Dimensionality Reduction in Multiobjective Optimization: The Minimum Objective Subset Problem

Abstract

The number of objectives in a multiobjective optimization problem strongly influences both the performance of generating methods and the decision making process in general. On the one hand, with more objectives, more incomparable solutions can arise, the number of which affects the generating method’s performance. On the other hand, the more objectives are involved the more complex is the choice of an appropriate solution for a (human) decision maker. In this context, the question arises whether all objectives are actually necessary and whether some of the objectives may be omitted; this question in turn is closely linked to the fundamental issue of conflicting and non-conflicting optimization criteria. Besides a general definition of conflicts between objective sets, we here introduce the $\mathcal{N}\mathcal{P}$-hard problem of computing a minimum subset of objectives without losing information (MOSS). Furthermore, we present for MOSS both an approximation algorithm with optimum approximation ratio and an exact algorithm which works well for small input instances. We conclude with experimental results for a random problem and the multiobjective 0/1-knapsack problem.