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A maximum matching is also a maximal matching, and hence it is possible to find a largest maximal matching in polynomial time. However, no polynomial-time algorithm is known for finding a minimum maximal matching , that is, a maximal matching that contains the smallest possible number of edges.
For sparse bipartite graphs, the maximum matching problem can be solved in ~ (/) with Madry's algorithm based on electric flows. [ 3 ] For planar bipartite graphs, the problem can be solved in time O ( n log 3 n ) where n is the number of vertices, by reducing the problem to maximum flow with multiple sources and sinks.
In computer science, the Hopcroft–Karp algorithm (sometimes more accurately called the Hopcroft–Karp–Karzanov algorithm) [1] is an algorithm that takes a bipartite graph as input and produces a maximum-cardinality matching as output — a set of as many edges as possible with the property that no two edges share an endpoint.
In graph theory, the blossom algorithm is an algorithm for constructing maximum matchings on graphs. The algorithm was developed by Jack Edmonds in 1961, [1] and published in 1965. [2] Given a general graph G = (V, E), the algorithm finds a matching M such that each vertex in V is incident with at most one edge in M and | M | is maximized. The ...
To find a maximum matching in a graph , the blossom algorithm starts with a small matching and goes through multiple iterations in which it increases the size of the matching by one edge. We can find the Gallai–Edmonds decomposition from the blossom algorithm's work in the last iteration: the work done when it has a maximum matching M ...
Just like a maximal matching is within factor 2 of a maximum matching, [9] a maximal 3-dimensional matching is within factor 3 of a maximum 3-dimensional matching. For any constant ε > 0 there is a polynomial-time (4/3 + ε)-approximation algorithm for 3-dimensional matching. [10]
An example of a bipartite graph, with a maximum matching (blue) and minimum vertex cover (red) both of size six. In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (), describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs.
A matching in G' induces a schedule for F and obviously maximum bipartite matching in this graph produces an airline schedule with minimum number of crews. [29] As it is mentioned in the Application part of this article, the maximum cardinality bipartite matching is an application of maximum flow problem.