Unit
13B
The Traveling Salesman
Problem
Copyright © 2008 Pearson Education, Inc.
Slide 13-1
13-B
Hamiltonian Circuits
A Hamiltonian circuit is a path that passes through
every vertex of a network exactly once and returns
to the starting vertex. The paths indicated by arrows
in (a) and (b) are Hamiltonian circuits, while (c) has
no Hamiltonian circuits.
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Slide 13-2
13-B
The Traveling Salesman
Problem
Which path is a Hamilton circuit?
A
a) A  B  C  D
b) A  B  C  D  A
D
C
B
c) A  B  C  A  D
d) A  B  C  A
Copyright © 2008 Pearson Education, Inc.
Slide 13-3
13-B
The Traveling Salesman
Problem
Which path is a Hamilton circuit?
A
a) A  B  C  D
b) A  B  C  D  A
D
C
B
c) A  B  C  A  D
d) A  B  C  A
Copyright © 2008 Pearson Education, Inc.
Slide 13-4
13-B
Hamiltonian Circuits in
Complete Networks
The number of Hamiltonian Circuits in a complete
network of order n is
.
The twelve Hamiltonian circuits for a complete
network of order 5.
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Slide 13-5
Calculating the Number of
Hamiltonian Circuits

Network of order 3:

Network of order 4:

Network of order 5:

Network of order 6:

Network of order 7:
Copyright © 2008 Pearson Education, Inc.
Slide 13-6
13-B
The Traveling Salesman
Problem
How many Hamilton circuits are possible in a
complete network of order 8?
a) 7!/2
b) 8!/2
c) 8!
d) 9!/2
Copyright © 2008 Pearson Education, Inc.
Slide 13-7
13-B
The Traveling Salesman
Problem
How many Hamilton circuits are possible in a
complete network of order 8?
a) 7!/2
b) 8!/2
c) 8!
d) 9!/2
Copyright © 2008 Pearson Education, Inc.
Slide 13-8
Solving Traveling Salesman
Problems
13-B
The solution to a traveling salesman problem is the
shortest path (smallest total of the lengths) that
starts and ends in the same place and visits each
city once.
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Slide 13-9
Five National Parks – Planning a
Vacation
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13-B
Slide 13-10
Hamiltonian Circuits and Five
National Parks
13-B
Map of the five national parks and a complete
network representing the parks.
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Slide 13-11
Hamiltonian Circuits and Five
National Parks
The circuit in (a) has a total length of 664 miles,
while (b) has a total length of 499 miles.
Copyright © 2008 Pearson Education, Inc.
Slide 13-12
13-B
The Nearest Neighbor Method
Beginning at any vertex, travel to the nearest
vertex that has not yet been visited. Continue
this process of visiting “nearest neighbors” until
the circuit is complete.
The solution to the national
park network using the
nearest neighbor method
starting at Bryce has a total
length of 515 miles.
Copyright © 2008 Pearson Education, Inc.
Slide 13-13
13-B
The Traveling Salesman
Problem
Starting at vertex A, which vertex would be the
next one visited using the nearest neighbor
algorithm?
a) It doesn’t matter
4
A
B
3
b) B
5
c) C
D
d) D
Copyright © 2008 Pearson Education, Inc.
6
1
4
C
Slide 13-14
13-B
The Traveling Salesman
Problem
Starting at vertex A, which vertex would be the
next one visited using the nearest neighbor
algorithm?
a) It doesn’t matter
4
A
B
3
b) B
5
c) C
D
d) D
6
1
4
C
Continue with the nearest Neighbor Algorithm
Copyright © 2008 Pearson Education, Inc.
Slide 13-15
13-B
The Nearest Neighbor Method
and the Traveling Salesman Problem
The near-optimal solution to finding the shortest
path among 13,509 cities with populations over 500.
Courtesy of Bill Cook, David Applegate and
Robert Bixby, Rice University and
Vasek Chvatal, Rutgers University.
Copyright © 2008 Pearson Education, Inc.
Slide 13-16