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.

翻译：我们研究比例公平聚类问题，其中需从度量空间中选择$k$个中心来代表$n$个智能体，且任何足够大的智能体群体不应被集体低估。该背景下公平性的核心概念之一是$α$-核。Chen等人[2019]证明了$(1+\sqrt{2})$-核聚类的存在性，同时给出了$α<2$时$α$-核为空的实例。七年来，填补这一差距始终是开放问题。我们从下界角度取得进展，构造了一个对所有$α<2.1508$均使$α$-核为空的实例。我们的技术建立于以下方法：建立核心变体（即Hare核与Droop核）之间的关联；将最优空核实例的搜索约简为高度结构化的聚类实例族；以及使用混合整数线性规划（MILP）在此约简空间中搜索最优下界实例。基于该框架，我们还确定了候选中心数较少且仅需选择一个中心时Droop配额聚类实例的精确边界。对每个中心数$m \in \{3,4,5,6\}$，我们给出精确阈值$α_m^*$，使得$α_m^*$-核聚类总是存在，而对任何$α<α_m^*$，存在$m$个中心的实例使其$α$-核为空。尽管这些值最初通过计算机辅助搜索获得，我们也提供了不依赖MILP证书的直接证明。