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 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 accomplish this, we propose a model where large distances imply high satisfaction (in contrast to the classical setting where shorter 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 λ between 1 and k, the committee size, a voter derives satisfaction from only the λth favorite committee member; the goal is to maximize the satisfaction of the least satisfied voter. For the special case of λ = 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 λ is 1 and k, the problem is polynomial-time solvable in ℝ² and general metric space, respectively. However, for λ = k-1, it is NP-hard even in ℝ². Thus, we have "double-dichotomy" in ℝ² with respect to the value of λ, where the extreme cases are solvable in polynomial time but an intermediate case is NP-hard. Furthermore, this phenomenon appears to be "tight" for ℝ² because the problem is NP-hard for general metric space, even for λ = 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.