Inapproximability results for the inverse shortest paths problem with integer lengths and unique shortest paths

We study the complexity of two Inverse Shortest Paths (ISP) problems with integer arc lengths and the requirement for uniquely determined shortest paths. Given a collection of paths in a directed graph, the task is to find positive integer arc lengths such that the given paths are uniquely determined shortest paths between their respective terminals. The first problem seeks for arc lengths that minimize the length of the longest of the prescribed paths. In the second problem, the length of the longest arc is to be minimized. We show that it is NP-hard to approximate the minimal longest path length within a factor less than 8/7 or the minimal longest arc length within a factor less than 9/8. This answers the (previously) open question whether these problems are NP-hard or not. We also present a simple algorithm that achieves an O(|V |)-approximation guarantee for both variants. Both ISP problems arise in the planning of telecommunication networks with shortest path routing protocols. Our results imply that it is NP-hard to decide whether a given path set can be realized with a real shortest path routing protocol such as OSPF, IS-IS, or RIP.