The Traveling Salesman Problem with Few Inner Points

  • Vladimir G. Deineko
  • Michael Hoffmann
  • Yoshio Okamoto
  • Gerhard J. Woeginger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3106)


We study the traveling salesman problem (TSP) in the 2-dimensional Euclidean plane. The problem is NP-hard in general, but trivial if the points are in convex position. In this paper, we investigate the influence of the number of inner points (i.e., points in the interior of the convex hull) on the computational complexity of the problem. We give two simple algorithms for this problem. The first one runs in O(k!kn) time and O(k) space, and the second runs in O(2 k k 2 n) time and O(2 k kn) space, when n is the total number of input points and k is the number of inner points. Hence, if k is taken as a parameter, this problem is fixed-parameter tractable (FPT), and also can be solved in polynomial time if k=O(log n). We also consider variants of the TSP such as the prize-collecting TSP and the partial TSP in this setting, and show that they are FPT as well.


Polynomial Time Convex Hull Short Tour Linear Order Travel Salesman Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Arora, S.: Polynomial-time approximation schemes for Euclidean TSP and other geometric problems. J. ACM 45, 753–782 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Balas, E.: The prize collecting traveling salesman problem. Networks 19, 621–636 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bellman, R.: Dynamic programming treatment of the traveling salesman problem. J. ACM 9, 61–63 (1962)zbMATHCrossRefGoogle Scholar
  4. 4.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to algorithms, 2nd edn. MIT Press, Cambridge (2001)zbMATHGoogle Scholar
  5. 5.
    Deĭneko, V., van Dal, R., Rote, G.: The convex-hull-and-line traveling salesman problem: a solvable case. Inf. Process. Lett. 59, 295–301 (1996)Google Scholar
  6. 6.
    Deĭneko, V., Woeginger, G.J.: The convex-hull-and-k-line traveling salesman problem. Inf. Process. Lett. 59, 295–301(1996) Google Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)Google Scholar
  8. 8.
    Flood, M.M.: Traveling salesman problem. Oper. Res. 4, 61–75 (1956)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Garey, M.R., Graham, R.L., Johnson, D.S.: Some NP-complete geometric problems. In: Proc. 8th STOC, pp. 10–22 (1976)Google Scholar
  10. 10.
    Held, M., Karp, R.: A dynamic programming approach to sequencing problems. J. SIAM 10, 196–210 (1962)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Hoffmann, M., Okamoto, Y.: The minimum weight triangulation problem with few inner points (submitted)Google Scholar
  12. 12.
    Hwang, R.Z., Chang, R.C., Lee, R.C.T.: The searching over separators strategy to solve some NP-hard problems in subexponential time. Algorithmica 9, 398–423 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mitchell, J.: Guillotine subdivisions approximate polygonal subdivisions: a simple polynomial- time approximation scheme for geometric k-MST, TSP, and related problems. SIAM J. Comput. 28, 1298–1309 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Niedermeier, R.: Invitation to fixed-parameter algorithms. Habilitation Thesis, Universität Tübingen (2002)Google Scholar
  15. 15.
    Papadimitriou, C.H.: Euclidean TSP is NP-complete. Theor. Comput. Sci. 4, 237–244 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Rao, S., Smith, W.: Approximating geometric graphs via spanners and banyans. In: Proc. 30th STOC, pp. 540–550 (1998)Google Scholar
  17. 17.
    Sedgewick, R.: Permutation generation methods. ACM Comput. Surveys 9, 137–164 (1977)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Vladimir G. Deineko
    • 1
  • Michael Hoffmann
    • 2
  • Yoshio Okamoto
    • 2
  • Gerhard J. Woeginger
    • 3
  1. 1.Warwick Business SchoolThe University of WarwickConventryUnited Kingdom
  2. 2.Institute of Theoretical Computer ScienceETH ZurichZurichSwitzerland
  3. 3.Department of Mathematics and Computer ScienceTU EindhovenEindhovenThe Netherlands

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