Determine the exact complexity of OSCS optimization

Determine whether deciding whether a cooperative integer programming game admits a stable coalition structure with total payoff at least a given threshold is a2^p-complete, or otherwise characterize the exact computational complexity of the OSCS problem.

Background

The paper proves that deciding whether an SCS with total payoff at least a threshold exists belongs to the second level of the polynomial hierarchy, a2^p, and is NP-hard. The upper bound follows from existentially selecting a coalition structure and payoff vector and verifying stability through an oracle-level procedure.

The exact classification is not established. The authors identify a2^p-completeness as the natural conjecture, while noting that existing hardness constructions for core problems over coalition structures rely on polynomial-time worth functions and do not directly reach the second level when coalition values are generated by integer programs.