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The Mathematics of P vs NP


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Title Page ……………………………………………………………………………………………………………………0
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Table of Contents…………………………………………………………………………………………………………...1
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Authors profile………………………………………………………………………………………………………………2
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Cover letter……………………………………………………………………………………………………………………3
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Abstract and title……………………………………………………………………………………………………………4
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Meaning of symbols and definition………………………………………………………………………………... 5
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Introduction............................................................................................................................................................. 6
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1.1 The complexity class of P and NP ............................................................................................................ 6
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1.2 The concept of NP completeness ............................................................................................................ 7
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2. The Problem........................................................................................................................................................ 7
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2.1 Meaning and definition of P & NP: ......................................................................................................... 8
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2.2 Proposed Result ............................................................................................................................................. 9
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3.Proposed Proof………..……………………………………………………...............................................................9
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3.1 The issue of shortest route-Special case…………………………………......10
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3.2 Theorem………………………………………………………………...……10
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4.The issue of shortest route or the optimal tour-General case………...……11
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4.1The General Domain……………………………………………….………..12
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4.2 The Next network case……………………………………………….…….15
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4.3 The Case of hypothetical diagonals/Virtual Segments…………….……….16
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4.4 The Segment Connection…………………………………………………..17
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5. The last check……………………………………………………………….17
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6. The Proof of the route being the shortest…………………………………18
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6.1 Properties of the shortest route…………………………………….……… 18
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7. Final concerns about the optimal tour.........................................................19
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8.One step check for the optimal tour……………………………………………..20
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9. Few comparisons with standard known heuristics…………….…………20
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10. Algorithm/ heuristics for finding the shortest route…………….………..21
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11. Mathematical Equivalence……………………………………….……….22
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12. Few comparisons with actual solution _ a deeper insight……………...23
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13. Further progress and consequences………………………….…………..24
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Final call for the reader …………………………………………………..24
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