Exercises Section 3
1. Consider the directed graph given below (Attention: not all arcs point from left to
right.) Numbers proceeded by a $sign indicate the costs, and the other numbers
give the capacities of the corresponding arcs.
a. Use one of the treated shortest path algorithms to find a minimum cost path
from
A
to
H
.
b. Determine a maximum flow from
A
to
H
with minimum cost and also give a
minimum cut.
u
A
u
C
u
F
u
H
u
B
u
D
u
E
u
G
2
$1

2
$6
@
@
@
@
@
@
@
@
@
@
@
@
@R
2
$2

5
$3
@
@
@
@
@
@
@
@
@
@
@
@
@
R
4
$2
1
$4
@
@
@
@
@
@
@
@
@
@
@
@
@
@I
3
$2
2
$3

3
$1
@
@
@
@
@
@
@
@
@
@
@
@
@R
2
$2

2
$4
1
$1
2. The table below give the expenses for persons W, X, Y and Z to travel to places A,
B, C en D. The objective is to send each person to one of the four places such that
all places will be visited, whilst the total costs are as small as possible. Translate
this problem into a maximum flow problem and solve it with the maximum flow
algorithm.
A
B
C
D
W
16
12
11
12
X
13
11
8
14
Y
10
6
7
9
Z
11
15
10
8
3. Given the digraph below with source
r
and sink
s
. The two values between brackets
given at each arc give the demand and the capacity of that arc. Find a feasible flow
by use of the method presented in the course and increase this flow to a maximum
flow. Determine also the minimum cut.
1
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u
u
u
u
u
u
@
@
@
@
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@
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@
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@
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@
@
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R

@R
6

@R
6
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 Fall '14
 Graph Theory, $1, Shortest path problem, Flow network, $1 2

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