  The 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) 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 $\Theta_{2}^{P}$. We
also demonstrate efficient algorithms for the second problem when the input
consists of single-peaked rankings. Our contribution fills an essential gap in
the literature for these important multi-winner rules.
