Deciding Membership in Minimal Upward Covering Sets is Hard for Parallel Access to NP

A common thread in the social sciences is to identify the “most desirable” elements of a set of alternatives according to some binary dominance relation. Examples can be found in areas as diverse as voting theory, game theory, and argumentation theory. Brandt and Fischer [BF08] proved that MCu-MEMBER, the problem of deciding whether an alternative is contained in some inclusion-minimal upward covering set—a subset of alternatives that satisfies certain notions of internal and external stability—is NP-hard. We raise their NPhardness lower bound to the Θ 2 level of the polynomial hierarchy and provide a Σ p 2 upper bound. Relatedly, we show that other problems regarding minimal upward covering sets, such as deciding whether the set of all alternatives is a minimal upward covering set, are coNP-hard. As a consequence, minimal upward covering sets cannot be found in polynomial time unless P = NP.