In this note, we show that the Traveling Salesman Problem cannot be solved in
polynomial-time on a classical computer.Comment: 1 page, made a minor correctio

Feinstein, Craig Alan

English

arXiv

In this note, we show that the Traveling Salesman Problem cannot be solved in polynomial-time on a classical computer

Craig Alan Feinstein

Global Journal of Computer Science and Technology (GJCST)

© 2011I . Craig Alan Feinstein. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non-commercial use, distribution, and reproduction inany medium, provided the original work is properly cited. Global Journal of Computer Science and Technology Volume 11 Issue 23 Version 1.0 December 2011 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 0975-4172 & Print ISSN: 0975-4350  The Computational Complexity of the Traveling Salesman Problem  By Craig Alan Feinstein  Abstract - In this note, we show that the Traveling Salesman Problem cannot be solved in polynomial-time on a classical computer.  The Computational Complexity of the Traveling Salesman Problem   Strictly as per the compliance and regulations of:    GJCST Classification : F.2.2, F.2.3The Computational Complexity of the Traveling Salesman Problem Craig Alan Feinstein                                                                © 2011 Global Journals Inc.  (US)Global Journal of Computer Science and Technology  Volume XI Issue XXIII  Version I            1   2011DecemberAbstract - In this note, we show that the Traveling Salesman Problem cannot be solved in polynomial-time on a classical computer.Traveling Salesman Problem - A traveling salesman starts at city 1, travels to cities 2, . . . , n −1 in any order that the salesman chooses, and then ends his trip in city n. Let us denote (i, j) to be the distance from city to city j. The goal of the Traveling Salesman Problem is to find the minimum total distance possible for the traveling salesman to travel. There are no restrictions on the possible distances (i, j) between each of the cities other than the requirement that each (i, j) is a positive integer and (i, j) =   (j, i) [1, 3, 4].We give a simple proof that no deterministic Problem in o(2n) time:For any nonempty subset S    {2, . . . , n}and for any city i   S, let us define (S, i) to be the length of theshortest path that starts at city 1, visits all cities in the set Salesman Problem is equivalent to the problem ofcomputing ({2, . . . , n}, n). Clearly, ({i}, i) =   (1, i) andwhen  This recursive formula cannot be simplified, so the fastest way to compute ({2, . . . , n}, n) is to apply this recursive formula to ({2, . . . , n}, n). Since this subsets S {2, . . . , n} and each i   S, we obtain a lower Traveling Salesman Problem.This lower bound is confirmed by the fact that the fastest known deterministic and exact algorithm which solves the Traveling Salesman Problem was first time of (2  ) [2, 4].Author : 2712 Willow Glen Drive, Baltimore, Maryland 21209.1. T.H. Cormen, C.E. Leiserson, and R.L. Rivest, Introduction to Algorithms, McGraw-Hill, 1990.2. M. Held and R.M. Karp, “A Dynamic ProgrammingApproach to Sequencing Problems”, Journal of SIAM 10, pp. 196-210, 1962.3. C.H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, NJ, 1982.δδδ δ δ⊆∈ ∆∆ ∆ δ∆(S, i) = min{∆(S − {i}, j) + δ(j, i) | j ∈ S − {i}},|S| ≥ 2.∆∆∆( ∈n∗ΘConsider the following well-known NP-hard problem :4. G.J. Woeginger, “Open problems around exact algorithms”, Discrete Applied Mathematics, 156(3): 397- 405, 2008.E-mail : cafeinst@msn.comiand exact algorithm can solve the Traveling Salesman S − {i}, and finally stops at city i. Then the Traveling involves computing (S, i) for all (2 ) nonempty Θbound of for the worst-case running-time of any deterministic and exact algorithm that solves the published in 1962 and has a running-REFERENCES REFERENCES REFERENCIASn(2 )Θ n

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