Output-sensitive Complexity of Multi-Objective Integer Network Flow Problems

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