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

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

European Journal of Operational Research, 2018, vol. 268, issue 2, 473-485

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.

Keywords: Combinatorial optimization; Shortest path problem; Quadratic 0–1 optimization; Computational complexity; Branch-and-Bound (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (12)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:ejores:v:268:y:2018:i:2:p:473-485

DOI: 10.1016/j.ejor.2018.01.054

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