### The Network Flow problem

• Transportation network

• Definition: Basic Network

• basic network = a directed graph N with the following properties:

• Graph N has exactly one source node

• Source node = a node with 0 (zero) in-degree edges

• Graph N has exactly one sink node

• Sink node = a node with 0 (zero) out-degree edges

• Each edge e of N has a positive capacity c(e)

 c(e) = the maximum amount of commodity that can flow through the edge e in a given time

• Example:

• Network flow (of some commodity)

• Definition: flow

• Flow = an assignment f(e) to each edge e of the network that satisfies:

• The feasibility condition:

 f(e)   ≤   c(e)

I.e.: flow on an edge cannot exceed the capacity of the edge

• The flow conservation law:

 ``` Total flow going into node x = Total flow going out from node x ```

Note:

 The source node and the sink node are except from the flow conservation law !!!

• Example:

It represents the following transportation of a commodity:

• Saturated and Unsaturated edges

• Saturated edge

• An edge e is saturated if:

 ``` f(e) = c(e) (flow through edge = capacity of edge) ```

Example:

• Unsaturated edge

• An edge e is unsaturated if:

 ``` f(e) < c(e) (flow through edge = capacity of edge) ```

• The Maximum flow Problem

• Maximum flow problem:

 For a given basic network, find a flow of the largest possible value from the source node S to sink node T