Justifying Groups in Multiwinner Approval Voting

TL;DR

This paper investigates the existence and computation of small groups that justify representation in multiwinner approval voting, providing theoretical insights and algorithms to find such groups efficiently.

Contribution

It introduces new theoretical bounds on the size of justifying groups, and presents algorithms for computing small, effective groups in approval voting.

Findings

Small $n/k$-justifying groups are likely to exist under the impartial culture model.

Efficient approximation algorithms can find small $n/k$-justifying groups.

Small groups can help achieve gender-balanced committees despite NP-hardness.

Abstract

Justified representation (JR) is a standard notion of representation in multiwinner approval voting. Not only does a JR committee always exist, but previous work has also shown through experiments that the JR condition can typically be fulfilled by groups of fewer than $k$ candidates. In this paper, we study such groups -- known as $n/k$-justifying groups -- both theoretically and empirically. First, we show that under the impartial culture model, $n/k$-justifying groups of size less than $k/2$ are likely to exist, which implies that the number of JR committees is usually large. We then present efficient approximation algorithms that compute a small $n/k$-justifying group for any given instance, and a polynomial-time exact algorithm when the instance admits a tree representation. In addition, we demonstrate that small $n/k$-justifying groups can often be useful for obtaining a gender-balanced JR committee even though the problem is NP-hard.