Deterministic approximate counting without degree restrictions

Construct a deterministic fully polynomial-time approximation scheme for the monomer–dimer partition function $Z_G(\lambda)$ on arbitrary simple graphs, with running time polynomial in $n$ and $1/\varepsilon$ and no dependence on a bounded maximum-degree assumption.

Background

The paper provides a randomized approximate counting algorithm and notes that deterministic approximation schemes are known for bounded-degree graphs, while only subexponential-time approximation is available in general graphs.

The stated problem is to remove the degree restriction while retaining polynomial dependence on the graph size and accuracy parameter. The discussion suggests that a deterministic local sampler under feasible pinnings, together with controlled truncation and enumeration, could provide a route, but uniform control in high degree remains unresolved.

References

Can $Z_G(\lambda)$ be approximated deterministically within relative error $\varepsilon$ in time polynomial in $n$ and $1/\varepsilon$, without a degree restriction?

— Sampling Matchings in Near-linear Time  (2609.21936 - Miao et al., 18 Sep 2026) in Section 7, Discussion and open problems, paragraph “Deterministic approximate counting”