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The quadratic shortest path problem: complexity, approximability, and solution methods
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DOI：10.1016/j.ejor.2018.01.054.png)
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Abstract
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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. (C) 2018 Elsevier B.V. All rights reserved.
Keywords:
Combinatorial optimization
Shortest path problem
Quadratic 0-1 optimization
Computational complexity
Branch-and-Bound
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