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：本文探讨了线性多目标整数最小费用流（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。研究证明，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，针对双目标最小费用流（BOIMCF）问题，本文提出了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表现。通过引入一种新颖、等价且更紧凑的最小费用流整数线性规划模型（该模型应用于ε约束标量化方法），数值实验验证了其效率的提升。