Polynomial-time algorithm for independence in the bipartite rigidity matroid B_{m,n}(2,2)

Develop a polynomial-time algorithm to decide whether a given bipartite graph is independent in the bipartite rigidity matroid \mathcal{B}_{m,n}(2,2).

Background

Bernstein’s theorem gives a combinatorial characterization of independence in \mathcal{B}_{m,n}(2,2) via edge orientations without directed or alternating cycles. Despite this structural description, the authors note that no polynomial-time algorithm is known to test independence in this matroid.

This computational problem sits at the intersection of rigidity theory, matroid theory, and algorithmic graph theory, and resolving it would clarify the complexity of a central case of bipartite rigidity.

References

Despite the combinatorial nature of the description in Theorem~\ref{thm:Bernstein}, we do not know a polynomial time algorithm to check independence in $\mathcal{B}_{m,n}(2,2)$.

— Rigidity matroids and linear algebraic matroids with applications to matrix completion and tensor codes (2405.00778 - Brakensiek et al., 1 May 2024) in Introduction (following Proposition 1.7)