Complexity of some inverse shortest path lengths problems

The input to an inverse shortest path lengths problem (ISPL) consists of a graph G with arc weights, and a collection of source‐sink pairs with prescribed distances that do not necessarily conform to the shortest path lengths in G. The goal is to modify the arc weights, subject to a penalty on the deviation from the given weights, so that the shortest path lengths are equal to the prescribed values. We show that although ISPL is an NP‐hard problem, several ISPL classes are polynomially solvable. These cases include ISPL where the collection of the pairs share a single source and all other nodes as destinations (the single‐source all‐sink problem SAISPL). For the case where the collection contains a single node pair (the single‐source single‐sink problem SSISPL), we identify conditions on the uniformity of the penalty functions and on the original arc weights, which make SSISPL polynomially solvable. These results cannot be strengthened significantly as the general single‐source ISPL is NP‐hard and the all‐sink case, with more than one source, is also NP‐hard. We further provide a convex programming formulation for a relaxation of ISPL in which the shortest path lengths are only required to be no less than the given values (LBISPL). It is demonstrated how this compact formulation leads to efficient algorithms for ISPL.