Sonar, ChinmayChinmaySonarDey, PalashPalashDeyMisra, NeeldharaNeeldharaMisra2025-08-312025-08-312020-01-01[9780999241165]2-s2.0-85095697031http://repository.iitgn.ac.in/handle/IITG2025/24336The Chamberlin-Courant and Monroe rules are fundamental and well-studied rules in the literature of multi-winner elections. The problem of determining if there exists a committee of size k that has a Chamberlin-Courant (respectively, Monroe) dissatisfaction score of at most r is known to be NP-complete. We consider the following natural problems in this setting: a) given a committee S of size k as input, is it an optimal k-sized committee, and b) given a candidate c and a committee size k, does there exist an optimal k-sized committee that contains c? In this work, we resolve the complexity of both problems for the Chamberlin-Courant and Monroe voting rules in the settings of rankings as well as approval ballots. We show that verifying if a given committee is optimal is coNP-complete whilst the latter problem is complete for T^P<inf>2</inf>. Our contribution fills an essential gap in the literature for these important multi-winner rules.falseConference Paper89-9520206cpConference Proceeding