1. traveling salesman problem ZIP-Method deductive approach of an optimal solution of symmetrical Traveling-salesman-problems http://www.jochen-pleines.de

2. new ideas... –das Rundreiseproblem –bisherige Lösungen –New ideas –Beispiel mit 6 Knoten –Beispiel mit 10 Knoten –Beispiel mit 26 Knoten (Ergebnisse) –Schlussfolgerungen und Ausblick

3. new ideas... but first a task for you: Please note in arbitrary order the numbers from 1 to 6. 5 2 1 4 3 6

4. new ideas... If you connect the nodes, then it develops: 1-component G with 6 edges with 6 nodes each node has the grade 2 5 5 2 1 4 3 6

5. new ideas... we remember: if n = 6then: n! = 720 (n-1)! = 120 (n-1)! / 2= 60 5 5 2 1 4 3 6

6. new ideas... now we add the values of the nodes: as well each node accurs twice: 5 5 2 1 4 3 6 –at the beginning of an edge (1-6) –and at the end of an edge (4-1). and that is evil ! 1

7. 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, 2-4 5 5 2 1 4 3 6

8. new ideas...... a complement partial- graph with the same structure remains like the edges: 1-4, 2-3, 5-6 5 5 2 1 4 3 6

9. 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 :1-4,2-3,5-6 5 5 2 1 4 3 6

10. 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 - 5 10. partial-graph 1 - 5 2 - 3 4 - 6 11. partial-graph 1 - 5 2 - 4 3 - 6 12. partial-graph 1 - 5 2 - 6 3 - 4 13. partial-graph 1 - 6 2 - 3 4 - 5 14. partial-graph 1 - 6 2 - 4 3 - 5 15. partial-graph 1 - 6 2 - 5 3 - 4 not more !

11. 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. 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 - 5 10. partial-graph 1 - 5 2 - 3 4 - 6 11. partial-graph 1 - 5 2 - 4 3 - 6 12. partial-graph 1 - 5 2 - 6 3 - 4 13. partial-graph 1 - 6 2 - 3 4 - 5 14. partial-graph 1 - 6 2 - 4 3 - 5 15. partial-graph 1 - 6 2 - 5 3 - 4 Please try it with your own example !

12. new ideas... 5 5 2 1 4 3 6 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)

13. new ideas... 5 5 2 1 4 3 6 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)

14. 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, 5-6 No.5, 6, 8, 9, 10, 11, 13, 14 altogether 8 (= 2 x 4) 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 - 5 10. partial-graph 1 - 5 2 - 3 4 - 6 11. partial-graph 1 - 5 2 - 4 3 - 6 12. partial-graph 1 - 5 2 - 6 3 - 4 13. partial-graph 1 - 6 2 - 3 4 - 5 14. partial-graph 1 - 6 2 - 4 3 - 5 15. partial-graph 1 - 6 2 - 5 3 - 4

15. 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.

16. 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 - 5 10. partial-graph 1 - 5 2 - 3 4 - 6 11. partial-graph 1 - 5 2 - 4 3 - 6 12. partial-graph 1 - 5 2 - 6 3 - 4 13. partial-graph 1 - 6 2 - 3 4 - 5 14. partial-graph 1 - 6 2 - 4 3 - 5 15. partial-graph 1 - 6 2 - 5 3 - 4

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

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

19. 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

20. example with 6 nodes –das Rundreiseproblem –bisherige Lösungen –neue Überlegungen –example with 6 nodes –Beispiel mit 10 Knoten –Beispiel mit 26 Knoten (Ergebnisse) –Schlussfolgerungen und Ausblick

21. example with 6 nodes Given 6 nodes with the values of the edges: to 1 to 2 to 3 to 4 to 5 to 6 from 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 -

22. example with 6 nodes 60 symmetric graphs 60 partial-graphs minimal graph to divide to sort to compute 15 partial-graphs

23. example with 6 nodes Nr.1.K. 2.K. 3.K. K.-Länge 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 66 10. 1 - 5 2 - 3 4 - 6 62 11. 1 - 5 2 - 4 3 - 6 69 12. 1 - 5 2 - 6 3 - 4 61 13. 1 - 6 2 - 3 4 - 5 69 14. 1 - 6 2 - 4 3 - 5 68 15. 1 - 6 2 - 5 3 - 4 67 Partial-graph Nr.1.K. 2.K. 3.K. K.-Länge 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 66 10. 1 - 5 2 - 3 4 - 6 62 11. 1 - 5 2 - 4 3 - 6 69 12. 1 - 5 2 - 6 3 - 4 61 13. 1 - 6 2 - 3 4 - 5 69 14. 1 - 6 2 - 4 3 - 5 68 15. 1 - 6 2 - 5 3 - 4 67 compl.-partial-graph Length of the graph: 111 (49 + 62) : 2 = 55,5

24. example with 10 nodes –das Rundreiseproblem - Fragestellung –Problem und bisherige Lösungen –neue Überlegungen –Beispiel mit 6 Knoten –Example with 10 nodes –Beispiel mit 26 Knoten (Ergebnisse) –Schlussfolgerungen und Ausblick

25. 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! · 2 4 (10 – 1)! = (5 – 1)! · 2 (5-1)

26. example with 10 nodes (n – 1)! ( n/2 – 1 ) ! · 2 n/2 - 1 by 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 = x 1

