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 well-known and classic scoring rule for achieving diverse representation: the Chamberlin-Courant (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 well-studied 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 polynomial-time algorithm even when the optimal score is a polynomial factor smaller (under standard computational complexity assumptions). We show that another well-studied 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.

翻译：我们考虑的是从一组美元候选人中选择一个特定规模的候选人的亚基数的算法问题。 我们考虑的是从一组美元候选人中选择一个特定规模的候选人的亚基数 $k$,并了解所有候选人的选民或亚基数排名。 我们考虑的是众所周知和经典的评分规则,以达到多样化的代表性:Camberlin-Courant(CC)或$$-Borda规则,其中委员会的得分是选民的平均值,是该选民委员会中最佳候选人的级别;它被普遍化为最高数额的美元,称为美元-博尔达规则。我们的第一个结果是对自然的和受广泛研究的贪婪的比值进行更好的分析。我们显示贪婪(1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\r\rrrrr>规则的得分数, 而这个比值的比值,我们的最佳比值的比值是比值的比值, 我们的比值最终的比值也显示我们最好的比值。