Pi_2^p-completeness of Implied Integer Recognition

Prove that the decision problem Implied Integer Recognition—given a rational matrix A ∈ Q^{M×N}, a rational vector b ∈ Q^M, and index sets S,T ⊆ N, decide whether conv({x ∈ R^N : Ax ≤ b} ∩ M^S) equals conv({x ∈ R^N : Ax ≤ b} ∩ M^{S∪T}), where M^S denotes the set of vectors whose components indexed by S are integral—is Π_2^p-complete by establishing Π_2^p-hardness.

Background

The paper studies the complexity of recognizing implied integrality, generalizing the classical Integrality Recognition problem of Papadimitriou and Yannakakis. For the binary case, the authors prove coNP-completeness. For the general (non-binary) case, they establish membership in the second level of the polynomial hierarchy (Π_2p), but the corresponding hardness remains unsettled.

This conjecture asks to complete the classification by proving Π_2p-hardness, thereby matching the established upper bound and yielding Π_2p-completeness for the general Implied Integer Recognition problem.

References

While we were able to treat the binary case, our certificate does not work for the general integer case. However, we establish containment in $\Pi_2p$ and conjecture hardness.

— Implied Integrality in Mixed-Integer Optimization  (2504.07209 - Hulst et al., 9 Apr 2025) in Section 6 (Complexity of recognizing implied integers)