# traveling salesman problem

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traveling salesman problem. ZIP-Method deductive approach of an optimal solution of symmetrical Traveling-salesman-problems. http://www.jochen-pleines.de. new ideas. das Rundreiseproblem bisherige Lösungen New ideas Beispiel mit 6 Knoten Beispiel mit 10 Knoten - PowerPoint PPT Presentation

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• traveling salesman problemZIP-Methoddeductive approach of an optimal solution of symmetrical Traveling-salesman-problemshttp://www.jochen-pleines.de

• new ideas ...

das Rundreiseproblembisherige LsungenNew ideasBeispiel mit 6 KnotenBeispiel mit 10 KnotenBeispiel mit 26 Knoten (Ergebnisse)Schlussfolgerungen und Ausblick

• new ideas ...but first a task for you: Please note in arbitrary order the numbers from 1 to 6.521436

• new ideas ...If you connect the nodes, then it develops:

1-component Gwith 6 edges with 6 nodes each node has the grade 25521436

• new ideas ... we remember: if n = 6then:n! = 720(n-1)! = 120(n-1)! / 2= 605521436

• new ideas ... now we add the values of the nodes:

as well each node accurs twice:5521436at the beginning of an edge (1-6)and at the end of an edge (4-1). and that is evil !1

• new ideas ...If each node accurs only once, then originate from a graph: a partial-graph with all of the 6 nodes, but only with 3 edges:for example: edges: 1-6, 5-3, 2-45521436

• new ideas ...... a complement partial- graph with the same structure remains like theedges: 1-4, 2-3, 5-65521436

• new ideas ...So the graph gets together: from the partial-graph with edges:1-6,5-3,2-4 and the partial-graph with edges :1-4,2-3,5-65521436

• new ideas ...we remember: With n = 6 nodes there exist 120 graphs respectively 60 symmetric graphs How many partial-graphs actual exist ? 1. partial-graph 1 - 2 3 - 4 5 - 6 2. partial-graph 1 - 2 3 - 5 4 - 6 3. partial-graph 1 - 2 3 - 6 4 - 5 4. partial-graph 1 - 3 2 - 4 5 - 6 5. partial-graph 1 - 3 2 - 5 4 - 6 6. partial-graph 1 - 3 2 - 6 4 - 5 7. partial-graph 1 - 4 2 - 3 5 - 6 8. partial-graph 1 - 4 2 - 5 3 - 6 9. partial-graph 1 - 4 2 - 6 3 - 510. partial-graph 1 - 5 2 - 3 4 - 611. partial-graph 1 - 5 2 - 4 3 - 612. partial-graph 1 - 5 2 - 6 3 - 413. partial-graph 1 - 6 2 - 3 4 - 514. partial-graph 1 - 6 2 - 4 3 - 515. partial-graph 1 - 6 2 - 5 3 - 4 not more !

• new ideas ...By using the the Symmetry-rule and the Sort-rule each of the 120 Graphs can be devorced into 2 of the 15 partial-graphs.Please try it with your own example !

• new ideas ...Symmetry-Rule:The begin-node of an edge has the lower number then the end-node: i < j:-> f(1-6) + f(5-3) + f(2-4) = f(1-6) + f(3-5) + f(2-4) 5521436

• new ideas ...Sort-Rule:die edges will be sorted by the begin-node of the edges. 1. edge 2. edge 3. edge -> f(1-6) + f(3-5) + f(2-4) = f(1-6) + f(2-4) + f(3-5) 5521436

• new ideas ...How many partial graphs matching to a partial graph, i.o. do all of the graphs form one together again ? Example: 1-2, 3-4, 5-6No.5, 6, 8, 9, 10, 11, 13, 14altogether 8 (= 2 x 4)

• new ideas ...Further considerations:

The smallest (complete-)graph gets together: either: from the two found partial-graphs (smallest partial-graph with accompanying smallest comp-partial-graph) or: from two partial-graphs lying between this.

