Randomized FPRAS for general nonbipartite perfect matchings

Establish a fully polynomial randomized approximation scheme for the number of perfect matchings in arbitrary general nonbipartite graphs.

Background

The paper distinguishes its deterministic FPTAS results for dense graph classes from the broader approximate-counting problem. Exact counting is hard even for restricted dense instances, whereas randomized approximation is known in some settings, including graphs with minimum degree at least half the order and regular strong expanders.

The status of an FPRAS for arbitrary general nonbipartite graphs is explicitly identified as unresolved in the discussion of prior work.

References

For general nonbipartite graphs, the existence of an FPRAS remains open, although positive results cover several restricted families.

— Hidden Circuits and Exact Counting in Ordered Graphs  (2609.18132 - Liu et al., 16 Sep 2026) in Section 1, Introduction

For general nonbipartite graphs, an FPRAS for the number of perfect matchings remains open.

— Diffuse Gaussian Truncation For Deterministic Approximate Counting  (2609.04079 - Yi, 3 Sep 2026) in Section 1, Introduction, subsection “Previous work,” paragraph “Randomized matching algorithms”