Proportional Representation in Metric Spaces and Low-Distortion Committee Selection

We introduce a novel definition for a small set$R$ of$k$ points being representative of a larger set in a metric space. Given a set$V$ (e.g., documents or voters) to represent, and a set$C$ of possible representatives, our criterion requires that for any subset$S$ comprising a$\theta$ fraction of $V$, the average distance of$S$ to their best$\theta \cdot k$ points in$R$ should not be more than a factor$\gamma$ compared to their average distance to the best$\theta \cdot k$ points among all of $C$. This definition is a strengthening of proportional fairness and core fairness, but - different from those notions - requires that large cohesive clusters be represented proportionally to their size.

Since there are instances for which - unless$\gamma$ is polynomially large - no solutions exist, we study this notion in a resource augmentation framework, implicitly stating the constraints for a set$R$ of size$k$ as though its size were only $k/\alpha$, for $\alpha > 1$. Furthermore, motivated by the application to elections, we mostly focus on the ordinal model, where the algorithm does not learn the actual distances; instead, it learns only for each point$v$ in$V$ and each candidate pairs$c, c'$ which of$c, c'$ is closer to $v$. Our main result is that the Expanding Approvals Rule of Aziz and Lee is$(\alpha, \gamma)$ representative with $\gamma \leq 1 + 6.71 \cdot \frac{\alpha}{\alpha-1}$.

Our results lead to three notable byproducts. First, we show that the Expanding Approvals Rule achieves constant proportional fairness in the ordinal model, giving the first positive result on metric proportional fairness with ordinal information. Second, we show that for the core fairness objective, the Expanding Approvals Rule achieves the same asymptotic tradeoff between resource augmentation and approximation as the recent results of Li et al., which used full knowledge of the metric. Finally, our results imply a very simple single-winner voting rule with metric distortion at most 44.