How to find the maximum flow in a bipartite network?

To find the maximum flow in a bipartite network, we can use the Ford-Fulkerson algorithm.

The Ford-Fulkerson algorithm is an iterative method that finds the maximum flow in a network by repeatedly finding augmenting paths and increasing the flow along those paths. In a bipartite network, the algorithm can be simplified by using the fact that the flow can only go from one partition to the other.

To apply the Ford-Fulkerson algorithm to a bipartite network, we first need to represent the network as a flow network. We can do this by adding a source node connected to all nodes in one partition, and a sink node connected to all nodes in the other partition. The capacity of each edge is set to 1, since we can only send one unit of flow along each edge.

Next, we can use the Ford-Fulkerson algorithm to find the maximum flow in the network. We start with an initial flow of 0, and then repeatedly find an augmenting path from the source to the sink using a depth-first search or a breadth-first search. Along this path, we increase the flow by 1, and update the residual capacities of the edges.

We continue this process until we can no longer find an augmenting path, at which point we have found the maximum flow in the network. The time complexity of the Ford-Fulkerson algorithm is O(Ef), where E is the number of edges in the network and f is the maximum flow.

In summary, to find the maximum flow in a bipartite network, we can use the Ford-Fulkerson algorithm by representing the network as a flow network and finding augmenting paths from the source to the sink.