• new ideas ...How to find out the lowest graph ? 1. step: on find out the lowest partial - graph 2. step: on find out the belonging lowest Comp-partial-graph. 1. partial-graph 1 - 2 3 - 4 5 - 6 2. partial-graph 1 - 2 3 - 5 4 - 6 3. partial-graph 1 - 2 3 - 6 4 - 5 4. partial-graph 1 - 3 2 - 4 5 - 6 5. partial-graph 1 - 3 2 - 5 4 - 6 6. partial-graph 1 - 3 2 - 6 4 - 5 7. partial-graph 1 - 4 2 - 3 5 - 6 8. partial-graph 1 - 4 2 - 5 3 - 6 9. partial-graph 1 - 4 2 - 6 3 - 510. partial-graph 1 - 5 2 - 3 4 - 611. partial-graph 1 - 5 2 - 4 3 - 612. partial-graph 1 - 5 2 - 6 3 - 413. partial-graph 1 - 6 2 - 3 4 - 514. partial-graph 1 - 6 2 - 4 3 - 515. partial-graph 1 - 6 2 - 5 3 - 4

• new ideas ...There perhaps the lowest graph is found ! But only: perhaps!

• new ideas ...example:lowest partial-graph has a length of edge:20der lowest belonging Compl.-partial-graph:40result a graph: 60that means, (a+b) < c -> a and/or b < (c/2) and a < b or a = b -> a < (c/2)

• new ideas ...Further iteration steps: (up to half the edge length of the so far smallest found complet graph) starting out from the smallest partial-graph it this one is respectively greater partial-graph with his complement partial-graph checked most nearly, wether from this a smaler complete-graph can be put together.if yes: the new complete-graph is initial value for further iteration steps.if no: the smallest graph is already found

• example with 6 nodes

das Rundreiseproblembisherige Lsungenneue berlegungenexample with 6 nodesBeispiel mit 10 KnotenBeispiel mit 26 Knoten (Ergebnisse)Schlussfolgerungen und Ausblick

• example with 6 nodesGiven 6 nodes with the values of the edges:

to 1 to 2 to 3 to 4 to 5 to 6from 1 - 12 25 30 28 22

from 2 - 16 20 22 10

from 3 - 23 26 21

from 4 - 31 18

from 5 - 14

from 6 -

• example with 6 nodes60 symmetric graphsminimal graph

• example with 6 nodesNr.1.K. 2.K. 3.K. K.-Lnge 1.1 - 2 3 - 4 5 - 649 2.1 - 2 3 - 5 4 - 6 56 3.1 - 2 3 - 6 4 - 5 64 4.1 - 3 2 - 4 5 - 659 5.1 - 3 2 - 5 4 - 6 65 6. 1 - 3 2 - 6 4 - 5 66 7. 1 - 4 2 - 3 5 - 6 60 8. 1 - 4 2 - 5 3 - 6 73 9. 1 - 4 2 - 6 3 - 5 6610. 1 - 5 2 - 3 4 - 6 6211. 1 - 5 2 - 4 3 - 6 6912. 1 - 5 2 - 6 3 - 4 6113. 1 - 6 2 - 3 4 - 5 6914. 1 - 6 2 - 4 3 - 5 6815. 1 - 6 2 - 5 3 - 4 67 Partial-graphLength of the graph: 111 (49 + 62) : 2 = 55,5

• example with 10 nodes

das Rundreiseproblem - FragestellungProblem und bisherige Lsungenneue berlegungenBeispiel mit 6 KnotenExample with 10 nodesBeispiel mit 26 Knoten (Ergebnisse)Schlussfolgerungen und Ausblick

• example with 10 nodes development of the ZIP-term with n = 10: 1 3 5 7 9 = 9! = (1 2) (2 2) (3 2) (4 2) 9! = ( 1 2 3 4 ) ( 2 2 2 2 )

1 2 3 4 5 6 7 8 9 = 2 4 6 8

9! = 4! 24

(10 1)! = (5 1)! 2(5-1)

• example with 10 nodes (n 1)! ( n/2 1 ) ! 2 n/2 - 1by that: { ( n/2 1 ) ! } (Sort-rule) Partial-graph: (number of edges =n/2) 1 x 2 x 2 x 2 x 2 (Symmetry)

