Maximum Flow Problem in Graph Theory

In graph theory, the maximum flow problem involves determining the path that achieves the highest possible flow rate within a network. This problem can be solved using various algorithms such as the Ford-Fulkerson algorithm and Dinic's algorithm. For those seeking a MATLAB implementation, the following code provides a practical solution: [Insert MATLAB code here]. The implementation begins by constructing a network graph data structure, typically represented through adjacency matrices or edge lists with capacity attributes. The Ford-Fulkerson algorithm operates by continuously augmenting paths through residual graphs, using either depth-first or breadth-first search to find augmenting paths while updating residual capacities. Key functions include capacity initialization, residual graph updates, and path augmentation loops. Once the algorithm identifies the maximum flow value and corresponding flow distribution, it outputs the results and terminates the program execution.