On the Parallelizability of Approval-Based Committee Rules

TL;DR

This paper investigates the computational complexity of parallelizing approval-based committee rules, revealing that many are P-hard to parallelize, but some can be parallelized under specific voter preference restrictions.

Contribution

It proves P-hardness for parallelizing several polynomial-time ABC rules and identifies conditions under which parallelization is feasible.

Findings

Computing committees with certain ABC rules is P-hard, preventing parallelization.

Parallelization is possible for Chamberlin-Courant under single-peaked or single-crossing preferences.

The Method of Equal Shares cannot be parallelized in large elections due to P-hardness.

Abstract

Approval-Based Committee (ABC) rules are an important tool for choosing a fair set of candidates when given the preferences of a collection of voters. Though finding a winning committee for many ABC rules is NP-hard, natural variations for these rules with polynomial-time algorithms exist. The recently introduced Method of Equal Shares, an important ABC rule with desirable properties, is also computable in polynomial time. However, when working with very large elections, polynomial time is not enough and parallelization may be necessary. We show that computing a winning committee using these polynomial-time ABC rules (including the Method of Equal Shares) is P-hard, thus showing they cannot be parallelized. In contrast, we show that finding a winning committee can be parallelized when the votes are single-peaked or single-crossing for the important ABC rule Chamberlin-Courant.