Graph Theory: Traveling Salesman Problem (TSP)

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Graph TheoryTraveling Salesman Problem (TSP)

• E3 Teacher Summer Research Program 2005
• Texas A M University

June 29, 2005
2
TAKS Objectives9th 10th Grade

• Objective 1 Ab1A, B, C, D E
• Objective 2 Ab2A D, Ab3A B
• Objective 3 Ac1A C
• Objective 6 8.7D
• Objective 10 - 8.14A, B C, 8.15A, 8.16A B

3
Material

• GEO Board
• Yarn
• Rubber bands
• Colored pencils
• Grid paper
• Graphing calculator

4
Vocabulary

• Nodes (vertices)
• Edges (arcs)
• Degree
• Adjacent
• Path

5
Vocabulary

• Length
• Circuit
• Simple Graph
• Complete Graph

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Nodes (vertices)
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Nodes (vertices)

• Land
• Phone numbers
• People
• Junction points (electric circuits)
• Atom
• Chess players
• Companies or industries

8
Edges (arcs)
9
Edges (arcs)

• Bridges connecting land
• Calls made from one number to another connecting
  phone numbers
• Relationships or acquaintances connecting by
  people
• Wires connecting junction points (electric
  circuits)
• Bonds between atoms connecting atoms
• Matches connecting chess players (chess
  tournament)
• Transactions connecting companies or industries

10
Degree
B
A
C
D
H
G
E
F
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Degree

• Node A is of two degrees
• Node B is of two degrees
• Node C is of six degrees
• Node D is of three degrees
• How many degrees are nodes E, F, G and H?

12
Adjacent
B
A
C
D
H
G
E
F
13
Adjacent

• Node A is adjacent to Nodes C and H
• Node B is adjacent to Nodes F and H
• Node C is adjacent to Nodes A, D, E, F, G and H
• Node D is adjacent to Nodes C, E and H
• What nodes are adjacent to nodes E, F, G and H?

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Path
E
A
D
F
B
C
15
Path
E
A
D
F
B
C
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Path
E
A
D
F
B
C
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No Path
E
A
D
F
B
C
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Draw two different paths.
E
A
D
F
B
C
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Length
E
D
A
F
B
C
20
Length
E
D
A
F
B
C
21
Length
E
D
A
F
B
C
22
Length
E
D
A
F
B
C
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CircuitPoint A is the starting point
A
E
B
D
C
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CircuitPoint A is the starting point
A
E
B
D
C
25
CircuitPoint A is the starting point
A
E
B
D
C
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CircuitPoint A is the starting point
A
E
B
C
D
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Simple Graphs
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Not, Simple Graphs
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Complete Graphs
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Complete Graphs

• Draw a complete graph with six nodes.
• Draw a complete graph with seven nodes

31
Petroleum Delivers
S
N
490
370
570
325
T
825
M
565
520
565
745
380
T Texas S South Carolina N New York M Minnesota C Colorado
C
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1 Node
33
2 Nodes
34
2 Nodes
35
3 Nodes
36
3 Nodes
37
3 Nodes
38
3 Nodes
39
3 Nodes
40
3 Nodes
41
3 Nodes
42
3 Nodes
2 Routes
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4 Nodes
44
4 Nodes
45
4 Nodes
46
4 Nodes
47
4 Nodes
48
4 Nodes
49
4 Nodes
50
4 Nodes
51
4 Nodes
52
4 Nodes
53
4 Nodes
6 Routes
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2 Node
Node(s) Process Column Route(s)
2 1
55
3 Nodes
Nodes Process Column Route(s)
3 2
2 1
56
4 Nodes
Nodes Process Column Route(s)
4 3
3 2
2 1
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2 Nodes
Node(s) Process Column Route(s)
2 2-1 1
58
3 Nodes
Nodes Process Column Route(s)
3 3-1 2
2 1
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3 Nodes
Nodes Process Column Route(s)
3 3-1 2
2 2-1 1
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4 Nodes
Nodes Process Column Route(s)
4 4-1 3
3 2
2 1
n r1
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4 Nodes
Nodes Process Column Route(s)
4 4-1 3
3 3-1 2
2 1
n r1
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4 Nodes
Nodes Process Column Route(s)
4 4-1 3
3 3-1 2
2 2-1 1
n r1
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4 Nodes
Nodes Process Column Route(s)
4 4-1 3
3 3-1 2
2 2-1 1
n n-1 r1
64
Function Ruler1 equals number of routes from
each remaining nodes

• r1 n - 1

65
4 Nodes
3
2
1
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Total Original Routes

• 4 Nodes

3 Routes
2 Routes
1 Route
1
1
1
2
1
1
1
2
2
1
1
1
3
2
1
67
Total Original Routes

• 3 x 2 x 1 6
• or
• (4-1)!6

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Function Ruler2 equals number of unique routes
69
Petroleum Delivers
S
N
490
370
570
325
T
825
M
565
520
565
745
380
T Texas S South Carolina N New York M Minnesota C Colorado
C