Proportional Clustering, the $eta$-Plurality Problem, and Metric Distortion

2025-02-14

AI Summary

This paper studies fair clustering and voting-based facility location under ordinal preferences—without access to underlying metric distances. For proportional clustering (under the Droop quota), β-majority points, and center selection with metric distortion, we establish, for the first time, the equivalence between proportional clustering and β-majority points when (k = 1). We propose the Plurality Veto rule, which constructs a ((sqrt{5} - 2) approx 0.236)-majority point using only ordinal preference data. Moreover, we provide the first theoretical guarantee achieving ((2 + sqrt{5}) approx 4.236)-proportional fairness in clustering solely from ordinal inputs—resolving an open problem posed by Kalayci et al. All results are tight and require no distance information whatsoever. Our work introduces a novel paradigm for ordinal fair clustering, unifying proportional representation, majority concepts, and metric-free design.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Machine Learning: Learning Preferences or Rankings
• Knowledge Representation and Reasoning: Preferences

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Security and Privacy: Data transparency and provenance

Abstract

We show that the proportional clustering problem using the Droop quota for $k = 1$ is equivalent to the $eta$-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.

Problem

• Proportional clustering equivalence to β-plurality
• Plurality Veto rule for metric space selection
• Ordinal information for proportionally fair clusterings

Innovation

• Proportional clustering with Droop quota
• Plurality Veto rule application
• Ordinal information for fair clustering