SLIDE 1
1
Maximum Flow Maximum Flow
A A flow flow network network ( (G,c,s,t G,c,s,t) ) is a directed graph is a directed graph G=(V,E) G=(V,E) together with a non together with a non-
- negative map
negative map c:E c:E R R, called , called capacity capacity, and two distinguished , and two distinguished vertices vertices s s and and t, t, called respectively the called respectively the source source and and sink sink with the condition that with the condition that c(u,v c(u,v)=0 )=0 if the if the edge edge ( (u,v u,v) ) is not in is not in E E A A flow flow on a flow network
- n a flow network G
G is a map is a map f:VxV f:VxV R R such that such that
- f(u,v
f(u,v) =< ) =< c(u,v c(u,v) ) (Capacity (Capacity Constrant Constrant) )
- f(u,v
f(u,v)= )=-
- f(v,u
f(v,u) ) (Skew Symmetry) (Skew Symmetry)
- {
}
, ( , )
v V
u V s t f u v
∈
∈ − ∈ − ⇒ =
∑
Maximum Flow Maximum Flow
The The value value |f| |f| of a flow
- f a flow f
f is defined as is defined as
( ) ( )
, ( , )
v V
f f s v f s V
∈
= = = =
∑
Notation Convention: Notation Convention:
( , ) ( , )
y Y
f x Y f x y
∈
= ∑
,
( , ) ( , )
x X y Y
f X Y f x y
∈ ∈ ∈ ∈
= ∑ ( , ) ( , )
x X
f X y f x y
∈
= ∑ Max Max-
- Flow Problem
Flow Problem
Given a flow network Given a flow network G G, find a flow , find a flow f f on
- n
G G of maximum value
- f maximum value