Network Flows
Peter Schröder
CS138A 1999
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Flow Networks
Definitions
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a flow network G=(V,E) is a directed graph
in which each edge (u,v) in E has a
nonnegative capacity c(u,v)
there are two distinguished vertices: a
source s and a sink t
for simplicity assume that every vertex lies
on some path from s to t (this implies that
the graph is connected)
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Flow
Definition
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a flow on a graph is a real valued function
VxV>R with the following properties
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capacity constraint: for all u,v in V
f u, v   c u, v 
skew symmetry: for all u,v in V
f u, v   f v, u
flow conservation: for all u in V\{s,t}
 f u, v   0
vV
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Flow
Definitions

the value of a flow is defined as
f 
 f s, v 
vV


in the maximum flow problem we wish to
find the maximum flow from s to t in some
graph G=(V,E)
non zero net flow from u to v implies (u,v)
in E or (v,u) in E
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Flow
Multiple sources and sinks
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introduce super source and super sink with
infinite capacity connections to the
individuals sources and sinks respectively
implicit set summation convention
Lemma 1: Let G=(V,E) be a flow network
and let f be a flow in G. Then for X subset
Vf X, X  0 X, Y  V f X, Y   f Y, X
X, Y, Z  V
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XY  
f X  Y, Z  f X, Z  f Y, Z
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Example
Use implicit set summation to prove
 f  f V, t 
f  f s, V 
 f V, V   f V  s, V 
 f V, V  s 
 f V, t   f V, V  s  t 
 f V, t 
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by definition
by Lemma 1
by Lemma 1
by Lemma 1
by flow conservation
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Ford-Fulkerson Method
Principles behind differing
implementations
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repeated update through augmenting
paths
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Ford-Fulkerson-Method(G,s,t)
if1.there
is unused capacity left, use it…
2.
3.
4.
5.
CS138A 1999
initialize flow to 0
while( exists augmenting path )
augment flow f along path
return f
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Residual Networks
Formalization of remaining flow
opportunities
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given flow network G=(V,E) and flow f
define residual capacity
c f u, v   cu, v   f u, v 

the residual flow network is induced by
they positive residual capacity Gf=(V,Ef)
where Ef consists of all pairs u,v in V with
positive residual flow capacity (note that it
can have as many as twice as many
edges)
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Properties
Residual networks
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Lemma 2: Let G=(V,E) be a flow network
with source s and sink t and f be a flow; let
Gf be the residual network induced and let
f’ be a flow in Gf then the flow sum f+f’ is a
flow in G with value |f+f’|=|f|+|f’|
Augmenting path
simple path p in residual network from s to
t
 residual capacity cf(p) is minimum capacity
along
path
p
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Residual Capacity
Property
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Lemma 3: let G=(V,E) be a flow network
with flow f and p be an augmenting path in
Gf. Define
 c f p  if u, v  is on p

fp u, v    c f p  if v, u is on p

0 otherwise

then fp is a flow in Gf with value |fp|=cf(p)
positive
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Residual Capacity
Adding an augmenting flow
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Corollary 4: Let G=(V,E) be a flow network
with flow f and p be an augmenting path in
Gf Let fp be defined as before; define
f’=f+fp then f’ is a flow in G with value
|f’|=|f|+|fp| greater than |f|

immediate from previous two lemmas
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Cuts of Flow Networks
What is a cut?
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a cut(S,T) is a partition of V into S and T
with s in S and t in T with net flow f(S,T)
and capacity c(S,T) (picture)
Lemma 5: let f be a flow in a flow network
G with source s and sink t and let (S,T) be
a cut of G; then f(S,T)=|f|
Corollary 6: the value of any flow f in a flow
network G is bounded from above by the
capacity of any cut of G
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Cuts in Flow Networks
Max-flow min-cut theorem
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Theorem 7: if f is a flow in a flow network
G=(V,E) with source s and sink t, then the
following conditions are equivalent:
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f is a maximum flow in G
the residual network Gf contains no
augmenting paths
|f|=c(S,T) for some cut (S,T) of G
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Basic Ford-Fulkerson
Iteratively increase flow by residual
capacity
1.
2.
3.
4.
5.
6.
7.
8.
Ford-Fulkerson(G,s,t)
for( (u,v) in E[G] )
f[u,v] = 0; f[v,u] = 0;
while( exists path p from s to t in Gf )
cf(p) = min{cf(u,v): (u,v) in p}
for( (u,v) in E[p] )
f[u,v] = f[u,v]+cf(p);
f[v,u] = -f[u,v]
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Ford-Fulkerson
Running time: depends on method to pick
augmenting path (use BFS)
Edmonds-Karp algorithm
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pick p with BFS looking for the shortest path
with unit edge lengths in the residual
network; O(VE2)
Lemma 8: when Edmunds-Karp is run on G
then for all vertices v in V\{s,t} the shortest
path distancef s, v  in the residual
network Gf increases monotonically with
augmentation
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Edmunds-Karp
Properties
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Theorem 9: if the Edmonds-Karp algorithm
is run on a flow network G=(V,E) with
source s and sink t, then the total number
of flow augmentations performed by the
algorithm is at most O(VE)
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