Massive Algorithms: Maximum Flow - Ford Fulkerson algorithm

http://en.wikipedia.org/wiki/Ford%E2%80%93Fulkerson_algorithm
http://www.ugrad.cs.ubc.ca/~cs490/sec202/notes/flow/Flow%20Intro.pdf

The Ford–Fulkerson method is an algorithm which computes the maximum flow in a flow network. The name "Ford–Fulkerson" is often also used for the Edmonds–Karp algorithm, which is a specialization of Ford–Fulkerson.

The idea behind the algorithm is as follows: As long as there is a path from the source (start node) to the sink (end node), with available capacity on all edges in the path, we send flow along one of these paths. Then we find another path, and so on. A path with available capacity is called an augmenting path.

Let

G(V,E)

be a graph, and for each edge from

u

v

, let

c(u,v)

be the capacity and

f(u,v)

be the flow. We want to find the maximum flow from the source

s

to the sink

t

. After every step in the algorithm the following is maintained:

Capacity constraints:	$\forall (u, v) \in E \ f(u,v) \le c(u,v)$	The flow along an edge can not exceed its capacity.
Skew symmetry:	$\forall (u, v) \in E \ f(u,v) = - f(v,u)$	The net flow from $u$ to $v$ must be the opposite of the net flow from $v$ to $u$ (see example).
Flow conservation:	$\forall u \in V: u \neq s \text{ and } u \neq t \Rightarrow \sum_{w \in V} f(u,w) = 0$	That is, unless $u$ is $s$ or $t$ . The net flow to a node is zero, except for the source, which "produces" flow, and the sink, which "consumes" flow.
Value(f):	$\sum_{(s,u) \in E} f(s, u) = \sum_{(v,t) \in E} f(v, t)$	That is, the flow leaving from $s$ or arriving at $t$ .

This means that the flow through the network is a legal flow after each round in the algorithm. We define the residual network

G_f(V,E_f)

to be the network with capacity

c_f(u,v) = c(u,v) - f(u,v)

and no flow. Notice that it can happen that a flow from

v

u

is allowed in the residual network, though disallowed in the original network: if

f(u,v)>0

and

c(v,u)=0

then

c_f(v,u)=c(v,u)-f(v,u)=f(u,v)>0

Algorithm Ford–Fulkerson

Inputs Given a Network

G = (V,E)

with flow capacity

c

, a source node

s

, and a sink node

t

Output Compute a flow

f

from

s

t

of maximum value

$f(u,v) \leftarrow 0$ for all edges $(u,v)$
While there is a path from to in , such that for all edges :
1. Find $c_f(p) = \min\{c_f(u,v) : (u,v) \in p\}$
2. For each edge
  1. $f(u,v) \leftarrow f(u,v) + c_f(p)$ (Send flow along the path)
  2. $f(v,u) \leftarrow f(v,u) - c_f(p)$ (The flow might be "returned" later)

The path in step 2 can be found with for example a breadth-first search or a depth-first search in

G_f(V,E_f)

. If you use the former, the algorithm is calledEdmonds–Karp.

When no more paths in step 2 can be found,

s

will not be able to reach

t

in the residual network. If

S

is the set of nodes reachable by

s

in the residual network, then the total capacity in the original network of edges from

S

to the remainder of

V

is on the one hand equal to the total flow we found from

s

t

, and on the other hand serves as an upper bound for all such flows. This proves that the flow we found is maximal. See also Max-flow Min-cut theorem.

If the graph