Introduction

In the realm of network flow optimization, the Ford-Fulkerson algorithm stands as one of the most powerful and influential methods. It plays a vital role in solving a wide range of real-world problems, from maximizing data transmission rates in computer networks to optimizing transportation systems and more.

Understanding the Ford-Fulkerson Algorithm

• Augmenting Paths: The core idea behind the Ford-Fulkerson algorithm lies in finding augmenting paths from the source to the sink within the network. An augmenting path is a path along which additional flow can be sent without violating capacity constraints. The algorithm repeatedly searches for such augmenting paths until none can be found.

• Residual Graph: To efficiently find augmenting paths, the Ford-Fulkerson algorithm introduces the concept of a residual graph. The residual graph represents the remaining capacity of each edge in the original network and the potential for additional flow along reverse edges.

Algorithm Steps

• Initialize the flow to zero for all edges in the network.
• While there exists an augmenting path in the residual graph: Find the minimum capacity edge on the augmenting path (known as the bottleneck). Update the flow values along the augmenting path, increasing the flow through forward edges and decreasing it through reverse edges.
• Compute the maximum flow as the sum of all flow values leaving the source node.

Termination and Correctness

The Ford-Fulkerson algorithm terminates when no more augmenting paths can be found. At this point, the flow computed by the algorithm is guaranteed to be the maximum flow in the network, thanks to the Max Flow Min Cut Theorem.

Applications of the Ford-Fulkerson Algorithm

• Network Capacity Planning: Maximizing data transmission rates in computer networks. Optimizing traffic flow on transportation networks.
• Resource Allocation: Distributing resources efficiently in supply chain management.
• Image Segmentation: In computer vision applications, it can be utilized for image segmentation.

The Ford-Fulkerson algorithm has proven to be a cornerstone in the world of network flow optimization. By skillfully finding augmenting paths and leveraging residual graphs, it efficiently computes the maximum flow in a network. Its applications span across numerous fields, including computer networks, transportation systems, supply chain management, and image processing.

To use network flows and Ford-Fulkerson algorithm as databases you can use PostgreSQL's extension Apache AGE: -
More about apache age here: https://age.apache.org/
Github here: https://github.com/apache/age/
To implement Ford-Fulkerson algorithms in Apache AGE, you can use drivers given here and use AGE with programming languages such as python.: https://github.com/apache/age/tree/master/drivers