Membership in NP of simultaneous PROPm and Pareto-optimal orientation existence

Determine whether deciding the existence of an orientation that is simultaneously PROPm and Pareto optimal belongs to the complexity class NP.

Background

The paper proves that, for additive goods valuations, deciding whether a graphical orientation is both PROPm and Pareto optimal is NP-hard, even when all valuation values belong to {0,1,2,4}. However, the paper does not establish that the problem is in NP. PROPm can be verified in polynomial time for a given orientation, but verifying Pareto optimality is coNP-complete in general and remains weakly coNP-complete even for two agents. Consequently, the precise complexity classification of the simultaneous PROPm-and-Pareto-optimality existence problem remains unresolved.

References

It is not clear whether the above decision problem belongs to NP.

— Settling PROPm and PROPavg in Graphical Resource Allocation  (2610.10336 - Li et al., 7 Oct 2026) in Section 3.3, “Compatibility of PROPm with Pareto Optimality”