For every ε>0 there exists an election in which every approximately dominating committee has size at least about 1/(32π ε^2), matching the known upper bound up to a constant factor.
Choosing collectively optimal sets of alternatives based on the Condorcet criterion
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Pessimal Elections for Approximately Dominating Sets
For every ε>0 there exists an election in which every approximately dominating committee has size at least about 1/(32π ε^2), matching the known upper bound up to a constant factor.