Fault Tolerance in Euclidean Committee Selection

TL;DR

This paper investigates fault-tolerance in Euclidean committee selection, proposing algorithms and complexity results for replacing failed candidates and optimizing committee robustness in multi-dimensional spaces.

Contribution

It introduces the first complexity and approximation results for fault-tolerant committee selection in Euclidean spaces, including polynomial algorithms in one dimension and NP-hardness in higher dimensions.

Findings

Polynomial-time solutions in 1D for all problems

NP-hardness in dimensions ≥ 2

Constant-factor approximation algorithms in fixed dimensions

Abstract

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.