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 (LIR), which aims to capture this intuition; subsequently, Sanchez-Fernandez et al. (2017) proposed a weaker variant of this axiom called proportional justified representation (RIR). 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.