We identify a whole family of approval-based multi-winner voting rules that satisfy PJR. Moreover, we identify a subfamily of voting rules within this family that satisfy EJR. All these voting rules can be computed in polynomial time as long as the subalgorithms that characterize each rule within the family are polynomial. One of the voting rules that satisfy EJR can be computed in O(nmk).
@article{arxiv.1704.00356,
title = {Committees providing EJR can be computed efficiently},
author = {Luis Sánchez-Fernández and Edith Elkind and Martin Lackner},
journal= {arXiv preprint arXiv:1704.00356},
year = {2017}
}