Deck 6: The Mathematics of Touring: Traveling Salesman Problems

Question 1

How many vertices are in a complete graph with 45 edges?

Answer 1

10

Question 2

For the weighted graph shown below, find the weight of the Hamilton Circuit A, E, D, B, C, A.

Answer 2

11.8

Question 3

Consider the graph above ; does the graph contain a Hamilton Circuit? Explain.

Answer 3

No. Once you travel from vertex C to vertex E, you are never able to return to vertex

Question 4

For the graph shown below, how many distinct Hamilton Paths start at A and end at F?

Question 5

The chart below shows the one-way charge for a taxi-cab ride between five different cities. Give the cost of a tour found by applying the repetitive nearest-neighbor algorithm.

Question 6

For the weighted graph below, apply the repetitive nearest-neighbor algorithm and write the resulting tour using vertex D as a starting and ending vertex.

Question 7

The chart below shows the one-way charge for a taxi-ride between five different cities. Give the cost of the tour found by applying the cheapest-link algorithm.

Question 8

For the graph shown below, find a Hamilton Circuit using vertex A as your starting and ending point.

Question 9

Consider the graph above ; how many Hamilton Paths are there starting at vertex A and ending at vertex F?

Question 10

For the weighted graph shown below, apply the nearest-neighbor algorithm with C as a starting vertex and find the cost of the resulting tour.

Question 11

How many distinct Hamilton Circuits, if any, are there in the graph shown below? (Mirror images of a circuit are considered distinct or different from the circuit.)

Question 12

How many edges are there in the complete graph K12 ?

Question 13

Draw a graph which has a Hamilton Circuit but does not have an Euler Circuit.

Question 14

For the weighted graph below, apply the nearest-neighbor algorithm with D as the starting point and write the resulting tour.

Question 15

For the weighted graph shown below, apply the cheapest-link algorithm and find the cost of the resulting tour.

Question 16

Consider the graph shown below; does the graph contain a Hamilton Circuit? Explain.

Question 17

For the weighted graph shown below, find the optimal tour and write your answer using A as the starting and ending vertex.

Question 18

Is the graph given below a Complete Graph? Explain.

Question 19

For the weighted graph given below, use the brute-force algorithm to find the optimal tour. Write the tour using vertex A as the starting and ending vertex.

Question 20

Suppose that in solving a TSP (traveling salesman problem) you use the cheapest-link algorithm and find that the length of the tour is 15,000 miles. If the optimal tour has a length of 12,000 miles, then what is the relative error of your cheapest-link tour? Express your answer as a percentage.

Question 21

How many distinct Hamilton circuits, if any, are there in the graph shown below? (Mirror images of a circuit are considered distinct from the circuit.)

A) 2
B) 8
C) 28
D) 7 !
E) None of the above.

Question 22

The game Doodle-Chip consists of moving a marker around a board. Once on a square, the marker can move to any other square which is diagonal (kitty-corner) with the square occupied. Is it possible to place the marker on the board at position A, move to every other square once and only once, and end
on the square labeled B?
(a) Yes
(b) No

Question 23

For the weighted graph below, apply the cheapest-link algorithm and give its cost.

Question 24

Which of the following is true?

A) The brute-force algorithm is an inefficient algorithm.
B) The nearest-neighbor algorithm is an efficient algorithm.
C) The brute-force algorithm always gives the optimal tour.
D) (a), (b), and (c) are all true.
E) None of the above.

Question 25

Which of the graphs below has a Hamilton Circuit?

A) Graph 1
B) Graph 2
C) Graph 3
D) All of the graphs.
E) None of the above.

Question 26

Which algorithm will always give the optimal tour for a weighted complete graph?

A) Brute-force algorithm
B) Nearest-neighbor algorithm
C) Repetitive nearest-neighbor algorithm
D) Cheapest-link algorithm
E) None of the above.

Question 27

How many edges are in the complete graph K15 ?

A) 15
B) 105
C) 210
D) 14!
E) None of the above.

Question 28

Consider the graph shown below; how many Hamilton Paths are there from vertex A to vertex F ?

A) 0
B) 1
C) 2
D) 5 !
E) None of the above.

Question 29

Consider the weighted graph given below; which tour is the optimal tour? - 58 -

A) A, B, C, D, A
B) A, C, D, B, A
C) A, D, B, C, A
D) A, B, D, C, A
E) None of the above.

Question 30

The chart below shows the average travel time in minutes between five different cities. Give the total time (cost) of the tour found by applying the cheapest-link algorithm.

A) 81
B) 85
C) 92
D) 100
E) None of the above.

Question 31

Consider the weighted graph given below; which tour is the optimal tour?

A) A, D, C, B, A
B) A, C, D, B, A
C) A, D, B, C, A
D) A, E, C, D, A
E) None of the above.

Question 32

Consider the weighted graph given below; apply the nearest-neighbor algorithm with vertex F as a starting vertex and give the resulting tour.

A) F, B, C, A, D, E, F
B) F, C, A, B, E, D, F
C) F, E, D, C, B, A, F
D) F, C, A, E, B, D, F
E) None of the above.

Question 33

How many distinct Hamilton Circuits, if any, are there in the graph shown below? (Mirror images of a circuit are considered distinct from the circuit.)

A) 5
B) 25
C) 100
D) 5!
E) None of the above.

Question 34

Consider the weighted graph given below; apply the nearest-neighbor algorithm with vertex C as a starting vertex and give the resulting tour?

A) C, B, D, F, E, A, C
B) C, F, B, A, D, E, C
C) C, E, B, F, D, A, C
D) C, D, E, F, A, B, C
E) None of the above.

Question 35

Consider the graph shown below; how many Hamilton Paths are there from vertex A to vertex F ?

A) 1
B) 2
C) 3
D) 5 !
E) None of the above.

Question 36

The game Doodle-Chip consists of moving a marker around a board. Once on a square, the marker can move to any other square which is adjacent (shares a side) with the square occupied. Is it possible to
Place the marker on the board at position A, move to every other square once and only once, and end
On square labeled B?

A) Yes
B) No

Question 37

How many vertices are in the complete graph with 66 edges?

A) 12
B) 33
C) 66
D) 66!
E) None of the above.

Question 38

Which of the graphs below has a Hamilton Circuit but does not have an Euler Circuit?

A) Graph 1
B) Graph 2
C) Graph 3
D) Both Graph 1 and Graph 2
E) None of the above.

Question 39

For the weighted graph below, apply the repetitive nearest-neighbor algorithm and give its cost.

A) 11.2
B) 11.4
C) 11.6
D) 11.7
E) None of the above.

Question 40

The chart below shows the average travel time in minutes between five different cities in Vermont. Give the total time of a tour found by applying the repetitive nearest-neighbor algorithm.

A) 110
B) 115
C) 120
D) 122
E) None of the above.