  We present an almost optimal algorithm for the classic Chamberlin-Courant
multiwinner voting rule (CC) on single-peaked preference profiles. Given $n$
voters and $m$ candidates, it runs in almost linear time in the input size,
improving the previous best $O(nm^2)$ time algorithm of Betzler et al. (2013).
We also study multiwinner voting rules on nearly single-peaked preference
profiles in terms of the candidate-deletion operation. We show a
polynomial-time algorithm for CC where a given candidate-deletion set $D$ has
logarithmic size. Actually, our algorithm runs in $2^{|D|} \cdot poly(n,m)$
time and the base of the power cannot be improved under the Strong Exponential
Time Hypothesis. We also adapt these results to all non-constant Thiele rules
which generalize CC with approval ballots.
