25. Flow in Networks
Flow Network, Maximal Flow, Cut, Rest Network, Max-flow Min-cut Theorem, Ford-Fulkerson Method, Edmonds-Karp Algorithm, Maximal Bipartite Matching [Ottman/Widmayer, Kap. 9.7, 9.8.1], [Cormen et al, Kap. 26.1-26.3]
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Motivation
Modelling flow of fluents, components on conveyors, current in electrical networks or information flow in communication networks.
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Flow Network
Flow networkG= (V, E, c): directed graph withcapacities
Antiparallel edges forbidden:
(u, v) ∈E ⇒ (v, u)6∈ E. Model a missing edge(u, v)by c(u, v) = 0.
Sourcesandsinkt: special nodes.
Every nodev is on a path betweens andt: s v t
s
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v4
t 16
13
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4
4 9 7
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Flow
A Flowf : V ×V → Rfulfills the following conditions:
Bounded Capacity:
For allu, v∈ V: 0≤f(u, v) ≤ c(u, v). Conservation of flow:
For allu∈V \ {s, t}: X
v∈V
f(v, u)−X
v∈V
f(u, v) = 0.
s
v1
v2
v3
v4
t 16/8
13/10
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4/4 9/4
4/4 7/6
Valueof the flow:
w(f) =P
v∈Vf(s, v)−P
v∈V f(v, s).
Herew(f) = 18.
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How large can a flow possibly be?
Limiting factors: cuts
cut separatingsfromt: Partition ofV intoS andT withs∈S, t ∈T.
Capacityof a cut: c(S, T) =P
v∈S,v0∈Tc(v, v0) Minimal cut: cut with minimal capacity.
Flow over the cut: f(S, T) =P
v∈S,v0∈T f(v, v0)−P
v∈S,v0∈Tf(v0, v)
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How large can a flow possibly be?
For each flow and each cut it holds thatf(S, T) =w(f):
f(S, T) = X v∈S,v0∈T
f(v, v0)− X v∈S,v0∈T
f(v0, v)
= X
v∈S,v0∈V
f(v, v0)− X v∈S,v0∈S
f(v, v0)− X v∈S,v0∈V
f(v0, v) + X v∈S,v0∈S
f(v0, v)
= X
v0∈V
f(s, v0)− X v0∈V
f(v0, s) Second equality: amendment, last equality: conservation of flow.
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Maximal Flow ?
In particular, for each cut(S, T) ofV. f(S, T) ≤ X
v∈S,v0∈T
c(v, v0) =c(S, T)
Will discover that equality holds forminS,Tc(S, T).
s
v1
v2
v3
v4
t 16
13
12
14
20
4
4 9 7
Maximal Flow ?
Naive Procedure
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4/4 7/6 s
v1
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t 16/8
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20/15
4/4 9/4
4/4 7/7
s
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t 16/8
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4/4 9/2
4/4 7/7 s
v1
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v4
t 16/10
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4/4 9/0
4/2 7/7
Conclusion: greedy increase of flow does not solve the problem.
The Method of Ford-Fulkerson
Start withf(u, v) = 0for allu, v∈ V
Determine rest network*Gf and expansion path inGf Increase flow via expansion path*
Repeat until no expansion path available.
*Will now be explained
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Increase of flow, negative!
Let some flowf in the network be given.
Finding:
Increase of the flow along some edge possible, when flow can be increased along the edge,i.e. iff(u, v) < c(u, v).
Rest capacitycf(u, v) =c(u, v)−f(u, v).
Increase of flowagainst the directionof the edge possible, if flow can be reduced along the edge, i.e. iff(u, v) > 0.
Rest capacitycf(v, u) =f(u, v).
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Rest Network
Rest networkGf provided by the edges with positive rest capacity:
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t 8
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3 10
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6 14
4 5
4
4 1 6
Rest networks provide the same kind of properties as flow networks with the exception of permitting antiparallel edges
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Observation
Theorem
LetG = (V, E, c) be a flow network with sourcesand sinktandf a flow inG. LetGf be the corresponding rest networks and letf0be a flow inGf. Thenf ⊕f0defines a flow in Gwith valuew(f) +w(f0).
(f ⊕f0)(u, v) =
(f(u, v) +f0(u, v)−f0(v, u) (u, v) ∈E
0 (u, v) 6∈E.
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Proof
Limit of capacity:
(f ⊕f0)(u, v) =f(u, v) +f0(u, v)−f0(v, u)
≥ f(u, v) +f0(u, v)−f(u, v) =f0(u, v) ≥ 0 (f ⊕f0)(u, v) =f(u, v) +f0(u, v)−f0(v, u)
≤ f(u, v) +f0(u, v)
≤ f(u, v) +cf(u, v)
= f(u, v) +c(u, v)−f(u, v) =c(u, v).
