Optimal Algorithms for Multiwinner Elections and the ChamberlinCourant Rule
Abstract
We consider the algorithmic question of choosing a subset of candidates of a given size $k$ from a set of $m$ candidates, with knowledge of voters' ordinal rankings over all candidates. We consider the wellknown and classic scoring rule for achieving diverse representation: the ChamberlinCourant (CC) or $1$Borda rule, where the score of a committee is the average over the voters, of the rank of the best candidate in the committee for that voter; and its generalization to the average of the top $s$ best candidates, called the $s$Borda rule. Our first result is an improved analysis of the natural and wellstudied greedy heuristic. We show that greedy achieves a $\left(1  \frac{2}{k+1}\right)$approximation to the maximization (or satisfaction) version of CC rule, and a $\left(1  \frac{2s}{k+1}\right)$approximation to the $s$Borda score. Our result improves on the best known approximation algorithm for this problem. We show that these bounds are almost tight. For the dissatisfaction (or minimization) version of the problem, we show that the score of $\frac{m+1}{k+1}$ can be viewed as an optimal benchmark for the CC rule, as it is essentially the best achievable score of any polynomialtime algorithm even when the optimal score is a polynomial factor smaller (under standard computational complexity assumptions). We show that another wellstudied algorithm for this problem, called the Banzhaf rule, attains this benchmark. We finally show that for the $s$Borda rule, when the optimal value is small, these algorithms can be improved by a factor of $\tilde \Omega(\sqrt{s})$ via LP rounding. Our upper and lower bounds are a significant improvement over previous results, and taken together, not only enable us to perform a finer comparison of greedy algorithms for these problems, but also provide analytic justification for using such algorithms in practice.
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 arXiv:
 arXiv:2106.00091
 Bibcode:
 2021arXiv210600091M
 Keywords:

 Computer Science  Computer Science and Game Theory;
 Computer Science  Data Structures and Algorithms;
 Economics  Theoretical Economics
 EPrint:
 Accepted by the TwentySecond ACM Conference on Economics and Computation (EC 2021)