Polynomial-time verification of PMMS allocations

Determine whether verifying that a given allocation satisfies the PMMS condition for additive goods or additive chores is solvable in polynomial time, despite the stated coNP-completeness of the decision problem.

Background

The paper proves that deciding whether an additive-goods or additive-chores instance admits a PMMS allocation is NP-hard. It also observes that the corresponding verification problem—checking whether a specified allocation satisfies all PMMS inequalities—requires computing pairwise maximin-share benchmarks.

The computational status of this verification task is explicitly unresolved in the sense that it is not known to be polynomial-time solvable; the paper further notes that the decision problem is coNP-complete. Establishing an efficient verification algorithm, or otherwise clarifying the precise complexity under the relevant input representation, remains an open complexity question.