We study the problem of fair $k$-committee selection under an egalitarian objective. Given $n$ agents partitioned into $m$ groups (\eg, demographic quotas), the goal is to aggregate their preferences to form a committee of size $k$ that guarantees minimum representation from each group while minimizing the maximum \emph{cost} incurred by any agent. We model this setting as the ordinal fair $k$-center problem, where agents are embedded in an unknown metric space, and each agent reports a complete preference ranking (i.e., ordinal information) over all agents, consistent with the underlying distance metric (i.e., cardinal information). The cost incurred by an agent with respect to a committee is defined as its distance to the closest committee member. The quality of an algorithm is evaluated using the notion of distortion, which measures the worst-case ratio between the cost of the committee produced by the algorithm and the cost of an optimal committee, when given complete access to the underlying metric space. When cardinal information is not available, no constant distortion is possible for the ordinal $k$-center problem, even without fairness constraints, when $k\geq 3$ [Burkhardt et.al., AAAI'24]. To overcome this hardness, we allow limited access to cardinal information by querying the metric space. In this setting, our main contribution is a factor-$5$ distortion algorithm that requires only $O(k \log^2 k)$ queries. Along the way, we present an improved factor-$3$ distortion algorithm using $O(k^2)$ queries.

翻译：我们研究在平等主义目标下的公平$k$委员会选举问题。给定$n$个智能体划分为$m$个群体（例如，基于人口统计配额），目标是通过聚合其偏好来组建一个规模为$k$的委员会，该委员会需保证每个群体获得最低限度的代表席位，同时最小化任何智能体所承受的最大\emph{成本}。我们将此场景建模为序数公平$k$中心问题，其中智能体嵌入一个未知的度量空间，每个智能体报告一个关于所有智能体的完整偏好排序（即序数信息），该排序与底层距离度量（即基数信息）一致。一个智能体相对于委员会的成本定义为其与最近委员会成员的距离。算法的质量通过扭曲度的概念进行评估，该概念衡量在完全访问底层度量空间时，算法产生的委员会成本与最优委员会成本之间的最坏情况比率。当基数信息不可用时，即使没有公平性约束，对于$k \geq 3$的情况，序数$k$中心问题也无法实现常数扭曲度[Burkhardt et.al., AAAI'24]。为了克服这一困难，我们允许通过查询度量空间来有限地访问基数信息。在此设置下，我们的主要贡献是提出了一种仅需$O(k \log^2 k)$次查询即可实现因子$5$扭曲度的算法。此外，我们还提出了一种使用$O(k^2)$次查询即可实现改进的因子$3$扭曲度的算法。