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On the Quadratic Shortest Path Problem

  • Borzou RostamiEmail author
  • Federico Malucelli
  • Davide Frey
  • Christoph Buchheim
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9125)

Abstract

Finding the shortest path in a directed graph is one of the most important combinatorial optimization problems, having applications in a wide range of fields. In its basic version, however, the problem fails to represent situations in which the value of the objective function is determined not only by the choice of each single arc, but also by the combined presence of pairs of arcs in the solution. In this paper we model these situations as a Quadratic Shortest Path Problem, which calls for the minimization of a quadratic objective function subject to shortest-path constraints. We prove strong NP-hardness of the problem and analyze polynomially solvable special cases, obtained by restricting the distance of arc pairs in the graph that appear jointly in a quadratic monomial of the objective function. Based on this special case and problem structure, we devise fast lower bounding procedures for the general problem and show computationally that they clearly outperform other approaches proposed in the literature in terms of their strength.

Keywords

Shortest Path Problem Quadratic 0–1 optimization Lower bounds 

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Borzou Rostami
    • 1
    Email author
  • Federico Malucelli
    • 2
  • Davide Frey
    • 3
  • Christoph Buchheim
    • 1
  1. 1.Fakultät für MathematikTU DortmundDortmundGermany
  2. 2.Department of Electronics, Information, and BioengineeringPolitecnico di MilanoMilanItaly
  3. 3.INRIA-Rennes Bretagne AtlantiqueRennesFrance

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