Separations in the polynomial hierarchy

Ascertain whether the polynomial hierarchy separates at each level; specifically, prove or refute for k ∈ ℕ that Σ_k^p ≠ Σ_{k+1}^p (and analogously Π_k^p ≠ Π_{k+1}^p), or determine whether the hierarchy collapses at some level.

Background

In discussing computational complexity background relevant to bilevel optimization hardness, the paper recalls a central open question in complexity theory: whether the polynomial hierarchy is strict at each level or collapses.

This question underpins the interpretation of Σ_2^p-hardness results, since separations between levels imply that such problems are strictly harder than NP-complete problems.

References

For any $ k\in \NN$, it remains unknown whether $\Sigma_k^p \neq \Sigma_{k+1}^p$ or $\Sigma_k^p = \Sigma_{k+1}^p$, and similarly for $\Pi_{k}^p$ and $\Pi_{k+1}^p$.

— Geometric and computational hardness of bilevel programming (2407.12372 - Bolte et al., 17 Jul 2024) in Section 4.1 (Polynomial hierarchy)

Separations in the polynomial hierarchy

Background

References

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