Matchingalgoritmes

Matchingalgoritmes are a class of computational methods designed to find a matching in a graph. A matching is a set of edges with no shared endpoints, and a maximum matching has the largest possible number of edges. Variants include maximum cardinality matching, maximum or minimum weight matching, and perfect matching. The field covers both bipartite graphs, where vertices split into two disjoint sets, and general graphs without such a partition. The problems are central in combinatorial optimization and have numerous practical applications.

In bipartite graphs, the primary problem is to find a maximum cardinality matching. Classic algorithms include Kuhn’s algorithm, which uses alternating paths and depth-first search, and the Hopcroft–Karp algorithm, which achieves O(E sqrt(V)) time and is efficient for large graphs. These methods underpin many applications such as task assignment and pairing in markets.

For general graphs, finding a maximum matching is more complex due to the possibility of odd cycles. The Edmonds blossom algorithm, introduced in 1965, handles these cycles through a contraction technique and established polynomial-time solvability. Improvements and refinements, such as the Micali–Vazirani algorithm, offer efficient performance for unweighted graphs. When weighted matching is required, the assignment problem in bipartite graphs is solved by the Hungarian algorithm, while general graphs rely on weighted extensions of blossom-type methods (for example, Blossom V).

Applications of matchingalgoritmes include job assignment, student-project pairing, organ donor matching, network design, and resource allocation. They are closely related to network flow problems and can often be reduced to or from such formulations, enabling a wide range of theoretical and practical solutions.