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

TL;DR

This paper introduces a near-optimal, fast algorithm for the Chamberlin-Courant voting rule on single-peaked preferences, and extends the approach to nearly single-peaked profiles and Thiele rules, improving computational efficiency.

Contribution

It provides the first almost linear time algorithm for CC on single-peaked preferences and develops polynomial-time algorithms for nearly single-peaked profiles with candidate deletion.

Findings

Algorithm runs in almost linear time for single-peaked preferences.

Polynomial-time algorithm for CC with small candidate-deletion sets.

Extensions to Thiele rules with approval ballots.

Abstract

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.