Explain the maximum cardinality matching in a network.

Maximum cardinality matching in a network is the largest possible set of edges that do not share a common vertex.

In a network, a matching is a set of edges that do not share a common vertex. The cardinality of a matching is the number of edges in the set. A maximum cardinality matching is the largest possible set of edges that do not share a common vertex.

To find a maximum cardinality matching in a network, we can use the Edmonds-Karp algorithm. This algorithm is a variation of the Ford-Fulkerson algorithm and uses breadth-first search to find an augmenting path, which is a path from the source to the sink that has unused edges. The algorithm then updates the matching by adding the augmenting path to it. This process is repeated until no more augmenting paths can be found.

Another way to find a maximum cardinality matching is to use the Hopcroft-Karp algorithm. This algorithm is more efficient than the Edmonds-Karp algorithm and uses a bipartite graph to find the matching. A bipartite graph is a graph in which the vertices can be divided into two disjoint sets such that every edge connects a vertex in one set to a vertex in the other set. The algorithm starts with an empty matching and then finds an augmenting path using depth-first search. The matching is updated by alternating the edges in the augmenting path.

In conclusion, a maximum cardinality matching in a network is the largest possible set of edges that do not share a common vertex. There are different algorithms that can be used to find a maximum cardinality matching, such as the Edmonds-Karp algorithm and the Hopcroft-Karp algorithm.