How to solve a network flow problem with multiple sources and sinks?

To solve a network flow problem with multiple sources and sinks, use the max-flow min-cut theorem.

The max-flow min-cut theorem states that the maximum flow in a network is equal to the minimum capacity of a cut in the network. A cut is a partition of the nodes in the network into two disjoint sets, one containing the sources and the other containing the sinks.

To apply the max-flow min-cut theorem, first construct a flow network with the sources and sinks as nodes, and the edges representing the capacity of the flow between them. Then, use an algorithm such as the Ford-Fulkerson algorithm to find the maximum flow in the network.

To find the minimum cut in the network, use the residual graph obtained from the Ford-Fulkerson algorithm. A residual graph is a graph that represents the remaining capacity of the edges after the flow has been sent through the network. The minimum cut is the cut that separates the sources from the sinks with the minimum capacity in the residual graph.

Once the minimum cut has been found, the maximum flow in the network is equal to the capacity of the minimum cut. This can be used to determine the maximum amount of flow that can be sent from the sources to the sinks in the network.

Overall, solving a network flow problem with multiple sources and sinks involves constructing a flow network, finding the maximum flow using an algorithm such as the Ford-Fulkerson algorithm, and then using the max-flow min-cut theorem to determine the maximum flow in the network.