On the Complexity of Extended and Proportional Justified Representation

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, Sánchez-Fernández 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 fixedparameter tractability results. Introduction Consider an election where voters have simple preferences: each voter approves some of the candidates and disapproves the remaining candidates; she is indifferent among the candidates in each group. Suppose that the goal is to select a fixed-size set of winners, or committee. This model captures a number of applications: the candidates could be potential members of a governing body, items to be shown on a seller’s homepage, or tunes to be played at a wedding. Accordingly, there is a number of natural voting rules that take approval ballots as their input and output a set of committees that are tied for winning (Kilgour 2010; Brams and Fishburn 2007; LeGrand, Markakis, and Mehta 2007; Aziz et al. 2015; Skowron, Faliszewski, and Lang 2016; Sánchez-Fernández, Fernández, and Fisteus 2016; Brill et al. 2017). Many of these voting rules attempt to ensure that all groups of voters are fairly represented in the selected committee. However, it has been far from clear how to best capture the representation requirements. Aziz et al. (2017) proposed a compelling representation axiom called justified representation (JR), as well as a Copyright c © 2018, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. stronger version of this axiom called extended justified representation (EJR). Intuitively, the JR axiom says that every sufficiently large group of voters who jointly approve at least one candidate should be represented in a committee; EJR additionally requires that very large groups whose preferences exhibit significant agreement should be allocated several representatives. While EJR is a rather demanding axiom, every election admits a committee that provides EJR. Aziz et al. (2017) show that it is easy to check if a given committee provides JR, and to find a committee with this property; indeed, many common approval-based voting rules are guaranteed to output committees that provide JR. The EJR axiom is considerably more challenging from a computational perspective: among the voting rules considered in the literature, there is only one rule, namely, Proportional Approval Voting (PAV) that satisfies EJR (in the sense that every committee output by PAV provides EJR), and computing the winners under this rule is NP-hard (Aziz et al. 2015). Moreover, Aziz et al. (2017) show that it is coNP-complete to check if a given committee provides EJR. Sánchez-Fernández et al. (2017) put forward an intermediate property called proportional justified representation (PJR): every committee that provides EJR also provides PJR, and every committee that provides PJR also provides JR, but converse implications are not true. An attractive property of PJR is that it is compatible with another important axiom called perfect representation, while for EJR this is not the case. Sánchez-Fernández et al. (2017) argued that two well-studied approval-based committee selection rules satisfy PJR when the target committee size k divides the number of voters n; one of these rules is polynomial-time computable. Other authors identified two polynomial-time computable rules that satisfy PJR for all values of k and n (Brill et al. 2017; Sánchez-Fernández, Fernández, and Fisteus 2016). However, the complexity of checking whether a given committee provides PJR remained open. Thus, the existing results seemed to suggest that, from a computational perspective, PJR is more tractable than EJR. Contributions In this paper, we resolve two main open problems regarding the computational complexity of EJR and PJR. First, we present two polynomial-time algorithms that output committees satisfying EJR. Our first algorithm is a simple local search algorithm that looks for committees with approximately optimal PAV scores (recall that PAV is the only voting rule that was known to satisfy EJR prior to our work). It exploits an interesting connection between extended justified representation and another important concept called average satisfaction (Sánchez-Fernández et al. 2017). Our second algorithm can be viewed as a variant of Single Transferable Vote with fractional vote transfers, adapted to the setting of approval ballots. This algorithm proceeds iteratively, adding candidates to the committee one by one and adjusting the voters’ weights according to how well they are represented by the already selected candidates. Second, we settle the complexity of testing PJR: we prove that checking whether a given committee provides PJR is coNP-complete. We complement this complexity result by showing that PJR and EJR can be tested efficiently if any of the following parameters are small: (1) n (number of voters) (2) m (number of candidates) (3) a (maximum number of candidates approved by a voter) (4) d (maximum number of voters approving a given candidate). More specifically, we provide FPT results for n and m, and XP results for a and d. Preliminaries An election is a pair E = (N,C), where N = {1, . . . , n} is a set of voters and C = {c1, . . . , cm} is a set of candidates. Each voter i ∈ N is associated with an approval ballotAi ⊆ C: this is the set of candidates approved by i. For each c ∈ C, we write Nc = {i ∈ N | c ∈ Ai}. We are interested in procedures that, given an election E and a positive integer k, 1 ≤ k ≤ |C|, output a non-empty collection of size-k subsets of candidates; such procedures are called committee selection rules. Given an election E = (N,C), we define the PAV-score of a committee W ⊆ C as