When far is better: The Chamberlin-Courant approach to obnoxious committee selection

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 asfar 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 havedouble-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.