Chapter 6: Test Practice Problems
1. The number of Hamilton circuits in
is
A) 15!
B) 105
C) 14!
D) 15
E) None of these
2. The number of edges in
is
A) 210
B) 15
C) 14!
D) 105
E) None of these
A
Questions 3 to 6 refer to the following situation: A delivery truck
must deliver furniture to 4 different locations (A, B, C, and D). The
trip must start and end at A. The graph below shows the distances
between locations (in miles). We want to minimize the total distance
traveled.
3. The nearest neighbor algorithm applied to the graph from vertex A
yields the following solution:
10
4
5
B
3
D
2
A)
B)
C)
D)
E)
A, B, D, C, A
A, D, B, C, A
A, C, B, D, A.
A, D, C, B, A
None of these
6
C
4. The cheapest link algorithm applied to the graph yields the following solution:
A)
B)
C)
D)
E)
A, B, D, C, A
A, D, B, C, A
A, C, B, D, A.
A, D, C, B, A
None of these
5. The repetitive nearest neighbor algorithm applied to the graph yields the following solution:
A)
B)
C)
D)
E)
A, B, D, C, A
A, D, B, C, A
A, C, B, D, A.
A, D, C, B, A
None of these
6. An optimal solution to this problem is given by
A)
B)
C)
D)
E)
A, B, D, C, A
A, B, C, D, A
A, C, D, B, A.
A, D, C, B, A
None of these
Questions 7 to 9 refer to the following situation: A traveling salesman’s territory consists of the 5 cities
shown on the following mileage chart. The salesman must organize a round trip that starts and ends at
city E (his hometown) and will pass through each of the other four cities exactly once
A
B
C
D
E
7.
A
**
446
963
735
941
E, D, C, B, A, E
E, A, C, B, D, E
E, C, B, A, D, E
E, D, B, A, C, E
None of these
E, D, C, B, A, E
E, A, B, C, D, E
E, C, B, A, D, E
E, D, B, A, C, E
None of these
E, D, C, B,A, E
E, A, B, C, D, E
E, C, B, A, D, E
E, D, B, A, C, E
None of these
Circle the correct words:
The repetitive nearest neighbor algorithm for solving the Traveling Salesman Problem is
( Approximate
11.
Optimal )
and
( Efficient
or
Inefficient )
or
Optimal )
and
( Efficient
or
Inefficient )
or
Inefficient )
Circle the correct words:
The brute force algorithm for solving the Traveling Salesman Problem is
( Approximate
13.
or
Circle the correct words:
The nearest neighbor algorithm for solving the Traveling Salesman Problem is
( Approximate
12.
E
941
532
292
209
**
The repetitive nearest neighbor algorithm applied to this problem yields the following solution
A)
B)
C)
D)
E)
10.
D
735
326
308
**
209
The cheapest link algorithm applied to this problem yields the following solution
A)
B)
C)
D)
E)
9.
C
963
522
**
308
292
The nearest neighbor algorithm applied to this problem from city E yields the following solution:
A)
B)
C)
D)
E)
8.
B
446
**
522
326
532
or
Optimal )
and
( Efficient
Circle the correct words:
The cheapest link algorithm for solving the Traveling Salesman Problem is
( Approximate
or
Optimal )
and
( Efficient
or
Inefficient )
14.
The number of edges in K10 is
A) 10!
B) 90
E) None of these
15.
D) 45
In a complete graph with 14 vertices (A through N), the total number of Hamilton circuits (including
mirror-image circuits) that start at vertex A is
A) 14!
B)13!
E) None of these
16.
C) 10
C)15!
D)91
In a complete graph with 6 vertices, the total number of Hamilton circuits, not including mirror image
circuits is: (do not count the same circuit traveled backwards)
A) 15
B) 120
E) None of these
C) 60
D) 30
Questions 17 through 22 refer to the following situation. A delivery truck must
deliver furniture to 4 different locations (A,B,C, and D). The trip must start and
end at A. The graph in Figure 6.1 shows the distances between locations (in
miles). We want to minimize the total distance traveled.
17
A
11
6
4
The nearest neighbor algorithm applied to the graph from vertex A yields
the following solution
B
3
D
A)
B)
C)
D)
E)
18.
C
A, D, B, C, A
A, D, C, B, A
A, C, B, D, A.
A, B, D, C, A
None of these
A, D, B, C, A
A, D, C, B, A
A, B, C, D, A.
A, B, D, C, A
None of these
How many different Hamilton circuits would we have to check if we use the brute force algorithm?
(Do not count the same circuit traveled backward.)
