Polynomial-time computation of exterior algebraic shifting

Develop a polynomial-time algorithm for computing the exterior algebraic shift of an arbitrary finite simplicial complex, preferably a deterministic or Las Vegas algorithm that is correct over fields of arbitrary characteristic.

Background

The paper studies algorithms for computing partial and full exterior algebraic shifts of hypergraphs and simplicial complexes. Although computing a shift reduces formally to row-echelon computations, the relevant coefficient field is typically a rational-function extension containing sufficiently many algebraically independent parameters, making the arithmetic expensive.

The authors state that no polynomial-time algorithm is known for arbitrary simplicial complexes. They also recall Kalai’s question concerning a general deterministic or Las Vegas polynomial-time algorithm. The algorithms developed in the paper improve practical performance and provide verification procedures, but they do not resolve the general polynomial-time complexity problem.

References

In fact, no algorithm is known to compute the exterior shift of an arbitrary simplicial complex $K$ in polynomial time. However, in characteristic zero there is a way out. A transformation matrix with coefficients picked at random in $=$ is sufficiently generic with high probability. This yields a fast Monte--Carlo algorithm which is standard \S2.6; an implementation based on by Keehn can be found here . Kalai raised the question if there is a general deterministic or at least a Las Vegas polynomial time algorithm for exterior shifting \S2.6.

— Faster Algebraic Shifting  (2501.17908 - Vecchia et al., 29 Jan 2025) in Section 1, Introduction