blossomalgoritm

The blossom algorithm is a family of algorithms for finding a maximum cardinality matching in general undirected graphs. It is most commonly known through Edmonds’ Blossom Algorithm, developed by Jack Edmonds in 1965. The key idea is to handle odd cycles, or blossoms, by contracting them into single vertices so that augmenting paths can be found in the contracted graph. After an augmenting path is identified, the blossoms are expanded and the matching is augmented in the original graph.

In operation, the algorithm repeatedly constructs alternating trees starting from unmatched vertices and searches for augmenting paths. When an odd cycle is encountered during this search, the cycle’s vertices form a blossom, which is contracted into a single vertex to continue the search. If an augmenting path is found in the contracted graph, the path is projected back to the original graph by expanding the blossoms and flipping the matched/unmatched edges along the path. This process preserves correctness and ultimately yields a maximum matching.

The blossom algorithm is significant because it provides a polynomial-time method for maximum matching in general graphs, beyond bipartite graphs. Over time, various implementations and refinements have been developed. The basic approach inspired several practical algorithms, and modern implementations, such as Blossom V by Kolmogorov, are widely used due to their efficiency on large graphs. The algorithm has influenced related techniques in combinatorial optimization and serves as a foundational tool in graph theory.

Applications of the blossom algorithm appear in areas that require pairing without restricted graph structure, including scheduling, resource allocation, network design, and certain computational biology problems. Its neutral, general-purpose nature makes it a standard reference method for maximum matching in non-bipartite graphs.