  In this short note, we describe an approval-based committee selection rule
that admits a polynomial-time algorithm and satisfies the Extended Justified
Representation (EJR) axiom. This rule is based on approximately maximizing the
PAV score, by means of local search. Our proof strategy is to show that this
rule provides almost optimal average satisfaction to all cohesive groups of
voters, and that high average satisfaction for cohesive groups implies extended
justified representation.
