Inverse Shortest Path Models Based on Fundamental Cycle Bases

The inverse shortest path problem has received considerable attention in the literature since the seminal paper by Burton and Toint from 1992. Given a graph and a set of paths the problem is to find arc costs such that all specified paths are shortest paths. The quality of the arc costs is measured by the deviation from some ideal arc costs. Our contribution is a novel modeling technique for this problem based on fundamental cycle bases. For ’LP compatible’ norms we present a cycle basis model equivalent to the LP dual. The LP dual of our cycle basis model is a path based model that only requires a polynomial number of path constraints. This model is valid also for ’LP incompatible’ norms. This yields the first polynomial sized

path

formulation of the original problem.