1. Traveling-Salesman Problems
Ch 6

2. More examples of Complete GraphsB C D E A B C D E F
Graph1
Graph 2
No. of edges with n=5 vertices
= n(n-1)/2
= 5(5-1)/2
= 10
No. of edges with n=6 vertices
= n(n-1)/2
= 6(6-1)/2
= 15
n represents the number of vertices in a complete graph

3. Not a Complete graph No edge between the pair of edges A, C
No edge between the pair of edges B, D

4. Not a Complete graph No edge between the pair of vertices A, D
No edge between the pair of vertices B, C
No edge between the pair of vertices A, C
No edge between the pair of vertices B, D

5. Not a Complete graph We cannot apply the formula forB
We cannot apply the formula for
number of edges for a graph
which is not a complete graph D C E
No edge between the pair of vertices A, E
No edge between the pair of vertices B, E

6. Modifying the previous graph: Complete graphB
We can apply the formula for
number of edges for a graph
which is a complete graph.
# of edges = n(n-1)
= 5x4/2
= 10 D C E
Add an edge between the pair of vertices A, E
Add an edge between the pair of vertices B, E
n represents the number of vertices in a complete graph

7. No of edges of a complete graph
No. of edges of the complete
graph
= n(n-1)/2
= 3(3-1)/2
= 3 B C
n represents the number of vertices in a complete graph

8. No. of Hamilton circuits of a complete graph
No. of Hamilton circuit of the
Complete graph
= (n-1)!
= (4-1)!
= 3!
= 1x2x3
= 6 A B C D
n represents the number of vertices in a complete graph

9. No. of Hamilton circuits of a complete graphB C D E
No. of Hamilton circuit of the
Complete graph
= (n-1)!
= (5-1)!
= 4!
= 1x2x3x4
= 24
n represents the number of vertices in a complete graph

10. No. of Hamilton circuits of a complete graph
No. of Hamilton circuit of the
Complete graph
= (n-1)!
= (6-1)!
= 5!
= 1x2x3x4x5
= 120 A B C D E F
n represents the number of vertices in a complete graph

11. Simple Strategies for solving TSPs
Method 1:
Make a list of all possible Hamilton circuits
Calculate the total cost for each circuit.
Select a circuit with least total cost for the answer.

12. Simple Strategies for solving TSPs
$185
$133
$200
$119
$152
$121
$120
$150
$199
$174

13. Simple Strategies for solving TSPs
Method 2:
Start at home (A)
From there go to the city to which the cost of travel is the cheapest.
Then from there go to the next city to which the cost of travel is the cheapest, and so on.
From the last city, return to A.

14. Simple Strategies for solving TSPs
The optimal circuit:
A,E,C,B,D,A => Cost $676 (Method1)
A,C,E, D,B, A => cost $773 (Method 2)