Polynomial-time certification of vertex-set completeness

Determine whether deciding, for a polytope P described by linear inequalities and a finite set V of points, whether V contains all vertices of P admits a polynomial-time certification method; equivalently, determine whether the vertex-set completeness problem belongs to NP.

Background

For a polytope P given by an H-representation and a finite candidate set V, the paper notes that deciding whether V equals the vertex set of P, or even whether V contains the vertex set, is known to lie in coNP. The unresolved issue is whether these problems also belong to NP, which would amount to the existence of polynomial-size certificates verifiable in polynomial time for vertex-set exactness or completeness.

The paper’s certificate-based method provides a formally verified approach, but the cited complexity-theoretic question concerns the general existence of polynomial-time certification methods and is not resolved by the presented results. It is therefore a distinct open problem motivating the development of efficient certification techniques.