Explain the concept of minimum cost network flow.

Minimum cost network flow is a mathematical model used to find the cheapest way to transport goods through a network.

In a minimum cost network flow problem, we are given a directed graph with nodes representing sources, sinks, and intermediate points, and edges representing the capacity and cost of transporting goods between nodes. The objective is to find the flow of goods that minimizes the total cost of transportation while satisfying the capacity constraints.

To solve this problem, we use the Ford-Fulkerson algorithm, which iteratively finds augmenting paths in the graph and increases the flow along those paths until no more augmenting paths can be found. An augmenting path is a path from the source to the sink that has available capacity and a non-negative cost.

The algorithm terminates when there are no more augmenting paths, and the resulting flow is optimal. The total cost of the flow is the sum of the costs of all the edges with positive flow.

To illustrate this concept, consider the following example:

We have a network with four nodes, A, B, C, and D, and five edges with capacities and costs as shown below:

A -> B: capacity 3, cost 2
A -> C: capacity 2, cost 4
B -> C: capacity 2, cost 1
B -> D: capacity 2, cost 5
C -> D: capacity 3, cost 3

We want to find the minimum cost flow from A to D with a total capacity of 4.

Using the Ford-Fulkerson algorithm, we start with an initial flow of zero and find an augmenting path from A to D with a capacity of 2 and a cost of 7 (A -> C -> D). We increase the flow along this path to 2, and update the residual graph accordingly:

A -> B: capacity 3, cost 2, flow 0
A -> C: capacity 2, cost 4, flow 2
B -> C: capacity 2, cost 1, flow 0
B -> D: capacity 2, cost 5, flow 0
C -> D: capacity 3, cost 3, flow 2

We then find another augmenting path from A to D with a capacity of 2 and a cost of 8 (A -> B -> D). We increase the flow along this path to 2, and update the residual graph again:

A -> B: capacity