Polynomial-time recognition of generic realisability for paving matroids

Determine whether there exists an algorithm polynomial in the ground-set size n and rank d that decides whether a rank-d paving matroid on [n] is generically realisable.

Background

The paper gives a polynomial-time procedure, with respect to n and d, for deciding whether a rank-d paving matroid is liftable by reducing the problem to the rank of a d-uniform hypergraph in a sparsity matroid. In contrast, generic realisability is characterized by a flatness condition in the relevant sparsity matroid, equivalently by a (1,d)-sparsity condition for the associated hypergraph in the sparse paving case.

The authors explicitly leave unresolved whether generic realisability can be recognized in polynomial time. They note that deciding (1,d)-sparsity is NP-hard with respect to n and d, suggesting that a polynomial-time algorithm is unlikely, but this complexity-theoretic expectation does not resolve the stated algorithmic question.

References

It is currently open whether there exists a polynomial-time algorithm (in n and d) for determining if the same such matroid is generically realisable.

— Matroid liftability and Oxley--Wang derived matroids, with connections to scene analysis  (2610.01854 - Dewar et al., 1 Oct 2026) in Section 1, subsection “Our contribution,” under “Generic realisability for paving matroids”