Multiwinner Elections under Minimax Chamberlin-Courant Rule in Euclidean Space

TL;DR

This paper studies the computational complexity of multiwinner elections in Euclidean space under the minimax Chamberlin-Courant rule, proving NP-hardness and providing approximation schemes with tight bounds.

Contribution

It introduces three polynomial-time approximation schemes for the problem, demonstrating their effectiveness and tightness, especially under the 1-Borda rule.

Findings

NP-hardness in dimensions d ≥ 2

Existence of three approximation schemes

Approximation bounds are tight or nearly tight

Abstract

We consider multiwinner elections in Euclidean space using the minimax Chamberlin-Courant rule. In this setting, voters and candidates are embedded in a $d$-dimensional Euclidean space, and the goal is to choose a committee of $k$ candidates so that the rank of any voter's most preferred candidate in the committee is minimized. (The problem is also equivalent to the ordinal version of the classical $k$-center problem.) We show that the problem is NP-hard in any dimension $d \geq 2$, and also provably hard to approximate. Our main results are three polynomial-time approximation schemes, each of which finds a committee with provably good minimax score. In all cases, we show that our approximation bounds are tight or close to tight. We mainly focus on the $1$-Borda rule but some of our results also hold for the more general $r$-Borda.