27. example with 10 nodes

28. cur.No.length1. edge2.edge3.edge4.edge5.edgeNotice 1761 - 23 - 74 - 65 - 89 -10 2771 - 52 - 93 - 84 - 67 -10 3791 - 52 -103 - 74 - 68 - 9 4831 - 52 - 73 - 84 - 69 -10 5 921 - 23 - 54 - 67 -108 - 9 6931 - 23 - 94 - 65 - 87 -10 7961 - 23 - 64 - 95 - 87 -10 8961 - 82 - 53 - 74 - 69 -10 9971 - 52 - 34 - 67 -108 - 9 101001 - 23 - 64 - 57 -108 - 9 111001 - 52 - 73 - 64 -108 - 9 min.p-g komp 121011 - 82 - 93 - 54 - 67 -10 131031 - 32 - 94 - 65 - 87 -10.... until 945 min.p-g min. p-g + min.p-g komp = 176; 176 / 2 = 88 Sorting of the 945 partial-graphs after edge-length example with 10 nodes

29. cur.No.length1. edge2.edge3.edge4.edge5.edgeNotice 1761 - 23 - 74 - 65 - 89 -10 2771 - 52 - 93 - 84 - 67 -10 3791 - 52 -103 - 74 - 68 - 9 4831 - 52 - 73 - 84 - 69 -10 5 921 - 23 - 54 - 67 -108 - 9 6931 - 23 - 94 - 65 - 87 -10 7961 - 23 - 64 - 95 - 87 -10 8961 - 82 - 53 - 74 - 69 -10 9971 - 52 - 34 - 67 -108 - 9 101001 - 23 - 64 - 57 -108 - 9 111001 - 52 - 73 - 64 -108 - 9 121011 - 82 - 93 - 54 - 67 -10 131031 - 32 - 94 - 65 - 87 -10.... until 945 example with 10 nodes

30. cur.No.length1. edge2.edge3.edge4.edge5.edgeNotice 1761 - 23 - 74 - 65 - 89 -10 2771 - 52 - 93 - 84 - 67 -10 3791 - 52 -103 - 74 - 68 - 9 4831 - 52 - 73 - 84 - 69 -10 5 921 - 23 - 54 - 67 -108 - 9 6931 - 23 - 94 - 65 - 87 -10 7961 - 23 - 64 - 95 - 87 -10 8961 - 82 - 53 - 74 - 69 -10 9971 - 52 - 34 - 67 -108 - 9 101001 - 23 - 64 - 57 -108 - 9 111001 - 52 - 73 - 64 -108 - 9 121011 - 82 - 93 - 54 - 67 -10 131031 - 32 - 94 - 65 - 87 -10.... until 945 example with 10 nodes

31. cur.No.length1. edge2.edge3.edge4.edge5.edgeNotice 1761 - 23 - 74 - 65 - 89 -10 2771 - 52 - 93 - 84 - 67 -10 3791 - 52 -103 - 74 - 68 - 9 4831 - 52 - 73 - 84 - 69 -10 5 921 - 23 - 54 - 67 -108 - 9 6931 - 23 - 94 - 65 - 87 -10 7961 - 23 - 64 - 95 - 87 -10 8961 - 82 - 53 - 74 - 69 -10 9971 - 52 - 34 - 67 -108 - 9 101001 - 23 - 64 - 57 -108 - 9 111001 - 52 - 73 - 64 -108 - 9 min.p-g komp 121011 - 82 - 93 - 54 - 67 -10 131031 - 32 - 94 - 65 - 87 -10.... until 945 min. p-g min. p-g + min.p-g komp = 175; 175 / 2 = 87,5 example with 10 nodes

32. cur.No.length1. edge2.edge3.edge4.edge5.edgeNotice 1761 - 23 - 74 - 65 - 89 -10 2771 - 52 - 93 - 84 - 67 -10 3791 - 52 -103 - 74 - 68 - 9 4831 - 52 - 73 - 84 - 69 -10 5 921 - 23 - 54 - 6 7 -10 8 - 9 6931 - 23 - 94 - 65 - 87 -10 7961 - 23 - 64 - 95 - 87 -10 8961 - 82 - 53 - 74 - 69 -10 9971 - 52 - 34 - 67 -108 - 9 101001 - 23 - 64 - 57 -108 - 9 111001 - 52 - 73 - 64 -108 - 9 121011 - 82 - 93 - 54 - 67 -10 131031 - 32 - 94 - 65 - 87 -10.... until 945 opt. p-g opt. p-g komp example with 10 nodes

33. Quantity of partial-graphs Sum of the values of edges after the 1. pass through: only 11 of 945 partial-graphs of altogether 181.440 graphs

34. example with 10 nodes From altogether 945 partial-graphs will eliminate with back tracking of the limited Enumeration: Stop after the 5th edge: 0 edge Stop after the 4th edge: 1 edge Stop after the 3th edge: 3 edges Stop after the 2th edge: 15 edges Stop after the first edge:105 edges earliest stop if possible!!

35. example with 10 nodes Relation between begin-node and place of the edge: 1. Edge 2. Edge 3. Edge 4. Edge 5. Edge 1-2...1-102-3...2-103-4...3-104-5...4-105-6...5-10 orororor 3-4...3-10 4-5...4-10 5-6...5-10 6-7...6-10 ororor 5-6...5-10 6-7...6-10 7-8...7-10 oror 7-8...7-10 8-9...8-10 or 9-10.

36. example with 10 nodes

37. example with 6 nodes Some more ideas: Numbering-rule (the greatest difference first) Minimal-edge-rule (calculation of the smallest edge still outstanding for every single edge-place; no more for everyone – see page before: relation between begin-node and place of the edge)

38. example with 6 nodes Example: node x i with his 5 edges Numbering-ruleMinimal-edge-rule

39. traveling salesman problem Thank you for your interest