Finding the maximum rate at which a material can flow through a network from a source to a sink.
The maximum flow problem is a classic optimization problem on a flow network. A flow network is a directed graph where each edge has a capacity, and two special vertices are designated as the source (S) and the sink (T). The problem is to find the maximum amount of 'flow' that can be sent from S to T without exceeding the capacity of any edge. The Ford-Fulkerson method is a general framework for solving this problem. It's a greedy approach that repeatedly finds an 'augmenting path'—a path from S to T in the residual graph with available capacity—and pushes flow along this path. The process continues until no more augmenting paths can be found. The value of the flow is then maximal. The efficiency of Ford-Fulkerson depends on how the augmenting paths are found. The Edmonds-Karp algorithm is a specific implementation of this method where Breadth-First Search (BFS) is used to find the shortest augmenting path (in terms of the number of edges) at each step. This guarantees that the algorithm terminates and provides a polynomial time complexity of O(V * E^2). The max-flow problem is closely related to the min-cut problem, as stated by the powerful max-flow min-cut theorem, and has wide-ranging applications in network design, logistics, and scheduling.