Polynomial-time algorithm for EF1+PO (or EF1+fPO) with indivisible goods

Develop a polynomial-time algorithm that, given a fair division instance of indivisible goods with additive valuations, computes an allocation that is envy-free up to one item (EF1) and Pareto optimal (PO), or computes an allocation that is EF1 and fractionally Pareto optimal (fPO).

Background

In the setting of indivisible goods with additive valuations, EF1 allocations are known to exist and Pareto-optimal allocations can be found via fractional Pareto optimality. Moreover, maximizing Nash social welfare yields EF1+PO allocations but is NP-hard, and prior work provides pseudo-polynomial or constant-agent polynomial-time methods. Despite this progress, a general polynomial-time algorithm for simultaneously achieving EF1 and PO (or EF1 and fPO) remains elusive.

This paper resolves the existence question for chores and provides polynomial-time algorithms when the number of agents is constant, but it does not close the general algorithmic question for goods. The quoted sentence highlights that obtaining a polynomial-time algorithm for EF1 combined with PO (or fPO) is still an open problem.