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,?,b_n) in order to minimize the overall transportation costs. Meusel and Burkard 6] recently proved that the TP-PD is strongly NP-hard. In their NP-hardness reduction, the used demand values b_j, j=1,?,n, are large integers. In this note we show that the TP-PD remains strongly NP-hard even for the case where b_j]{0,3} for j=1,?,n. As a positive result, we show that the TP-PD becomes strongly polynomial time solvable if b_j] {0,1,2} holds for j=1,?,n. This result can be extended to the case where b_j]{3,3+1,3+2} for an integer 3.