Advertisement

Journal of Combinatorial Optimization

, Volume 34, Issue 3, pp 891–915 | Cite as

The traveling salesman problem on grids with forbidden neighborhoods

  • Anja Fischer
  • Philipp Hungerländer
Article

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.

Keywords

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

Mathematics Subject Classification

90C27 05C38 

Notes

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.

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 0691129932Google Scholar
  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):515–544MathSciNetCrossRefzbMATHGoogle 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):531–566MathSciNetCrossRefzbMATHGoogle 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):582–605MathSciNetCrossRefzbMATHGoogle Scholar
  5. Arora S (1998) Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J ACM 45(5):753–782MathSciNetCrossRefzbMATHGoogle 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):501–555MathSciNetCrossRefzbMATHGoogle 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 MathSciNetCrossRefzbMATHGoogle Scholar
  8. Christofides, N (1976) Worst-case analysis of a new heuristic for the traveling salesman problem. Technical report, GSIA, Carnegie-Mellon University,Google Scholar
  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):125–134MathSciNetCrossRefzbMATHGoogle Scholar
  10. Cook WJ (2011) In pursuit of the traveling salesman: mathematics at the limits of computation. Princeton University Press, PrincetonGoogle Scholar
  11. Cull P, De Curtins J (1978) Knight’s tour revisited. Fibonacci Q 16:276–285MathSciNetzbMATHGoogle Scholar
  12. Dantzig G, Fulkerson R, Johnson S (1954) Solution of a large-scale traveling-salesman problem. Oper Res 2:393–410MathSciNetGoogle 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 15:310–337Google 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. ACMGoogle Scholar
  16. Gutin G, Punnen A (2002) The traveling salesman problem and its variations. Springer, BerlinzbMATHGoogle 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):676–686MathSciNetCrossRefzbMATHGoogle 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 UmformtechnikGoogle Scholar
  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):219–232MathSciNetCrossRefzbMATHGoogle Scholar
  24. MATLAB. version 7.10.0 (r2010a) (2010)Google Scholar
  25. Papadimitriou CH (1977) The Euclidean travelling salesman problem is NP-complete. Theor Comput Sci 4(3):237–244CrossRefzbMATHGoogle Scholar
  26. Parberry I (1997) An efficient algorithm for the knight’s tour problem. Discret Appl Math 73(3):251–260MathSciNetCrossRefzbMATHGoogle Scholar
  27. Reinelt G (1994) The traveling salesman: computational solutions for TSP applications. Springer, BerlinzbMATHGoogle Scholar
  28. Schwenk AJ (1991) Which rectangular chessboards have a knight’s tour? Math Mag 64(5):325–332MathSciNetCrossRefzbMATHGoogle Scholar
  29. Umans C., Lenhart W (1997) Hamiltonian cycles in solid grid graphs. In: 38th annual symposium on foundations of computer science, pp 496–505Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institute for Numerical and Applied MathematicsUniversity of GöttingenGöttingenGermany
  2. 2.Institute for MathematicsAlpen-Adria Universität KlagenfurtKlagenfurt Am WörtherseeAustria

Personalised recommendations