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=(b <Subscript>1</Subscript>,…,b <Subscript>n</Subscript>) 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 <Subscript>j</Subscript>, 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 <Subscript>j</Subscript>∈{0,3} for j=1,…,n. As a positive result, we show that the TP-PD becomes strongly polynomial time solvable if b <Subscript>j</Subscript>∈{0,1,2} holds for j=1,…,n. This result can be extended to the case where b <Subscript>j</Subscript>∈{κ,κ+1,κ+2} for an integer κ. Copyright Springer-Verlag Berlin Heidelberg 1999