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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Final published version
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - The Quadratic Shortest Path Problem
T2 - Complexity, Approximability, and Solution Methods
AU - Rostami, Borzou
AU - Chassein, André
AU - Hopf, Michael
AU - Frey, Davide
AU - Buchheim, Christoph
AU - Malucelli, Federico
AU - Goerigk, Marc
N1 - 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
PY - 2018/7/16
Y1 - 2018/7/16
N2 - 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.
AB - 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.
KW - Combinatorial optimization
KW - Shortest path problem
KW - Quadratic 0–1 optimization
KW - Computational complexity
KW - Branch-and-Bound
U2 - 10.1016/j.ejor.2018.01.054
DO - 10.1016/j.ejor.2018.01.054
M3 - Journal article
VL - 268
SP - 473
EP - 485
JO - European Journal of Operational Research
JF - European Journal of Operational Research
SN - 0377-2217
IS - 2
ER -