Classical work on metric space based committee selection problem interprets distance as ``near is better''. In this work, motivated by real-life situations, we interpret distance as ``far is better''. Formally stated, we initiate the study of ``obnoxious'' committee scoring rules when the voters' preferences are expressed via a metric space. To this end, we propose a model where large distances imply high satisfaction and study the egalitarian avatar of the well-known Chamberlin-Courant voting rule and some of its generalizations. For a given integer value $1 \le \lambda \le k$, the committee size k, a voter derives satisfaction from only the $\lambda$-th favorite committee member; the goal is to maximize the satisfaction of the least satisfied voter. For the special case of $\lambda = 1$, this yields the egalitarian Chamberlin-Courant rule. In this paper, we consider general metric space and the special case of a $d$-dimensional Euclidean space. We show that when $\lambda$ is $1$ and $k$, the problem is polynomial-time solvable in $\mathbb{R}^2$ and general metric space, respectively. However, for $\lambda = k-1$, it is NP-hard even in $\mathbb{R}^2$. Thus, we have ``double-dichotomy'' in $\mathbb{R}^2$ with respect to the value of {\lambda}, where the extreme cases are solvable in polynomial time but an intermediate case is NP-hard. Furthermore, this phenomenon appears to be ``tight'' for $\mathbb{R}^2$ because the problem is NP-hard for general metric space, even for $\lambda=1$. Consequently, we are motivated to explore the problem in the realm of (parameterized) approximation algorithms and obtain positive results. Interestingly, we note that this generalization of Chamberlin-Courant rules encodes practical constraints that are relevant to solutions for certain facility locations.

翻译：基于度量空间的委员会选举问题的经典研究将距离解释为“近者更优”。在本工作中，受现实情境启发，我们将距离解释为“远者更优”。正式而言，我们首次研究了当选民偏好通过度量空间表达时的“厌恶型”委员会评分规则。为此，我们提出一个模型，其中较大的距离意味着较高的满意度，并研究了著名的Chamberlin-Courant投票规则及其部分推广的平等主义形式。对于给定的整数值 $1 \le \lambda \le k$（委员会规模为k），选民仅从其第$\lambda$偏好的委员会成员处获得满意度；目标是最大化最不满意选民的满意度。在$\lambda = 1$的特殊情况下，这产生了平等主义Chamberlin-Courant规则。本文中，我们考虑一般度量空间及$d$维欧几里得空间的特殊情况。我们证明当$\lambda$为$1$和$k$时，该问题分别在$\mathbb{R}^2$和一般度量空间中具有多项式时间解法。然而，对于$\lambda = k-1$的情况，即使在$\mathbb{R}^2$中也是NP难的。因此，在$\mathbb{R}^2$中关于{\lambda}的取值存在“双重二分性”：极端情况可在多项式时间内求解，而中间情况却是NP难的。此外，这一现象对$\mathbb{R}^2$而言似乎是“紧的”，因为即使在$\lambda=1$时，该问题在一般度量空间中也是NP难的。因此，我们尝试在（参数化）近似算法领域探索该问题，并获得了积极结果。值得注意的是，我们发现Chamberlin-Courant规则的这种推广蕴含了与某些设施选址解决方案相关的实际约束。