The Train Delivery Problem - Vehicle Routing Meets Bin Packing

We consider the train delivery problem which is a generalization of the bin packing problem and is equivalent to a one dimensional version of the vehicle routing problem with unsplittable demands. The train delivery problem is strongly NP-Hard and does not admit an approximation ratio better than 3/2. We design two types of approximation schemes for the problem. We give an asymptotic polynomial time approximation scheme, under a notion of asymptotic that makes sense even though scaling can cause the cost of the optimal solution of any instance to be arbitrarily large. Alternatively, we give a polynomial time approximation scheme for the case where W , an input parameter that corresponds to the bin size or the vehicle capacity, is polynomial in the number of items or demands. The proofs combine techniques used in approximating bin-packing problems and vehicle routing problems.