To show that multiobjective optimization problems like the multiobjective shortest path or assignment problems are hard, we often use the theory of NP-hardness. In this paper we rigorously investigate the complexity status of some well-known multiobjective optimization problems and ask the question if these problems really are NP-hard. It turns out, that most of them do not seem to be and for one we prove that if it is NP-hard then this would imply P = NP under assumptions from the literature. We also reason why NP-hardness might not be well suited for investigating the complexity status of intractable multiobjective optimization problems.
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