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

29 December 2022
Krzysztof Sornat
Virginia Vassilevska Williams
Yinzhan Xu
ArXiv (abs)PDFHTML
Abstract

We present an almost optimal algorithm for the classic Chamberlin-Courant multiwinner voting rule (CC) on single-peaked preference profiles. Given nnn voters and mmm candidates, it runs in almost linear time in the input size, improving the previous best O(nm2)O(nm^2)O(nm2) 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 DDD has logarithmic size. Actually, our algorithm runs in 2∣D∣⋅poly(n,m)2^{|D|} \cdot poly(n,m)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.

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