Sub-exponential connectivity algorithms without sparse symmetric representations

Determine whether a sub-exponential time algorithm can be developed to decide connectivity between two points in a symmetric semi-algebraic set defined by symmetric polynomials when no sparse representation in power sums or elementary symmetric polynomials is available. The goal is to achieve sub-exponential complexity without relying on prior knowledge of a decomposition of the input symmetric polynomials into low-degree generators such as power sums or elementary symmetric polynomials.

Background

The paper presents polynomial-time algorithms for deciding connectivity in symmetric semi-algebraic sets when the defining symmetric polynomials have degree at most d < n, exploiting a dimension-reduction technique based on representing these polynomials using low-degree symmetric generators such as power sums.

In the conclusion, the authors note that their efficiency relies on the availability of sparse representations in these generators. They explicitly raise the question of whether similar sub-exponential running times can be achieved in settings where such sparse representations are not known, suggesting that recent algorithmic advances might be relevant. This targets extending the algorithmic reach beyond the representation-dependent regime.