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

Abstract.

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 ₁,…,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∈{κ,κ+1,κ+2} for an integer κ.