Proportional Clustering, the $\beta$-Plurality Problem, and Metric Distortion

TL;DR

This paper establishes equivalences between proportional clustering and the $eta$-plurality problem, demonstrating new methods for fair clustering using only ordinal metric information and resolving an open question in the field.

Contribution

It shows the equivalence of proportional clustering with the $eta$-plurality problem and introduces ordinal-based algorithms for fair clustering, answering an open research question.

Findings

Proportional clustering with Droop quota for k=1 is equivalent to the $eta$-plurality problem.

The Plurality Veto rule can find ($ ext{sqrt}(5) - 2$)-plurality points using only ordinal data.

$(2+ ext{sqrt}(5))$-proportionally fair clusterings can be achieved with purely ordinal information.

Abstract

We show that the proportional clustering problem using the Droop quota for $k = 1$ is equivalent to the $\beta$-plurality problem. We also show that the Plurality Veto rule can be used to select ($\sqrt{5} - 2$)-plurality points using only ordinal information about the metric space and resolve an open question of Kalayci et al. (AAAI 2024) by proving that $(2+\sqrt{5})$-proportionally fair clusterings can be found using purely ordinal information.