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费用流（MOIMCF）问题中的输出敏感复杂度，并深入分析了枚举所有支撑非支配向量的时间复杂性。论文证明：除非NP=P，否则不存在一种输出多项式时间算法，能够在结果空间中按序枚举所有决定向量的支撑非支配向量。此外，针对双目标最小费用流（BOIMCF）问题，提出了识别支撑非支配向量的新方法，并附带了决策空间方法与目标空间方法的数值比较。文中还引入了ε约束标量化方法中最小费用流整数线性规划（ILP）公式的一种新颖、等效且更紧凑的表述形式，数值测试表明该方法具有更高的效率。