CSE 202 - Algorithms
Max Flow Problems
11/21/02
CSE 202 - Max Flow
Max Flow problem
Input: a directed graph with a non-negative weight
(capacity) on each edge, and a designated source node s
and sink node t.
A flow assigns to each edge a number (from 0 to its
capacity) such that for each node except s and t, the
sum of the flow into the node equals sum of the flow
out. [The fine print: this is slightly different from the book’s definition.]
The value of a flow is the sum of the flow out of the
source (which equals total flow into the sink).
Maximum-Flow problem: find a flow with the maximum
possible value.
Note: A max flow can assign 0 to each edge going into
the source, and to each edge going out of the sink.
2
CSE 202 - Max Flow
Example
16
source
13
b
10
s
This is wasteful!
8
The text’s formulation
doesn’t even let it happen.
4
2
b
14
12
a
c
9
4
10
s
12
a
t
7
d
c
0
6
20
sink
4
14
t
2
d
4
A flow of value 18
3
CSE 202 - Max Flow
Leftover capacity
16-10
13-8
New backedge
showing we can reduce
forward flow by 10
12-12
c
9-0
10-2 4-4
s
source
a
6
b
a
10 8 2
8
s
5
b
14-6
0
12
9
6
8
20-14
t
7-2
d
4-4
c
6
14
5
2
4
d
sink
t
0
Residual network
4
CSE 202 - Max Flow
Augmenting Path
We can push 5 more
through network
on indicated path
(New flow has
s
value 23.)
6
a
10 8 2
8
5
b
6
a
10 8 2
13
s
0
b
0
12
9
6
8
0
12
9
11
3
c
6
14
5
2
4
t
d
0
c
1
19
0
7
4
d
t
0
New residual network
5
Now we’re stuck!
We can’t get any
more across cut.
CSE 202 - Max Flow
Duality: Max Flow = Min Cut
Original problem had a cut of value 23. We can’t possibly
get more from source to sink.
16
13
b
c
9
4
10
s
12
a
14
20
t
7
d
4
Theorem: Max flow = min cut
Proof: Obviously, any flow  min cut.
But if max flow < min cut, there would be an augmenting
path from source to sink, leading to a higher-valued flow.
This is a contradiction. Thus, max flow = min cut. QED.
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CSE 202 - Max Flow
Ford-Fulkerson Methods
initialize flow to 0;
while (there’s an augmenting path){
update flow;
compute new residual graph; }
If there are several augmenting paths, does it
matter which we pick??
a
100
100
7
1
1
s
100
b
t
100
CSE 202 - Max Flow
Edmonds-Karp Algorithm
Always choose an augmenting path with as few edges as
possible (say it has L edges).
– This might create a new backedge, e.g (v,u).
– This new edge can’t be in a new path of L edges..
• (Handle multiple new backedges by induction proof.)
– Meanwhile, at least one original edge has been eliminated.
– Thus, there are at most E iterations using L long paths.
v
k long path
t
s
j long path
8
u
j+1+k is a minimal
length path (L)
m long
v
t
s
u
n long
m  j+1 and n  k+1
so m+n > j+k+1 = L
CSE 202 - Max Flow
Edmonds-Karp Algorithm
Always choose an augmenting path with as few
edges as possible.
– At most E iterations use L-edge augmenting path
– Longest augmenting path has V-1 edges.
– Thus, there are at most O(VE) augmentations.
– How long does it take to find and process a
shortest augmenting path??
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CSE 202 - Max Flow
Max Flow Algorithms
Edmonds-Karp is O(VE2)
Some faster algorithms based on “Push-Relabel”.
– E.g. Goldberg’s algorithm is O(V2E)
• Start source at height V, all others at 0
• Add in flow from higher nodes to lower ones.
• If a non-sink node can’t push all its incoming flow
out, increase it’s height.
• When done, return excess back to source.
– Carefully choosing order gives O(V3) algorithm.
Fastest known algorithm is
O( min(V2/3, E1/2) E lg(1+V2/E) lg(max capacity) ).
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CSE 202 - Max Flow
An Application
Maximum matching problem:
– Given an (undirected) graph (V,E) find a
maximum-sized set of disjoint edges.
• Not the same as a maximal set of disjoint edges.
– Amazingly, not NP-complete (but it’s not easy)
Bipartite graph: Graph such that you can
partition the nodes V = V1 U V2, and every edge
goes between a node of V1 and one of V2.
Maximum matching for a bipartite graph can be
reduced to a max flow problem.
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CSE 202 - Max Flow
Glossary (in case symbols are weird)
      
 
 subset  element of
set

infinity  empty
 for all  there exists  intersection 
union
 big theta  big omega  summation
 >=
 <=
 about equal
 not equal  natural numbers(N)
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CSE 202 - Max Flow