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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
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