Consider a flow network, i.e., a directed graph where each arc has a nonnegative capacity value and an associated length, together with nonempty supply intervals for the sources and nonempty demand intervals for the sinks. The Maximum Min-Cost-Flow Problem (MaxMCF) is to find fixed supply and demand values within these intervals such that the optimal objective value of the induced Min-Cost-Flow Problem (MCF) is maximized. In this paper, we show that MaxMCF as well as its uncapacitated variant, the Maximum Transportation Problem (MaxTP), are NP-hard. Further, we prove that MaxMCF is APX-hard if a connectedness-condition regarding the sources and the sinks of the flow network is dropped. Finally, we show how the Minimum Min-Cost-Flow Problem (MinMCF) can be solved in polynomial time.

中文翻译：

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关于计算最大和最小最小成本流的复杂性

考虑一个流动网络，即一个有向图，其中每个弧都有一个非负容量值和一个相关长度，以及源的非空供应区间和汇的非空需求区间。最大最小成本流问题 (MaxMCF) 是在这些区间内找到固定的供需值，以使引发的最小成本流问题 (MCF) 的最优目标值最大化。在本文中，我们展示了 MaxMCF 及其无容量变体，即最大运输问题 (MaxTP)，是 NP-hard。此外，如果关于流网络的源和汇的连通性条件被丢弃，我们证明 MaxMCF 是 APX-hard。最后，我们展示了如何在多项式时间内解决最小最小成本流问题 (MinMCF)。