Resource Constrained Shortest Paths with a Super Additive Objective Function

We present an exact solution approach to the constrained shortest path problem with a super additive objective function. This problem generalizes the resource constrained shortest path problem by considering a cost function c(·) such that, given two consecutive paths P1 and P2, c(P1 ∪P2) ≥ c(P1)+c(P2). Since super additivity invalidates the Bellman optimality conditions, known resource constrained shortest path algorithms must be revisited. Our exact solution algorithm is based on a two stage approach: first, the size of the input graph is reduced as much as possible using resource, cost, and Lagrangian reduced-cost filtering algorithms that account for the super additive cost function. Then, since the Lagrangian relaxation provides a tight lower bound, the optimal solution is computed using a near-shortest path enumerative algorithm that exploits the lower bound. The behavior of the different filtering procedures are compared, in terms of computation time, reduction of the input graph, and solution quality, considering two classes of graphs deriving from real applications.