Max-flow Min-cut Theorem - Definition

Definition

Let be a network (directed graph) with and being the source and the sink of respectively.

The capacity of an edge is a mapping c: E→R+, denoted by cuv or c(u,v). It represents the maximum amount of flow that can pass through an edge.
A flow is a mapping f: E→R+, denoted by fuv or f(u,v), subject to the following two constraints:
  1. for each (capacity constraint)
  2. for each (conservation of flows).
The value of flow is defined by, where is the source of . It represents the amount of flow passing from the source to the sink.

The maximum flow problem is to maximize | f |, that is, to route as much flow as possible from s to t.

An s-t cut C = (S,T) is a partition of V such that s∈S and t∈T. The cut-set of C is the set {(u,v)∈E | u∈S, v∈T}. Note that if the edges in the cut-set of C are removed, | f | = 0.
The capacity of an s-t cut is defined by .

The minimum s-t cut problem is minimizing, that is, to determine S and T such that the capacity of the S-T cut is minimal.

Read more about this topic:  Max-flow Min-cut Theorem

Famous quotes containing the word definition:

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)

    “Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
    —Elaine Heffner (20th century)

    “The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
    —Jean Baudrillard (b. 1929)