Multiwinner Voting

Multiwinner voting rules pick a committee instead of a single winner, and rules like Chamberlin–Courant and Monroe are built around the idea that every voter should have a representative they actually like. Deciding whether a committee with a given score even exists is NP-hard for these rules. I look at the complexity of these rules through two natural questions: is a given committee optimal, and does some optimal committee contain a given candidate? For Chamberlin–Courant and Monroe (with rankings or approval ballots), the first is coNP-complete and the second is complete for Θ₂ᴾ, while single-peaked rankings allow efficient algorithms for the second.