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Proof
Conservation of flow X
u∈V
(f ⊕f0)(u, v) =X
u∈V
f(u, v) +X
u∈V
f0(u, v)−X
u∈V
f0(v, u)
(Flow conservation offandf0)= X
u∈V
f(v, u) +X
u∈V
f0(v, u)−X
u∈V
f0(u, v)
= X
u∈V
(f ⊕f0)(v, u)
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Beweis
Value off ⊕f0 (in the sequelN+ :=N+(s),N− :=N−(s)):
w(f⊕f0) = X
v∈N+
(f⊕f0)(s, v)− X
v∈N−
(f⊕f0)(v, s)
= X
v∈N+
f(s, v) +f0(s, v)−f0(v, s)− X
v∈N−
f(v, s) +f0(v, s)−f0(s, v)
= X
v∈N+
f(s, v)− X
v∈N−
f(v, s) + X
v∈N+∪N−
f0(s, v) + X
v∈N+∪N−
f0(v, s)
=X
v∈V
f(s, v)−X
v∈V
f(v, s) +X
v∈V
f0(s, v) +X
v∈V
f0(v, s)
=w(f) +w(f0).
Flow in G
fexpansion pathp: path fromstotin the rest network Gf. Rest capacitycf(p) = min{cf(u, v) : (u, v)edge inp} Theorem
The mappingfp :V ×V → R,
fp(u, v) =
(cf(p) if(u, v)edge inp 0 otherwise
provides a flow inGf with valuew(fp) =cf(p) >0. [Proof: exercise]
Consequence
Strategy for an algorithm:
With an expansion pathpin Gf the flowf ⊕fp defines a new flow with valuew(f ⊕fp) =w(f) +w(fp) > w(f)
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Max-Flow Min-Cut Theorem
Theorem
Letf be a flow in a flow networkG = (V, E, c) with sourcesand sinkt. The following statementsa are equivalent:
1 f is a maximal flow inG
2 The rest networkGf does not provide any expansion paths
3 It holds thatw(f) =c(S, T) for a cut(S, T) of G.
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Proof
(3) ⇒ (1):
It holds thatw(f) ≤c(S, T) for all cutsS, T. Fromw(f) =c(S, T) it follows thatw(f) is maximal.
(1) ⇒ (2):
f maximal Flow inG. Assumption: Gf has some expansion path w(f ⊕fp) =w(f) +w(fp) > w(f). Contradiction.
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Proof (2) ⇒ (3)
Assumption:Gf has no expansion path. Define
S={v∈V : there is a paths vinGf}.(S, T) := (S, V \S)is a cut:
s∈S, t6∈S. Letu∈Sandv∈T.
If(u, v)∈E, thenf(u, v) =c(u, v), otherwise it would hold that(u, v)∈Ef. If(v, u)∈E, thenf(v, u) = 0, otherwise it would hold that
cf(u, v) =f(v, u)>0and(u, v)∈Ef
If(u, v)6∈Eand(v, u)6∈E, thenf(u, v) =f(v, u) = 0. Thus
w(f) =f(S, T) =X
u∈S
X
v∈T
f(u, v)−X
v∈T
X
u∈s
f(v, u)
=X
u∈S
X
v∈T
c(u, v)−X
v∈T
X
u∈s
0 =X
u∈S
X
v∈T
c(u, v) =c(S, T).
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Algorithm Ford-Fulkerson( G, s, t )
Input : Flow networkG= (V, E, c) Output : Maximal flow f.
for (u, v)∈E do f(u, v)←0
whileExists path p:s tin rest networkGf do cf(p)←min{cf(u, v) : (u, v)∈p}
foreach(u, v)∈pdo if (u, v)∈E then
f(u, v)←f(u, v) +cf(p) else
f(v, u)←f(u, v)−cf(p)
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Analysis
The Ford-Fulkerson algorithm does not necessarily have to converge for irrational capacities. For integers or rational numbers it terminates.
For an integer flow, the algorithms requires maximallyw(fmax) iterations of the while loop.
Search a single increasing path (e.g. with DFS or BFSO(|E|)) ThereforeO(fmax|E|).
s
u
v
t 1000
1000 1
1000
1000
With an unlucky choice the al- gorithm may require up to 2000 iterations here.
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Edmonds-Karp Algorithm
Choose in the Ford-Fulkerson-Method for finding a path inGf the expansion path of shortest possible length (e.g. with BFS)
Edmonds-Karp Algorithm
Theorem
When the Edmonds-Karp algorithm is applied to some integer valued flow networkG = (V, E) with sourcesand sinktthen the number of flow increases applied by the algorithm is inO(|V| · |E|) [Without proof]
Application: maximal bipartite matching
Given: bipartite undirected graphG = (V, E).
MatchingM: M ⊆E such that|{m∈ M : v ∈ m}| ≤1for allv ∈ V. Maximal MatchingM: MatchingM, such that|M| ≥ |M0| for each matchingM0.
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Corresponding flow network
Construct a flow network that corresponds to the partitionL, Rof a bipartite graph with sourcesand sinkt, with directed edges froms toL, fromLto R and fromR tot. Each edge has capacity1.
L R
s t
L R
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Integer number theorem
Theorem
If the capacities of a flow network are integers, then the maximal flow generated by the Ford-Fulkerson method provides integer numbers for eachf(u, v),u, v ∈V.
[without proof]
Consequence: Ford-Fulkerson generates for a flow network that corresponds to a bipartite graph a maximal matching
M ={(u, v) :f(u, v) = 1}.
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