Maximum Flow

Residual Network

A "fake" network constructed from real network considering the flow in two ways, remaining flow capacity, and current flow capacity which allows to cancel.

Augmenting path

Path from s to t in residual network

Residual capacity

The maximum amount of net flow that we can ship along the edges of an augmenting path p, it can be called as bottleneck. Residual capacity on edge(u, v) = real capacity from u to v - used flow from u to v = c(u, v) - f(u, v) Residual capacity on path(u -> v -> s) = min(cf(u,v), cf(v, s))

Ford-Fulkerson algorithm

For each edge(u ,v)
  f[u, v] = 0 // no flow at start
  f[v, u] = 0
while there exist a augmenting path p from s to t
    find cf(p) = residual capacity(bottleneck) on the agumenting path
    for each edge(u, v) on the agumenting path
      f[u, v] += f[u, v] + cf(p)   // we used cf(p) capacity on that tunnel
      f[v, u] += f[v, u] - cf(p)   // to be redraw, when you compute cf(v, u) =  c[v, u] - (- xxx) = c[v, u] + xxxx

The Edmonds-Karp algorithm

Use breath first search to find a path

Maximum bipartite matching:

Link s to t wth two lines, each line weighted 1