Deterministic FPTAS at the Dirac threshold

Establish whether a deterministic fully polynomial-time approximation scheme exists for the number of perfect matchings in every graph of minimum degree at least half its order, including the exact graph Dirac threshold.

Background

The main matching algorithm requires a fixed positive margin above the Dirac minimum-degree threshold. At the endpoint, entropy scaling can assign order-one weight to an edge, violating the diffuseness condition needed by the truncation argument; scaled eigenvalues can also approach -1.

The paper gives a deterministic FPTAS for a structured endpoint family, but leaves the endpoint problem unresolved for arbitrary Dirac graphs.