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

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.