How can you find the maximum flow in a network with multiple sources and sinks?

1 Single-source single-sink networks

The simplest case of finding the maximum flow in a network is when there is only one source node and one sink node. The source node is where the flow originates, and the sink node is where the flow ends. The maximum flow is the maximum amount of flow that can go from the source to the sink without violating the capacity constraint. One way to find the maximum flow in this case is to use the Ford-Fulkerson algorithm, which iteratively augments the flow by finding paths from the source to the sink that have some residual capacity. The residual capacity of an edge is the difference between its capacity and its current flow. The algorithm stops when there is no more augmenting path, and the resulting flow is the maximum flow.

2 Multiple-source multiple-sink networks

The problem of finding the maximum flow in a network becomes more complex when there are multiple source nodes and multiple sink nodes. For example, suppose you want to find the maximum amount of goods that can be shipped from several factories to several markets, given the capacity of each road and the demand of each market. One way to handle this problem is to transform it into a single-source single-sink problem by adding a super-source node and a super-sink node. The super-source node is connected to all the original source nodes by edges with infinite capacity, and the super-sink node is connected to all the original sink nodes by edges with capacity equal to their demand. Then, you can apply the Ford-Fulkerson algorithm to the transformed network, and the maximum flow from the super-source to the super-sink will be the same as the maximum flow in the original network.

3 Applications and extensions

Finding the maximum flow in a network has many applications in operations research, such as scheduling, routing, matching, and allocation. For example, you can use it to find the optimal way to assign tasks to workers, to distribute water or electricity through a network of pipes or wires, or to match students to schools or donors to recipients. Moreover, there are many extensions and variations of the maximum flow problem, such as the minimum-cost maximum flow problem, which minimizes the total cost of the flow along the edges; the maximum concurrent flow problem, which maximizes the number of flows that can be sent simultaneously through the network; and the multi-commodity flow problem, which allows different types of flows with different capacities and demands.

4 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?