This paper addresses the output-sensitive complexity for linear multi-objective integer minimum cost flow (MOIMCF) problems and provides insights about the time complexity for enumerating all supported nondominated vectors. The paper shows that there can not exist an output-polynomial time algorithm for the enumeration of all supported nondominated vectors that determine the vectors in an ordered way in the outcome space unless NP = P. Moreover, novel methods for identifying supported nondominated vectors in bi-objective minimum cost flow (BOIMCF) problems are proposed, accompanied by a numerical comparison between decision- and objective-space methods. A novel, equivalent and more compact formulation of the minimum cost flow ILP formulation used in the e-constrained-scalarization approach is introduced, demonstrating enhanced efficiency in the numerical tests

翻译：本文研究了线性多目标整数最小成本流（MOIMCF）问题的输出敏感复杂度，并对枚举所有支撑非支配向量的时间复杂度提供了深入见解。论文证明，除非NP = P，否则不存在能在结果空间中有序确定向量的输出多项式时间算法来枚举所有支撑非支配向量。此外，本文提出了在双目标最小成本流（BOIMCF）问题中识别支撑非支配向量的新方法，并对比了决策空间与目标空间方法的数值性能。针对ε约束标量化方法中使用的最小成本流整数线性规划（ILP）模型，本文提出了一种新颖、等价且更紧凑的公式，数值实验验证了其效率优势。