Improved Lower Bounds for Proportionally Fair Clustering

2026-06-05

AI Summary

This work addresses the long-standing open problem of closing the gap in the existence lower bound for the α-core in proportionally fair clustering, where the best-known interval remained (1+√2, 2]. By establishing a connection between the Hare core and the Droop core, the authors reduce the search for optimal counterexamples to a structured family of clustering instances. Leveraging mixed-integer linear programming (MILP) within this reduced space, they construct the first instance with an empty α-core for some α < 2.1508, thereby breaking a seven-year theoretical barrier. Moreover, for settings with a fixed number of candidate centers m ∈ {3, 4, 5, 6}, they determine the exact existence thresholds αₘ* and provide direct proofs that do not rely on MILP.

Abstract

We study proportionally fair clustering, where a set of $k$ centers must be chosen from a metric space to represent $n$ agents, and no sufficiently large group of agents should be collectively underrepresented. One of the central notions of fairness in this setting is the $α$-core. The existence of clusterings in the $(1+\sqrt{2})$-core was established by Chen et al. [2019], who also showed instances where the $α$-core is empty for every $α< 2$. Closing this gap has remained an open problem for seven years. We make progress from the lower-bound side by providing an instance whose $α$-core is empty for every $α< 2.1508$. Our techniques rely on establishing connections between variants of the core, namely the Hare core and the Droop core; reducing the search for optimal empty-core instances to a highly structured family of clustering instances; and using a Mixed Integer Linear Program (MILP) to search for optimal lower-bound instances within this reduced space. Using this framework, we also determine tight bounds for Droop quota clustering instances with a small number of possible candidate centers and a single center to be selected. For each number of centers $m \in \{3,4,5,6\}$, we give the exact threshold $α_m^*$ such that an $α_m^*$-core clustering always exists, while for every $α< α_m^*$ there is an instance with $m$ centers whose $α$-core is empty. Although these values were originally found through computer-aided search, we also provide direct proofs that do not rely on MILP certificates.

Problem

• proportionally fair clustering
• α-core
• lower bounds
• fairness
• clustering

Innovation

• proportionally fair clustering
• α-core
• mixed integer linear programming
• Droop quota
• fair representation