A note on the complexity of the transportation problem with a permutable demand vector

In this note we investigate the computational complexity of the transportation problem with a permutable demand vector, TP-PD for short. In the TP-PD, the goal is to permute the elements of the given integer demand vector b = (b1; : : : ; bn) in order to minimize the overall transportation costs. Meusel and Burkard [4] recently proved that the TP-PD is strongly NP-hard. In their NP-hardness reduction, the used demand values bj, j = 1; : : : ; n, are large integers. In this note we show that the TP-PD remains strongly NP-hard even for the case where bj 2 f0; 3g for j = 1; : : : ; n. As a positive result, we show that the TP-PD becomes strongly polynomial time solvable if bj 2 f0; 1; 2g holds for j = 1; : : : ; n.