In the committee selection problem, the goal is to choose a subset of size $k$ from a set of candidates $C$ that collectively gives the best representation to a set of voters. We consider this problem in Euclidean $d$-space where each voter/candidate is a point and voters' preferences are implicitly represented by Euclidean distances to candidates. We explore fault-tolerance in committee selection and study the following three variants: (1) given a committee and a set of $f$ failing candidates, find their optimal replacement; (2) compute the worst-case replacement score for a given committee under failure of $f$ candidates; and (3) design a committee with the best replacement score under worst-case failures. The score of a committee is determined using the well-known (min-max) Chamberlin-Courant rule: minimize the maximum distance between any voter and its closest candidate in the committee. Our main results include the following: (1) in one dimension, all three problems can be solved in polynomial time; (2) in dimension $d \geq 2$, all three problems are NP-hard; and (3) all three problems admit a constant-factor approximation in any fixed dimension, and the optimal committee problem has an FPT bicriterion approximation.

翻译：在委员会选举问题中，目标是从候选人集合 $C$ 中选择一个大小为 $k$ 的子集，使其整体上为选民集合提供最佳代表。我们在欧几里得 $d$ 维空间中考虑该问题，其中每位选民/候选人代表一个点，选民的偏好隐含地由其与候选人的欧几里得距离决定。我们探索委员会选举中的容错性，并研究以下三种变体：（1）给定一个委员会和一组 $f$ 个失效候选人，找出其最优替代方案；（2）在 $f$ 个候选人失效情况下，计算给定委员会最坏情况下的替代得分；（3）在最坏情况失效条件下，设计具有最优替代得分的委员会。委员会得分采用著名的（最小-最大）Chamberlin-Courant 规则确定：最小化任意选民与其在委员会中最接近候选人之间的最大距离。我们的主要结果包括：（1）在一维情况下，所有三个问题均可在多项式时间内求解；（2）在维度 $d \geq 2$ 时，所有三个问题均为 NP 难问题；（3）所有三个问题在任意固定维度下均允许常数因子近似，且最优委员会问题具有 FPT 双准则近似。