Optimal Trees for the Parametric Fixed Charge Shortest Paths Problem

We investigate a variant of the Fixed Charge Shortest Paths problem which enumerates the sequence of optimal solutions that arise as the focus shifts from the fixed cost parameters, (generating a Minimum Spanning Tree problem instance), to the variable costs, (generating a Shortest Paths problem instance). Thus we provide insight into the behavior of the problem, which is NP-hard and requires some compromise between the two efficiently solved problem classes. We provide an algorithm that finds the optimal solutions to this Parametric version that increases the complexity over the non-parametric version by a factor of no more than O(log n), where n is the number of nodes in the given network.