Near-Tight Algorithms for the Chamberlin-Courant and Thiele Voting Rules

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. 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| * 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.