# Ford Fulkerson Algorithm

## Algorithm Details

Year : 1956

Family : Maximum Flow

Authors : Edsger W. Dijkstra

Paper Link : http://www-m3.ma.tum.de/foswiki/pub/MN0506/WebHome/dijkstra.pdf

Time Complexity :

## Problem Statement

In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximum flow problem can be seen as a special case of more complex network flow problems, such as the circulation problem.

## PseudoCode

1 flow = 0 2 for each edge (u, v) in G: 3 flow(u, v) = 0 4 while there is a path, p, from s -> t in residual network G_f: 5 residual_capacity(p) = min(residual_capacity(u, v) : for (u, v) in p) 6 flow = flow + residual_capacity(p) 7 for each edge (u, v) in p: 8 if (u, v) is a forward edge: 9 flow(u, v) = flow(u, v) + residual_capacity(p) 10 else: 11 flow(u, v) = flow(u, v) - residual_capacity(p) 12 return flow

## Applications

■ Disjoint paths and network connectivity.

■ Bipartite matchings.

■ Circulations with upper and lower bounds.

■ Census tabulation (matrix rounding).

■ Airline scheduling.

■ Image segmentation.

■ Project selection (max weight closure).

■ Baseball elimination.

## Implementations

JS : https://github.com/prabod/Graph-Theory-Ford-Fulkerson-Maximum-Flow