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CS 312: Algorithm Design &
Analysis
Lecture #29: Network Flow and
Cuts
Slides by: Eric Ringger, with contributions from Mike Jones
Announcements
 HW #20:
 Due now.
 HW #20.5
 Due Monday
 Project #6: Linear Programming
 In preparation: HW #21
 Formulate problem for Proj. #6
 NO 2-hour time limit
 Due Wednesday
Objectives
 Better understand the idea of “changing
the problem” in the max flow algorithm
 Understand the relationship between flows
and cuts
 Solve Bipartite Matching
 See example of Padded matrix-vector
representation
Max Flow Example
6
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b
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s
2
2
e
2
d
f
t
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4
Summarize the max flow algorithm.
Max Flow Example
6
4
b
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s
2
2
e
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1. Pick a simple path st and increase the flow along
that path as much as possible
2. Create the residual graph GR including reverse edges
3. Repeat until there is no path with remaining capacity.
Solve
6
4
b
4
s
2
2
e
2
d
f
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t
4
Max Flow Example
6
One solution
4
b
4
s
2
2
e
2
d
f
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t
4
Residual Graph, incl. reverse edges
b
e
s
t
d
f
Max Flow Example
b
e
s
t
d
f
Residual Graph, incl. reverse edges
b
e
s
t
d
No paths from s to t,
so we are done.
f
Key Points about our
Max Flow Algorithm
 When we transform the problem we create
“imaginary edges”
 However, in the end our selected paths
always produce a flow schedule that
involves only the original edges.
(s,t)-Cut
6
L:
Source
Side
4
b
4
s
2
2
e
2
d
f
t
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R:
Sink
Side
4
An (s,t)-cut partitions the vertices into two disjoint sets:
• L contains s
• R contains t
To compute the capacity of a cut, add up the capacities
of edges that flow across the cut from L to R.
Cut Property
What does
a min cut
represent?
6
4
b
4
s
2
2
e
2
d
f
t
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4
Max flow is less than or equal to the flow across any cut.
Max-Flow Min-Cut Theorem:
The maximum flow through a network is equal to the
minimum flow across all cuts.
Cut Property
6
4
b
4
s
2
2
e
2
d
f
4
t
4
We can call
a minimum cut
a “matching
cut”,
since its
capacity
matches the
maximum flow.
Max flow is less than or equal to the flow across any cut.
Max-Flow Min-Cut Theorem:
The maximum flow through a network is equal to the
minimum flow across all cuts.
Algorithms
 Our focus has been on gaining intuition
toward understanding the Simplex
algorithm.
 The general algorithm for solving max.
flow:
 Ford-Fulkerson
 An implementation thereof:
 Edmonds-Karp: 𝑂(𝑉 ⋅ 𝐸 2 )
Match-making
Match-making
Is there a perfect matching?
Bipartite Matching
Great
example of
problem
reduction.
Is there a perfect matching?
LP: Padded Matrices & Vectors
End
Assignment
 HW #20.5
 Due Monday
 HW #21:
 Part 1 of Project #6
 Just the problem formulation and
representations
 No time limit
 Due Wednesday