Flow Networks and Flows
Flow Network is a directed graph that is used for modeling material Flow. There are two different vertices; one is a source which produces material at some steady rate, and another one is sink which consumes the content at the same constant speed. The flow of the material at any mark in the system is the rate at which the element moves.
Some real-life problems like the flow of liquids through pipes, the current through wires and delivery of goods can be modeled using flow networks.
Definition: A Flow Network is a directed graph G = (V, E) such that
Let G = (V, E) be a flow network. Let s be the source of the network, and let t be the sink. A flow in G is a real-valued function f: V x V→R such that the following properties hold:
The quantity f (u, v), which can be positive or negative, is known as the net flow from vertex u to vertex v. In the maximum-flow problem, we are given a flow network G with source s and sink t, and we wish to find a flow of maximum value from s to t.
The three properties can be described as follows:
The value of the flow is the net flow from the source,
The positive net flow entering a vertex v is described by
The positive net flow leaving a vertex is described symmetrically. One interpretation of the Flow-Conservation Property is that the positive net flow entering a vertex other than the source or sink must equal the positive net flow leaving the vertex.
A flow f is said to be integer-valued if f (u, v) is an integer for all (u, v) ∈ E. Clearly, the value of the flow is an integer is an integer-valued flow.