Weighted FPTAS below the half-degree threshold

Develop a deterministic fully polynomial-time approximation scheme for weighted perfect-matching counts on pseudorandom graph supports below the half-degree threshold throughout the degree–codegree range covered by the unweighted pseudorandom-support result.

Background

The paper proves a deterministic FPTAS for unweighted pseudorandom supports whose density may be below one half, under degree, codegree, nonbipartiteness, and edge-extendability conditions.

It explicitly notes that the corresponding weighted result is not obtained over the full degree–codegree parameter range. Extending the entropy-scaling and diffuse-perturbation analysis to weighted inputs is therefore left unresolved.

References

Below the half-degree threshold, \cref{cor:pseudorandom-supports} controls unweighted pseudorandom supports, but the analogous weighted statement remains open in the full degree--codegree range.

— Diffuse Gaussian Truncation For Deterministic Approximate Counting  (2609.04079 - Yi, 3 Sep 2026) in Section 6, Scope and open problems, overviewstage 2 (Perfect matchings)