On the Complexity of Extended and Proportional Justified Representation

Abstract

We consider the problem of selecting a fixed-size committee based on approval ballots. It is desirable to have a committee in which all voters are fairly represented. Aziz et al. (2015a; 2017) proposed an axiom called extended justified representation (EJR), which aims to capture this intuition; subsequently, Sanchez-Fernandez et al. (2017) proposed a weaker variant of this axiom called proportional justified representation (PJR). It was shown that it is coNP-complete to check whether a given committee provides EJR, and it was conjectured that it is hard to find a committee that provides EJR. In contrast, there are polynomial-time computable voting rules that output committees providing PJR, but the complexity of checking whether a given committee provides PJR was an open problem. In this paper, we answer open questions from prior work by showing that EJR and PJR have the same worst-case complexity: we provide two polynomial-time algorithms that output committees providing EJR, yet we show that it is coNP-complete to decide whether a given committee provides PJR. We complement the latter result by fixed-parameter tractability results.

The Complexity of Proportionality Degree in Committee Elections

“…Proportionality Degree. We mostly focus on the notion of a proportionality degree of a committee, introduced by Aziz et al (2018). Let us consider some voter v and a committee S. We define v's satisfaction with S as |A(v)∩S|, i.e., the number of committee members that v approves.…”

“…As a consequence, seeking formal understanding of proportionality in multiwinner elections is among the most active branches of computational social choice (Lackner and Skowron 2020). In this paper we extend this line of work by analyzing the computational complexity of one of the recent measures of proportionality, the proportionality degree (Aziz et al 2018).…”

“…The key to the success of this notion is that committees providing EJR always exist and are selected by voting rules designed to provide proportional representation, such as the proportional approval voting rule (PAV) of Thiele (1895). Thus many researchers followed Aziz et al (2017) either in defining new variants of the justified representation axioms (Sánchez-Fernández et al 2017;Peters, Pierczynski, and Skowron 2020) or in analyzing and designing rules that would provide committees satsifying these properties (Aziz et al 2018;Sánchez-Fernández et al 2021;Brill et al 2017). Others study these notions experimentally (Bredereck et al 2019).…”

“…After all, to provide them it suffices that a single member of each cohesive group approves enough committee members, irrespective of all the other voters. To address this issue, Aziz et al (2018) introduced the notion of proportionality degree (PD). They said that the satisfaction of a voter is equal to the number of committee members that he or she approves and they considered average satisfactions of the voters in cohesive groups.…”

“…So, establishing the PD of a rule gives quantitative understanding of its proportionality, whereas JR and EJR only give qualitative information. That said, Aziz et al (2018) did show that if a committee provides EJR then it has PD at least f ( ) = −1 /2, so the two notions are related. Further, they showed that PAV committees have PD f ( ) = − 1 and Skowron (2021) established good bounds on the PDs of numerous other proportionality-oriented rules.…”