Journal of Combinatorial Optimization

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

The traveling salesman problem on grids with forbidden neighborhoods

  • Anja FischerEmail author
  • Philipp Hungerländer


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.


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

Mathematics Subject Classification

90C27 05C38 



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.


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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

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