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.

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