An Inapproximability of Transshipment Problems with Permutable Transit Vectors

We investigate the hardness of approximating a transshipment problem with a permutable transit vector. We have already known that the transshipment problem of minimizing the total shipping cost is NP-hard. In this paper, we prove that even if the unit shipping cost associated with every arc is given as a positive integer, there is no polynomial time constant factor approximation algorithm for the transshipment problem under the hypothesis that P =NP. We obtain the inapproximability result by a gap-introducing reduction from an NP-complete problem by the name of 3DM. On the other hand, we propose a preparatory heuristic algorithm to the transshipment problem, which is based on a relaxation into the minimum cost flow problem and the minimum weight perfect matching on a bipartite graph. We also provide an upper bound of the total shipping cost obtained by the proposed heuristic.