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A note on the complexity of the transportation problem with a permutable demand vector. (English) Zbl 0967.90081
Summary: 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 $$b=(b_1,\dots,b_n)$$ in order to minimize the overall transportation costs. S. G. Meusel and R. E. Burkard [ibid. 50, 1-7 (1999; Zbl 0967.90084)] recently proved that the TP-PD is strongly NP-hard. In their NP-hardness reduction, the used demand values $$b_j$$, $$j=1,\dots,n$$, are large integers. In this note we show that the TP-PD remains strongly NP-hard even for the case where $$b_j\in\{0,3\}$$ for $$j=1,\dots,n$$. As a positive result, we show that the TP-PD becomes strongly polynomial time solvable if $$b_j\in \{0, 1,2\}$$ holds for $$j=1,\dots,n$$. This result can be extended to the case where $$b_j\in \{\kappa, \kappa+1, \kappa+2\}$$ for an integer $$\kappa$$.

##### MSC:
 90C08 Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.) 90C60 Abstract computational complexity for mathematical programming problems