', 'ProjB'),  # Conflict between projects
]
G.add_edges_from(edges)

# Compute maximum matching (Edmonds' algorithm)
matching = nx.algorithms.matching.max_weight_matching(G, maxcardinality=True)

# Display results
print("Maximum Matching:")
for u, v in matching:
    print(f"{u} — {v}")

Sample Output:

Maximum Matching:
Alice — ProjB
Bob — ProjC

Explanation:

Although Project A and B conflict, the Blossom Algorithm can handle such cases.
Bob is matched with ProjC and Alice with ProjB → a valid matching of 2 pairs.

List of Application Examples

Domain	Application Context	Why a General Graph?
Consulting Project Assignment	Assign consultants based on skills and project conflicts	Projects may conflict—non-bipartite relationships
Reviewer–Paper Assignment	Reviewer conflict-of-interest constraints	Certain combinations must be avoided
Academic Conference Scheduling	Avoid concurrent session conflicts	Presenters and sessions have complex interrelations
Multi-party Recommendation	Assign based on stakeholder conflicts and constraints	Constraints can’t be modeled as simple bipartite

Complexity of Implementation

Edmonds’ Blossom Algorithm requires handling:

Detection, contraction, and expansion of odd-length cycles (blossoms).
These steps are complex, making the algorithm difficult to implement correctly from scratch.

Due to this complexity, it’s recommended in algorithm education and competitive programming to first study maximum matching in bipartite graphs, such as with the Hopcroft–Karp Algorithm.

Hopcroft–Karp is structurally simpler and efficient, making it an ideal entry point for understanding matching algorithms.

Real-World Applications of Edmonds’ Blossom Algorithm

Edmonds’ Blossom Algorithm is capable of efficiently solving the maximum matching problem in general graphs, making it particularly suitable for complex relationship structures that cannot be represented as bipartite graphs. Below are concrete application examples:

1. Scheduling Conference Presentation Sessions

Overview:

- Presenters may be involved in multiple sessions.
- Sessions may conflict due to time, topics, or equipment constraints.
- The relationships between presenters and sessions form a non-bipartite graph.