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Traveling Salesman Problem

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TSP

The goal is, to find the most economical way for a

select number of cities with the following

restrictions:

- Must visit each city once and only once

- Must return to the original starting point

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BASICS

Complete Graph vertices joined by a single edge

Weighted Graph edges carry a value

Hamiltonian Circuit - connects all points on a graph, passes

through each point only once, returns to origin

Hamiltonian Path - A route not returning to the beginning

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Finding an Approximate Solution

Cheapest Link Algorithm

Edge with smallest weight is drawn first

Edge with second smallest weight is drawn in

Continue unless it closes a smaller circuit or three

edges come out of one vertex

Finished once a complete Hamilton Circuit is drawn

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Cheapest Link Algorithm

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Cheapest Link Algorithm

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Cheapest Link Algorithm

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Cheapest Link Algorithm

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Cheapest Link Algorithm

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Finding an Approximate Solution

Nearest Neighbor Algorithm

Start at any given vertex

Travel to edge that yields smallest weight and has not

been traveled through yet

Continue until we have a complete Hamilton circuit

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Nearest Neighbor Algorithm

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Nearest Neighbor Algorithm

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Nearest Neighbor Algorithm

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Nearest Neighbor Algorithm

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Nearest Neighbor Algorithm

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Comparing Approximating Methods

Cheapest Link Nearest Neighbor

Total weight for

Cheapest Link 31

Nearest Neighbor 33

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17

A 42-City Problem (The Nearest Neighbour Method)

(Starting at City 1)

1

21

20

19

29

7

30

28

3137

3236

11

9

34

10

39

38

35

12

33

8

2726

624

25

14

1523

22

213

16

3

17 4

18

42

40

41

5

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18

The Nearest Neighbour Method (Starting at City 1)

1

21

20

19

29

7

30

28

3137

3236

11

9

34

10

39

38

35

12

33

8

2726

624

25

14

1523

22

213

16

3

17 4

18

42

40

41

5

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19

The Nearest Neighbour Method (Starting at City 1)

Length 1498

1

21

20

19

29

7

30

28

3137

3236

11

9

34

10

39

38

35

12

33

8

2726

624

25

14

1523

22

213

16

3

17 4

18

42

40

41

5

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20

29

Remove Crossovers

1

21

20

19

7

30

28

31 37

32 36

11

9

34

10

39

38

35

12

33

8

2726

624

25

14

15 23

22

213

16

317 4

18

42

40

41

5

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21

Remove Crossovers

1

21

20

19

29

7

30

28

31 37

32 36

11

9

34

10

39

38

35

12

33

8

2726

624

25

14

15 23

22

213

16

317 4

18

42

40

41

5

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22

Remove Crossovers Length 1453

1

21

20

19

29

7

30

28

31 37

32 36

11

9

34

10

39

38

35

12

33

8

2726

624

25

14

15 23

22

213

16

317 4

18

42

40

41

5

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Applications of the TSP

Computer Wiring - connecting together computer

components using minimum

wire length

Archaeological Seriation - ordering sites in time

Genome Sequencing - arranging DNA fragments in

sequence

Planning, logistics, and the manufacture of microchips

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THANK YOU