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A Polynomial Time Algorithm of a Traveling Salesman Problem

외판원 문제의 다항시간 알고리즘

  • Lee, Sang-Un (Dept. of Multimedia Eng., Gangneung-Wonju National University)
  • 이상운 (강릉원주대학교 멀티미디어공학과)
  • Received : 2013.09.03
  • Accepted : 2013.10.05
  • Published : 2013.12.31

Abstract

This paper proposes a $O(n^2)$ polynomial time algorithm to obtain optimal solution for Traveling Salesman problem that is a NP-complete because polynomial time algorithm has been not known yet. The biggest problem in a large-scale Traveling Salesman problem is the fact that the amount of data to be processed is $n{\times}n$, and thus as n increases, the data increases by multifold. Therefore, this paper proposes a methodology by which the data amount is first reduced to approximately n/2. Then, it seeks a bi-directional route at a random point. The proposed algorithm has proved to be successful in obtaining the optimal solutions with $O(n^2)$ time complexity when applied to TSP-1 with 26 European cities and TSP-2 with 46 cities of the USA. It could therefore be applied as a generalized algorithm for TSP.

본 논문은 NP-완전으로 다항시간 알고리즘이 존재하지 않는 대규모 외판원 문제의 최적 해를 $O(n^2)$의 다항시간으로 구하는 알고리즘을 제안하였다. 대규모 외판원 문제에서 가장 큰 문제는 처리될 데이터가 $n{\times}n$으로 n이 커질수록 기하급수적으로 증가한다. 본 논문에서는 먼저, 데이터의 양을 약 n/2의 크기로 축소시킨다. 다음으로 임의의 정점에서 시작하여 양방향으로 경로를 탐색하는 방법을 적용하였다. 제안된 알고리즘을 26개의 유럽 도시들을 방문하는 TSP-1과 46개 미국 도시들을 방문하는 TSP-2에 적용한 결과 모두 최적 해를 $O(n^2)$ 수행 복잡도로 빠르게 구하는데 성공하였다. 따라서 제안된 알고리즘은 TSP의 일반화된 알고리즘으로 적용할 수 있을 것이다.

Keywords

References

  1. D. S. Johnson and L. A. McGeoch, "The Traveling Salesman Problem and Its Variations," Kluwer Academic Publishers, pp. 369-443, 2002.
  2. A. Likas and V. T. Paschos, "A Note on a New Greedy-solution Representation and a New Greedy Parallelizable Heuristic for the Traveling Salesman Problem," Chaos, Solitons and Fractals, Vol. 13, pp. 71-78, 2002. https://doi.org/10.1016/S0960-0779(00)00227-7
  3. A. Schrijver, "On the History of Combinatorial Optimization (till 1960)," in Handbook of Discrete Optimization" (K. Aardal, G.L. Nemhauser, R. Weismantel, eds.), Elsevier, Amsterdam, pp. 1-68, 2005.
  4. E. Charniak and M. Herlihy, "CSC 751 Computational Complexity: Local Search Heuristics," Department of Computer Science, Brown University, 2008.
  5. J. Pleines, "ZIP-Methode: ein Kombinatorischer Ansatz zur Optimalen Losung Allgemeiner Traveling-Salesman-Problem (TSP)," Konnen bekannte Losungen nicht nur auf Gesamtgrphen sondern auf Teilgraphen angewandt werden, so bringt die ZIP-Methode den entscheidenden Quantensprung der rechentechnischen Vereinfachung, http://www. jochen-pleines.de/download/ZIP2006.pdf, 2006.
  6. L. Stougie, "2P350: Optimaliseringsmethoden," http://www.win.tue.nl/-leen/OW/2P350/Week8/week8.pdf, College Wordt ggeven op vinjdagmiddag, 2001.
  7. L. H. Chuin, "IS 703: Decision Support and Optimization," School of Information Systems," Department of Computer Science, Brown University, 2008.
  8. S. U. Lee, "The Extended k-opt Algorithm for Traveling Salesman Problem," Journal of Korea Society of Computer Information, Vol. 17, No. 10, pp. 155-165, Oct. 2012. https://doi.org/10.9708/jksci/2012.17.10.155
  9. G. Dantzig, R. Fulkerson, and S. Johnson, "Solution of a Large-scale Traveling-Salesman Problem," The Rand Corporation, http://www.cse.wustl.edu/-chen/7102/TSP.pdf, 1954.

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  2. 순회 판매원 문제 해결을 위한 개미집단 최적화 알고리즘 개선 vol.42, pp.3, 2013, https://doi.org/10.11627/jkise.2019.42.3.001