🤖 AI Summary
This paper investigates the enumeration complexity of supported nondominated vectors for the multi-objective integer minimum cost flow (MOIMCF) problem. Regarding the problem, it establishes—first time in the literature—that ordered enumeration of supported nondominated vectors admits no output-polynomial-time algorithm unless NP = P, revealing an intrinsic computational hardness. Methodologically, for the bi-objective case (BOIMCF), it introduces a novel characterization for identifying supported nondominated vectors and proposes a more compact ε-constraint integer programming reformulation. The contributions include: (i) a theoretical hardness result clarifying fundamental limits of enumeration; (ii) an efficient recognition method for supported solutions in BOIMCF; and (iii) an improved ε-constraint model whose tighter formulation substantially enhances computational efficiency. Comprehensive theoretical analysis and numerical experiments jointly delineate the performance boundaries of decision-space versus objective-space algorithms across problem scales.
📝 Abstract
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
