Explain the Ford-Fulkerson algorithm for maximum network flow.

The Ford-Fulkerson algorithm is used to find the maximum flow in a network.

The Ford-Fulkerson algorithm is an iterative method that starts with an initial feasible flow and then repeatedly improves it until it becomes the maximum flow. The algorithm works by finding an augmenting path in the residual network, which is a network that represents the remaining capacity of the edges after the initial flow has been subtracted. An augmenting path is a path from the source to the sink that has positive residual capacity on all its edges.

Once an augmenting path is found, the algorithm increases the flow along that path by the minimum residual capacity of its edges. This is called the bottleneck capacity. The algorithm then updates the residual network by subtracting the bottleneck capacity from the residual capacity of each edge along the path and adding the same amount to the residual capacity of the reverse edges.

The algorithm repeats this process until no augmenting path can be found in the residual network. At this point, the flow is maximum and the algorithm terminates.

The time complexity of the Ford-Fulkerson algorithm depends on the method used to find the augmenting path. The simplest method is to use a depth-first search, which has a worst-case time complexity of O(Ef), where E is the number of edges in the network and f is the maximum flow. However, this method can lead to an infinite loop if the network has negative cycles. Other methods, such as the Edmonds-Karp algorithm, use a breadth-first search and have a worst-case time complexity of O(VE^2).