Does APP contain NP?

Determine whether the complexity class APP contains NP; specifically, prove or refute that NP \subseteq APP.

Background

APP is defined via comparisons of GapP functions with an input-length–dependent GapP threshold under a nonzero promise gap, and it is known to be a subclass of PP and PP-low (PPAPP = PP). However, its relationship to NP is unresolved.

Establishing whether NP \subseteq APP would clarify APP’s expressive power among classical complexity classes and inform the boundaries of PP-low classes highlighted by the paper.

References

Compared with the class ${A_0PP}={SBQP} \subseteq PP$ , ${A_0PP}$ contains $QMA$ and is not known to be $PP$-low, while $APP$ is not known to contain even $NP$ but is $PP$-low.

— Even quantum advice is unlikely to solve PP (2403.09994 - Yirka, 15 Mar 2024) in Section 2 (Preliminaries), APP discussion following Definition 'APP'