Some directed weighted graphs can be seen as flow networks. In a flow network, edge weights represent capacities and each edge receives a flow that can't exceed the edge's capacity. The labels on the edges represent the used and total capacity of the edge. The maximum flow attempts to find a feasible flow through the network that is maximum, considering a single source (where the initial flow starts) and single sink (where the flow ends). The maximum flow problem allows one to solve related problems like pair wise assignment. There are various algorithms to solve the maximum flow problem. Three of the most famous ones are the FordFulkerson algorithm, the Edmonds-Karp algorithm, and Dinic's algorithm.
The idea behind the Ford-Fulkerson algorithm is to repeatedly find augmenting paths in the flow network. An augmenting path is a path that still has an available capacity. While it is possible to find augmenting paths, one can add a flow through the path equal to its capacity and repeat the process. Ford-Fulkerson algorithm runs in O(Ef), f being the maximum flow of the graph. The Edmonds-Karp algorithm improves on the Ford-Fulkerson algorithm by always selecting the augmenting path that is shortest. The runtime complexity of the Edmonds-Karp algorithm is O(VE2), independent of the maximum flow value. Dinic's algorithm runs in O(V2E) time, also building on shortest augmenting paths, but uses some concepts that make it more suitable for sparse graphs.