General polynomial-time algorithm for EF1+PO for both goods and chores

Develop a polynomial-time algorithm that, given a fair division instance with additive valuations (for goods) or additive costs (for chores), computes an allocation that is envy-free up to one item (for goods) or one chore (for chores) and Pareto optimal.

Background

This paper proves the existence of EF1+PO allocations for indivisible chores under additive costs and provides polynomial-time algorithms when the number of agents is constant. Nonetheless, the broader algorithmic problem of finding EF1+PO allocations in general, for both goods and chores, is not settled.

The concluding section explicitly frames the design of such a general polynomial-time algorithm as a major open question, extending the algorithmic challenge beyond goods to chores as well.

References

A major open question is whether one can design a polynomial-time algorithm to compute EF1 and PO allocations for both goods and chores.

— Existence of Fair and Efficient Allocation of Indivisible Chores  (2507.09544 - Mahara, 13 Jul 2025) in Section 7 Conclusion

The existence of an EF1 and PO allocation remains open even for unweighted mixed manna~\citep{AzizEtAl26BoBW}.

— Weighted Fair Division of Indivisible Mixed Manna  (2609.01580 - Teh, 1 Sep 2026) in Section 1, Related Work, paragraph “Other fairness and efficiency results for mixed manna”