The traveling salesman problem on grids with forbidden neighborhoods

Abstract

We introduce the traveling salesman problem with forbidden neighborhoods (TSPFN). This is an extension of the Euclidean TSP in the plane where direct connections between points that are too close are forbidden. The TSPFN is motivated by an application in laser beam melting. In the production of a workpiece in several layers using this method one hopes to reduce the internal stresses of the workpiece by excluding the heating of positions that are too close. The points in this application are often arranged in some regular (grid) structure. In this paper we study optimal solutions of TSPFN instances where the points in the Euclidean plane are the points of a regular grid. Indeed, we explicitly determine the optimal values for the TSPFN and its associated path version on rectangular regular grids for different minimal distances of the points visited consecutively. For establishing lower bounds on the optimal values we use combinatorial counting arguments depending on the parities of the grid dimensions. Furthermore we provide construction schemes for optimal TSPFN tours for the considered cases.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Notes

  1. 1.

    Note that we rotated the m-n-coordinate system of the Euclidean plane by 90\(^\circ \) in clockwise direction in order to work with the usual matrix numbering of the grid cells later on.

  2. 2.

    In slight abuse of notation we will often write in this paper that a path has a start (the first node mentioned) and an end vertex (the last node mentioned) although we are in the undirected case.

  3. 3.

    The only exception is in the upper left corner, when we go from (1, 3) to (1, 1) and then over (2, 2), to (3, 3) and (4, 4).

References

  1. Applegate DL, Bixby RE, Chvatal V, Cook WJ (2007) The traveling salesman problem: a computational study (Princeton series in applied mathematics). Princeton University Press, Princeton. ISBN 0691129932

  2. Arkin EM, Chiang Y-J, Mitchell JSB, Skiena SS, Yang T-C (1999) On the maximum scatter traveling salesperson problem. SIAM J Comput 29(2):515–544

    MathSciNet  Article  MATH  Google Scholar 

  3. Arkin E, Bender M, Demaine E, Fekete S, Mitchell J, Sethia S (2005) Optimal covering tours with turn costs. SIAM J Comput 35(3):531–566

    MathSciNet  Article  MATH  Google Scholar 

  4. Arkin EM, Fekete SP, Islam K, Meijer H, Mitchell JS, Núñez-Rodríguez Y, Polishchuk V, Rappaport D, Xiao H (2009) Not being (super)thin or solid is hard: a study of grid hamiltonicity. Comput Geom 42(6–7):582–605

    MathSciNet  Article  MATH  Google Scholar 

  5. Arora S (1998) Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J ACM 45(5):753–782

    MathSciNet  Article  MATH  Google Scholar 

  6. Arora S, Lund C, Motwani R, Sudan M, Szegedy M (1998) Proof verification and the hardness of approximation problems. J ACM 45(3):501–555

    MathSciNet  Article  MATH  Google Scholar 

  7. Chiang Y-J (2004) New approximation results for the maximum scatter TSP. Algorithmica 41(4):309–341. doi:10.1007/s00453-004-1124-z

    MathSciNet  Article  MATH  Google Scholar 

  8. Christofides, N (1976) Worst-case analysis of a new heuristic for the traveling salesman problem. Technical report, GSIA, Carnegie-Mellon University,

  9. Conrad A, Hindrichs T, Morsy H, Wegener I (1994) Solution of the knight’s Hamiltonian path problem on chessboards. Discret Appl Math 50(2):125–134

    MathSciNet  Article  MATH  Google Scholar 

  10. Cook WJ (2011) In pursuit of the traveling salesman: mathematics at the limits of computation. Princeton University Press, Princeton

    Google Scholar 

  11. Cull P, De Curtins J (1978) Knight’s tour revisited. Fibonacci Q 16:276–285

    MathSciNet  MATH  Google Scholar 

  12. Dantzig G, Fulkerson R, Johnson S (1954) Solution of a large-scale traveling-salesman problem. Oper Res 2:393–410

    MathSciNet  Google Scholar 

  13. Demaine E, Mitchell JSB, O’Rourke J (2004) The open problems project. http://cs.smith.edu/~jorourke/TOPP/

  14. Euler L (1759) Solution d’une question curieuse que ne paroit soumiseà aucune analyse. Mem Acad Dessciences Berl 15:310–337

    Google Scholar 

  15. Garey MR, Graham RL, Johnson DS (1976) Some NP-complete geometric problems. In: Proceedings of the eighth annual ACM symposium on theory of computing, STOC ’76, pp 10–22, New York, NY, USA. ACM

  16. Gutin G, Punnen A (2002) The traveling salesman problem and its variations. Springer, Berlin

    Google Scholar 

  17. Hoffmann I, Kurz S, Rambau J (2015) The maximum scatter TSP on a regular grid. https://epub.uni-bayreuth.de/2524/

  18. Itai A, Papadimitriou C, Szwarcfiter J (1982) Hamilton paths in grid graphs. SIAM J Comput 11(4):676–686

    MathSciNet  Article  MATH  Google Scholar 

  19. Jellen A, Fischer A, Hungerländer P (2016) Implementation of algorithms and illustration of optimal tours for the TSPFN with \(r \in \{0,1,\sqrt{2} \}\). http://philipphungerlaender.jimdo.com/tspfn-code/

  20. Kordaß R (2014) Untersuchungen zum Eigenspannungs- und Verzugsverhalten beim Laserstrahlschmelzen. Masterarbeit, Technische Universität Chemnitz, Fakultät für Maschinenbau, Professur für Werkzeugmaschinen und Umformtechnik

  21. Kozma L, Mömke T (2015) A PTAS for Euclidean maximum scatter TSP. CoRR, arXiv:1512.02963

  22. Kozma L, Mömke T (2017) Maximum scatter TSP in doubling metrics, pp 143–153. doi:10.1137/1.9781611974782.10

  23. Lin S-S, Wei C-L (2005) Optimal algorithms for constructing knight’s tours on arbitrary chessboards. Discret Appl Math 146(3):219–232

    MathSciNet  Article  MATH  Google Scholar 

  24. MATLAB. version 7.10.0 (r2010a) (2010)

  25. Papadimitriou CH (1977) The Euclidean travelling salesman problem is NP-complete. Theor Comput Sci 4(3):237–244

    Article  MATH  Google Scholar 

  26. Parberry I (1997) An efficient algorithm for the knight’s tour problem. Discret Appl Math 73(3):251–260

    MathSciNet  Article  MATH  Google Scholar 

  27. Reinelt G (1994) The traveling salesman: computational solutions for TSP applications. Springer, Berlin

    Google Scholar 

  28. Schwenk AJ (1991) Which rectangular chessboards have a knight’s tour? Math Mag 64(5):325–332

    MathSciNet  Article  MATH  Google Scholar 

  29. Umans C., Lenhart W (1997) Hamiltonian cycles in solid grid graphs. In: 38th annual symposium on foundations of computer science, pp 496–505

Download references

Acknowledgements

We thank Richard Kordaß and Thomas Töppel from the Fraunhofer IWU in Dresden for introducing us to this optimization problem in laser beam melting and for providing several real-world instances.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Anja Fischer.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fischer, A., Hungerländer, P. The traveling salesman problem on grids with forbidden neighborhoods. J Comb Optim 34, 891–915 (2017). https://doi.org/10.1007/s10878-017-0119-z

Download citation

Keywords

  • Shortest Hamiltonian path problem
  • Traveling salesman problem
  • Constrained Euclidean traveling salesman problem
  • Grid
  • Explicit solutions

Mathematics Subject Classification

  • 90C27
  • 05C38