Editing Max-flow min-cut theorem (section)

=== Flows ===
A '''flow''' through a network is a mapping <math>f:E\to\R^+</math> denoted by <math>f_{uv}</math> or <math>f(u, v)</math>, subject to the following two constraints:
# '''Capacity Constraint''': For every edge <math>(u, v) \in E</math>, <math>f_{uv} \le c_{uv}.</math>
# '''Conservation of Flows''': For each vertex <math>v</math> apart from <math>s</math> and <math>t</math> (i.e. the source and sink, respectively), the following equality holds:<br/><math>\sum\nolimits_{\{ u : (u,v)\in E\}} f_{uv} = \sum\nolimits_{\{w : (v,w)\in E\}} f_{vw}.</math>

A flow can be visualized as a physical flow of a fluid through the network, following the direction of each edge. The capacity constraint then says that the volume flowing through each edge per unit time is less than or equal to the maximum capacity of the edge, and the conservation constraint says that the amount that flows into each vertex equals the amount flowing out of each vertex, apart from the source and sink vertices. 

The '''value''' of a flow is defined by 

:<math>|f| = \sum\nolimits_{\{v : (s,v)\in E\}} f_{sv}=\sum\nolimits_{\{v : (v,t)\in E\}} f_{vt},</math> 

where as above <math>s</math> is the source and <math>t</math> is the sink of the network. In the fluid analogy, it represents the amount of fluid entering the network at the source. Because of the conservation axiom for flows, this is the same as the amount of flow leaving the network at the sink. 

The maximum flow problem asks for the largest flow on a given network. 
<blockquote>
'''[[maximum flow problem|Maximum Flow Problem.]]''' Maximize <math>|f|</math>, that is, to route as much flow as possible from <math>s</math> to <math>t</math>.
</blockquote>