Graph Theory Questions Medium
The Ford-Fulkerson algorithm is a method used to solve the maximum flow problem in a graph. This problem involves finding the maximum amount of flow that can be sent from a source vertex to a sink vertex in a directed graph.
The algorithm starts by initializing the flow in all edges to zero. Then, it repeatedly finds an augmenting path from the source to the sink using a depth-first search or a breadth-first search. An augmenting path is a path in the residual graph, which is a modified version of the original graph that keeps track of the remaining capacity in each edge.
Once an augmenting path is found, the algorithm determines the maximum amount of flow that can be sent along this path, which is the minimum capacity of all edges in the path. This amount is then added to the flow of each edge in the path, and subtracted from their residual capacities.
The process of finding augmenting paths and updating the flow continues until no more augmenting paths can be found. At this point, the algorithm terminates, and the maximum flow is equal to the sum of the flows along all edges leaving the source vertex.
The Ford-Fulkerson algorithm can be implemented using various methods to find augmenting paths, such as the Edmonds-Karp algorithm that uses a breadth-first search. It guarantees to find the maximum flow as long as the capacities of the edges are integers.
However, the Ford-Fulkerson algorithm may not terminate if the capacities are real numbers. In such cases, an additional step called the capacity scaling or the epsilon-scaling technique can be used to ensure termination.
Overall, the Ford-Fulkerson algorithm is a fundamental and widely used algorithm in graph theory for solving the maximum flow problem.