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

TL;DR

This work studies obnoxious committee selection, where voter satisfaction increases with distance in a metric space and the goal is to maximize the minimum satisfaction across voters using a parameterized egalitarian Chamberlin-Courant framework with distance defined by $d^{\lambda}(v,S)$. It develops a geometric reformulation for the $\lambda=1$ case in the plane that enables a polynomial-time dynamic program over arc boundaries, and shows polynomial-time solvability for $\lambda=k$ in general metric spaces, revealing a double-dichotomy in $\mathbb{R}^2$ between the two extremes. The paper also proves strong hardness results (NP-hardness and inapproximability) for intermediate $\lambda$ values, and provides a 1/4-approximation in general metric spaces along with a fixed-parameter tractable approximation scheme in Euclidean and doubling spaces. Beyond theoretical insights, these results connect obnoxious facility-location considerations with CC-type rules, offering practical implications for location planning where multiple facilities may be undesirable nearby. The combination of geometric DP techniques, net-based approximations, and parameterized algorithms yields a rich set of algorithms and complexity results across metric, planar, and doubling spaces.

Abstract

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 $1 \le λ\le k$, the committee size k, a voter derives satisfaction from only the $λ$-th 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 $\mathbb{R}^2$ and general metric space, respectively. However, for $λ= k-1$, it is NP-hard even in $\mathbb{R}^2$. Thus, we have ``double-dichotomy'' in $\mathbb{R}^2$ with respect to the value of 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 $\mathbb{R}^2$ 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.

Figures (1)

• Figure 1: Illustration for the proof of Proposition \ref{['prop:geom']} and the algorithm. Fig (a): intersection of boundaries of two unit disks is defined by two minor arcs. Fig (b): Two disjoint arcs $\mathsf{arc}(p_1, q_1, c)$ and $\mathsf{arc}(p_2, q_2, c)$ cannot appear on the boundary of a common intersection, since they correspond to disjoint regions. Fig (c): Illustration for the dynamic program. A region formed by intersection of $5$ disks is shown. $\mathsf{arc}(p, y, c)$ is shown in red. Blue region corresponds to the entry $A[x, p, z, c_1, c', 3]$, and green region corresponds to the newly added region to the blue region, corresponding to the entry $A[x, y, p, c_1, c, 4]$.

Theorems & Definitions (30)

• theorem 1
• proposition 1
• proof
• lemma 1
• lemma 2
• proof
• lemma 3
• theorem 2
• lemma 4
• proof