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31 Contents
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34 Title Page ……………………………………………………………………………………………………………………0
35 Table of Contents…………………………………………………………………………………………………………...1
36 Authors profile………………………………………………………………………………………………………………2
37 Cover letter……………………………………………………………………………………………………………………3
38 Abstract and title……………………………………………………………………………………………………………4
39 Meaning of symbols and definition………………………………………………………………………………... 5
40 Introduction............................................................................................................................................................. 6
41 1.1 The complexity class of P and NP ............................................................................................................ 6
42 1.2 The concept of NP completeness ............................................................................................................ 7
43 2. The Problem........................................................................................................................................................ 7
44 2.1 Meaning and definition of P & NP: ......................................................................................................... 8
45 2.2 Proposed Result ............................................................................................................................................. 9
46 3.Proposed Proof………..……………………………………………………...............................................................9
47 3.1 The issue of shortest route-Special case…………………………………......10
48 3.2 Theorem………………………………………………………………...……10
49 4.The issue of shortest route or the optimal tour-General case………...……11
50 4.1The General Domain……………………………………………….………..12
51 4.2 The Next network case……………………………………………….…….15
52 4.3 The Case of hypothetical diagonals/Virtual Segments…………….……….16
53 4.4 The Segment Connection…………………………………………………..17
54 5. The last check……………………………………………………………….17
55 6. The Proof of the route being the shortest…………………………………18
56 6.1 Properties of the shortest route…………………………………….……… 18
57 7. Final concerns about the optimal tour.........................................................19
58 8.One step check for the optimal tour……………………………………………..20
59 9. Few comparisons with standard known heuristics…………….…………20
60 10. Algorithm/ heuristics for finding the shortest route…………….………..21
61 11. Mathematical Equivalence……………………………………….……….22
62 12. Few comparisons with actual solution _ a deeper insight……………...23
63 13. Further progress and consequences………………………….…………..24
64 Final call for the reader …………………………………………………..24
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