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The quadratic shortest path problem: complexity, approximability, and solution methods

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  • Rostami, Borzou
  • Chassein, André
  • Hopf, Michael
  • Frey, Davide
  • Buchheim, Christoph
  • Malucelli, Federico
  • Goerigk, Marc

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.

Suggested Citation

  • Rostami, Borzou & Chassein, André & Hopf, Michael & Frey, Davide & Buchheim, Christoph & Malucelli, Federico & Goerigk, Marc, 2018. "The quadratic shortest path problem: complexity, approximability, and solution methods," European Journal of Operational Research, Elsevier, vol. 268(2), pages 473-485.
    • Handle: RePEc:eee:ejores:v:268:y:2018:i:2:p:473-485
    DOI: 10.1016/j.ejor.2018.01.054
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