Ford-Fulkerson Algorithm in Python

The Ford-Fulkerson algorithm is a widely used algorithm to solve the maximum flow problem in a flow network. The maximum flow problem involves determining the maximum amount of flow that can be sent from a source vertex to a sink vertex in a directed weighted graph, subject to capacity constraints on the edges.

The algorithm works by iteratively finding an augmenting path, which is a path from the source to the sink in the residual graph, i.e., the graph obtained by subtracting the current flow from the capacity of each edge. The algorithm then increases the flow along this path by the maximum possible amount, which is the minimum capacity of the edges along the path.

Problem:

Given a graph which represents a flow network where every edge has a capacity. Also, given two vertices source ‘s’ and sink ‘t’ in the graph, find the maximum possible flow from s to t with the following constraints:

• Flow on an edge doesn’t exceed the given capacity of the edge.
• Incoming flow is equal to outgoing flow for every vertex except s and t.

For example, consider the following graph from the CLRS book. 

👁 ford_fulkerson1

The maximum possible flow in the above graph is 23. 

👁 ford_fulkerson2

Ford-Fulkerson Algorithm in Python

 The following is simple idea of Ford-Fulkerson algorithm:

1. Start with initial flow as 0.
2. While there exists an augmenting path from the source to the sink:  
   • Find an augmenting path using any path-finding algorithm, such as breadth-first search or depth-first search.
   • Determine the amount of flow that can be sent along the augmenting path, which is the minimum residual capacity along the edges of the path.
   • Increase the flow along the augmenting path by the determined amount.
3. Return the maximum flow.

Below is the implementation of Ford-Fulkerson algorithm. To keep things simple, graph is represented as a 2D matrix.

The maximum possible flow is 23 

Time Complexity: Time complexity of the above algorithm is O(max_flow * E). We run a loop while there is an augmenting path. In worst case, we may add 1 unit flow in every iteration. Therefore the time complexity becomes O(max_flow * E).