Begin-edge = x1

• example with 10 nodes

nach von

1

2

3

4

5

6

7

8

9

10

1

-

16

44

93

1

30

30

5

78

42

2

-

68

61

42

77

41

79

22

32

3

-

39

48

21

36

28

40

80

4

-

43

8

66

46

30

35

5

-

67

69

11

84

91

6

-

97

43

63

67

7

-

85

89

18

8

-

2

85

9

-

5

• example with 10 nodescur.No.length1. edge2.edge3.edge4.edge5.edgeNotice1761 - 23 - 74 - 65 - 89 -102771 - 52 - 93 - 84 - 67 -103791 - 52 -103 - 74 - 68 - 94831 - 52 - 73 - 84 - 69 -105921 - 23 - 54 - 67 -108 - 96931 - 23 - 94 - 65 - 87 -107961 - 23 - 64 - 95 - 87 -108961 - 82 - 53 - 74 - 69 -109971 - 52 - 34 - 67 -108 - 9101001 - 23 - 64 - 57 -108 - 9111001 - 52 - 73 - 64 -108 - 9121011 - 82 - 93 - 54 - 67 -10131031 - 32 - 94 - 65 - 87 -10....until 945min. p-g + min.p-gkomp = 176; 176 / 2 = 88

• example with 10 nodescur.No.length1. edge2.edge3.edge4.edge5.edgeNotice1761 - 23 - 74 - 65 - 89 -102771 - 52 - 93 - 84 - 67 -103791 - 52 -103 - 74 - 68 - 94831 - 52 - 73 - 84 - 69 -105921 - 23 - 54 - 67 -108 - 96931 - 23 - 94 - 65 - 87 -107961 - 23 - 64 - 95 - 87 -108961 - 82 - 53 - 74 - 69 -109971 - 52 - 34 - 67 -108 - 9101001 - 23 - 64 - 57 -108 - 9111001 - 52 - 73 - 64 -108 - 9121011 - 82 - 93 - 54 - 67 -10131031 - 32 - 94 - 65 - 87 -10....until 945

• example with 10 nodescur.No.length1. edge2.edge3.edge4.edge5.edgeNotice1761 - 23 - 74 - 65 - 89 -102771 - 52 - 93 - 84 - 67 -103791 - 52 -103 - 74 - 68 - 94831 - 52 - 73 - 84 - 69 -105921 - 23 - 54 - 67 -108 - 96931 - 23 - 94 - 65 - 87 -107961 - 23 - 64 - 95 - 87 -108961 - 82 - 53 - 74 - 69 -109971 - 52 - 34 - 67 -108 - 9101001 - 23 - 64 - 57 -108 - 9111001 - 52 - 73 - 64 -108 - 9121011 - 82 - 93 - 54 - 67 -10131031 - 32 - 94 - 65 - 87 -10....until 945

• example with 10 nodescur.No.length1. edge2.edge3.edge4.edge5.edgeNotice1761 - 23 - 74 - 65 - 89 -102771 - 52 - 93 - 84 - 67 -103791 - 52 -103 - 74 - 68 - 94831 - 52 - 73 - 84 - 69 -105921 - 23 - 54 - 67 -108 - 96931 - 23 - 94 - 65 - 87 -107961 - 23 - 64 - 95 - 87 -108961 - 82 - 53 - 74 - 69 -109971 - 52 - 34 - 67 -108 - 9101001 - 23 - 64 - 57 -108 - 9111001 - 52 - 73 - 64 -108 - 9121011 - 82 - 93 - 54 - 67 -10131031 - 32 - 94 - 65 - 87 -10....until 945min. p-gmin. p-g + min.p-gkomp = 175; 175 / 2 = 87,5

• example with 10 nodescur.No.length1. edge2.edge3.edge4.edge5.edgeNotice1761 - 23 - 74 - 65 - 89 -102771 - 52 - 93 - 84 - 67 -103791 - 52

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