Submitted manuscript
Research output: Working paper
Research output: Working paper
}
TY - UNPB
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
PY - 2016/2/24
Y1 - 2016/2/24
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 - Shortest path problem
KW - Quadratic 0-1 optimization
KW - Computational complexity
KW - Branch and Bound
M3 - Working paper
BT - The quadratic shortest path problem
ER -