Group-Fair Metric Distortion of Facility Assignment Problems

2026-08-17

AI Summary

This study investigates group-fair facility location with affinity factors under unknown group structures. Focusing on Max-of-Sum and Sum-of-Max objectives, it systematically analyzes algorithmic performance in both full-information and ordinal settings by integrating metric space analysis, ordinal mechanism design, and approximation theory. The primary contributions include establishing distortion upper bounds for combinatorial social objectives alongside matching information-theoretic lower bounds, yielding exact or asymptotically tight results. Furthermore, this work achieves a theoretical unification of matching and clustering problems, providing a rigorous foundation for fair allocation under complex constraints. These findings advance the understanding of trade-offs between fairness and efficiency when group memberships are not exogenously given but must be inferred from agent preferences.

Abstract

We study the group-fair distortion of metric facility assignment problems, where a set of agents, partitioned into unknown groups, must be assigned to a collection of facilities, possibly subject to capacity or other feasibility constraints. Given an assignment, each agent incurs a cost that depends on both its distance to its assigned facility and, via an affinity factor, the average distance of the other members in its group to their assigned facilities. We consider full-information algorithms, which have complete knowledge of the metric space, and ordinal-information algorithms, which know the distances between facilities and only the rankings of the agents over facilities (sorted by increasing distance). We establish worst-case distortion upper bounds in terms of the Max-of-Sum and Sum-of-Max social objectives, which combine the classic utilitarian and egalitarian social cost measures. We also derive informational lower bounds for one-sided matching and clustering, two fundamental and well-studied problems captured by our model, that match our upper bounds exactly for Max-of-Sum and asymptotically for Sum-of-Max.

Problem

• Group-Fair Metric Distortion
• Facility Assignment
• Ordinal Information
• Social Objectives
• Worst-case Distortion

Innovation

• Group-Fair Distortion
• Metric Facility Assignment
• Ordinal Information
• Max-of-Sum
• Sum-of-Max