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    Rights statement: This is the author’s version of a work that was accepted for publication in European Journal of Operational Research. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in European Journal of Operational Research, 268, (2), 2018 DOI: 10.1016/j.ejor.2018.01.054

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The Quadratic Shortest Path Problem: Complexity, Approximability, and Solution Methods

Research output: Contribution to journalJournal articlepeer-review

Published
  • Borzou Rostami
  • André Chassein
  • Michael Hopf
  • Davide Frey
  • Christoph Buchheim
  • Federico Malucelli
  • Marc Goerigk
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<mark>Journal publication date</mark>16/07/2018
<mark>Journal</mark>European Journal of Operational Research
Issue number2
Volume268
Number of pages13
Pages (from-to)473-485
Publication StatusPublished
Early online date21/02/18
<mark>Original language</mark>English

Abstract

We consider the problem of finding a shortest path in a directed graph with a quadratic objective function (the QSPP). We show that the QSPP cannot be approximated unless P=NP . For the case of a convex objective function, an n-approximation algorithm is presented, where n is the number of nodes in the graph, and APX-hardness is shown. Furthermore, we prove that even if only adjacent arcs play a part in the quadratic objective function, the problem still cannot be approximated unless P=NP. In order to solve the problem we first propose a mixed integer programming formulation, and then devise an efficient exact Branch-and-Bound algorithm for the general QSPP, where lower bounds are computed by considering a reformulation scheme that is solvable through a number of minimum cost flow problems. In our computational experiments we solve to optimality different classes of instances with up to 1000 nodes.

Bibliographic note

This is the author’s version of a work that was accepted for publication in European Journal of Operational Research. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in European Journal of Operational Research, 268, (2), 2018 DOI: 10.1016/j.ejor.2018.01.054