A) 3
21.
7
The repetitive nearest neighbor algorithm applied to the graph yields the following solution:
A)
B)
C)
D)
E)
20.
5
The cheapest link algorithm applied to the graph yields the following solution
A)
B)
C)
D)
E)
19.
A, D, B, C, A
A, D, C, B, A
A, C, B, D, A.
A, B, D, C, A
None of these
B) 6
C) 24
An optimal solution to this problem is given by
A)
B)
C)
D)
E)
A, D, B, C, A
A, D, C, B, A
A, C, D, B, A.
A, B, D, C, A
None of these
D) 4
E) None of these
22. What is the length of the optimal route?
A)
B)
C)
D)
E)
22 miles
29 miles
23 miles
24 miles
None of these
Questions 23 through 26 refer to the
following situation: A traveling salesman’s
territory consists of the 5 cities shown on
the following mileage chart. The salesman
must organize a round trip that starts and
ends at Minneapolis (his hometown) and will
pass through each of the other four cities
exactly once.
Chicago Des Moines Fargo Minneapolis Indianapolis
Chicago
*
333
643
409
94
Des Moines 333
*
477
244
375
Fargo
643
477
*
235
571
Minneapolis 409
244
235
*
337
Indianapolis 94
375
571
337
*
23. The nearest neighbor algorithm applied
to this problem from Minneapolis yields the following solution
A) Minneapolis, Indianapolis, Chicago, Fargo, Des Moines, Minneapolis
B) Minneapolis, Chicago, Indianapolis, Des Moines, Fargo, Minneapolis
C) Minneapolis, Des Moines, Chicago, Indianapolis, Fargo, Minneapolis.
D) Minneapolis, Indianapolis, Chicago, Des Moines, Fargo, Minneapolis.
E) None of these
24. The cheapest link algorithm applied to this problem yields the following solution
A) Minneapolis, Indianapolis, Chicago, Fargo, Des Moines, Minneapolis
B) Minneapolis, Chicago, Indianapolis, Des Moines, Fargo, Minneapolis
C) Minneapolis, Des Moines, Chicago, Indianapolis, Fargo, Minneapolis.
D) Minneapolis, Indianapolis, Chicago, Des Moines, Fargo, Minneapolis.
E) None of these
25. The repetitive nearest neighbor algorithm applied to this problem yields the following solution
A) Minneapolis, Indianapolis, Chicago, Fargo, Des Moines, Minneapolis
B) Minneapolis, Chicago, Indianapolis, Des Moines, Fargo, Minneapolis
C) Minneapolis, Des Moines, Chicago, Indianapolis, Fargo, Minneapolis.
D) Minneapolis, Indianapolis, Chicago, Des Moines, Fargo, Minneapolis.
E) None of these
26.
At an average cost of 50 cents per mile, the cheapest possible trip (out of those found from #23 – 25
above) that starts at Minneapolis and passes through each of the other cities exactly once would cost
A) $738.00
B) $737.00
C)$738.50
D)$739.00
E) None of these
27.
Given an optimal value of 200 miles, what is the relative error of a Hamilton circuit of 245 miles, rounded
to the nearest whole percent?
A) 23%
B) 18%
E) None of these
28.
C) 82%
D) 123%
Given an optimal value of $63, what is the relative error of a Hamilton circuit with a value of $70,
rounded to the nearest whole percent?
A) 10%
B) 90%
C) 11%
D) 111%
To be successful on the Chapter 6 Test you need to be able to do/know the following:
For any graph:
 Know what a Hamilton Circuit is and find one if it exists

Know what a Hamilton Path is and find one if it exists
For a complete graph KN
 Determine the number of possible Hamilton Circuits (including and not including mirror images)

Determine the number of edges in the graph

Apply the following algorithms successfully and accurately
 Brute Force
 Nearest Neighbor
 Repetitive Nearest Neighbor
 Cheapest Link

Understand that the only way to find an optimal solution is by checking every possibility (Brute Force)

Determine the Relative error of a TSP (Traveling Salesman Problem)

Recognize the difference and know which algorithms are OPTIMAL or APPROXIMATE

Recognize the difference and know which algorithms are EFFICIENT of INEFFICIENT
SOLUTIONS TO REVIEW
1. C
2. D
3. D
4. A
5. A
6. E
7. A
8. D
9. C
10. Approximate/Efficient
11. Approximate/Efficient
12. Optimal/Inefficient
13. Approximate/Efficient
14. D
15. B
16 C
17. D
18. D
19. A
20. A
21. A
22. E
23. D
24. C
25. D
26. A
27. A
28. C