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A Polyhedral Study of the Quadratic Traveling Salesman Problem

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Operations Research Proceedings 2014

Part of the book series: Operations Research Proceedings ((ORP))

Abstract

In this paper we summarize some of the results of the author’s Ph.D.-thesis. We consider an extension of the traveling salesman problem (TSP). Instead of each path of two nodes, an arc, the costs depend on each three nodes that are traversed in succession. As such a path of three nodes, a 2-arc, is present in a tour if the two corresponding arcs are contained in that tour, we speak of a quadratic traveling salesman problem (QTSP). This problem is motivated by an application in biology, special cases are the TSP with reload costs as well as the angular-metric TSP. Linearizing the quadratic objective function, we derive a linear integer programming formulation and present a polyhedral study of the associated polytope. This includes the dimension as well as three groups of facet-defining inequalities. Some are related to the Boolean quadric polytope and some forbid conflicting configurations. Furthermore, we describe approaches to strengthen valid inequalities of TSP in order to get stronger inequalities for QTSP.

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Acknowledgments

This work was partially supported by the European Union and the Free State of Saxony funding the cluster eniPROD at Chemnitz University of Technology.

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Authors and Affiliations

  1. TU Dortmund, Vogelpothsweg 87, 44227, Dortmund, Germany

    Anja Fischer

Authors
  1. Anja Fischer
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Correspondence to Anja Fischer .

Editor information

Editors and Affiliations

  1. Operations Research, RWTH Aachen University, Aachen, Germany

    Marco Lübbecke

  2. Lehrstuhl II für Mathematik, RWTH Aachen University, Aachen, Germany

    Arie Koster

  3. RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Peter Letmathe

  4. E.ON Energy Research Center, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Reinhard Madlener

  5. Management Science, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Britta Peis

  6. Chair of Operations Management, RWTH Aachen University, Aachen, Germany

    Grit Walther

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Fischer, A. (2016). A Polyhedral Study of the Quadratic Traveling Salesman Problem. In: Lübbecke, M., Koster, A., Letmathe, P., Madlener, R., Peis, B., Walther, G. (eds) Operations Research Proceedings 2014. Operations Research Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-319-28697-6_21

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