NP membership of realizable chirotope recognition

Determine whether the decision problem of recognizing realizable chirotopes is in NP. Formally, given a function χ: (X)₃ → {−1,+1}, ascertain whether there exists a planar point set in general position realizing χ; equivalently, establish whether there exists a polynomial‑time pseudorandom generator that outputs realizable chirotopes of size n with nonzero probability for every instance.

Background

The recognition problem asks, for a given orientation map on triples, whether there exists a planar point set in general position that realizes it as a chirotope. This decision problem is known to be equivalent to the existential theory of the reals and NP-hard, underscoring its computational difficulty.

The authors note that NP membership remains unresolved; moreover, a positive resolution would imply the existence of a polynomial‑time pseudorandom generator that samples realizable chirotopes of size n with nonzero probability for every instance, highlighting the broader algorithmic significance of the question.

References

Deciding if a given function $(X)_3 \to {-1,1}$ is a realizable chirotope turns out to be a challenging problem: it is equivalent to the existential theory of the realsTheorem 8.7.2 and NP-hard. Whether this problem is in NP is open, and, interestingly, a positive answer is equivalent to the existence of a (pseudorandom) generator that produces a realizable chirotope of size $n$ in time polynomial in $n$ while ensuring that every element has nonzero probability.

— A canonical tree decomposition for order types, and some applications  (2403.10311 - Bouvel et al., 2024) in Subsubsection “Representing realizable chirotopes” within Section “Context and motivation”