Deterministic approximation for the volume of the truncated fractional matching polytope

We give a deterministic polynomial-time approximation scheme (FPTAS) for the volume of the truncated fractional matching polytope for graphs of maximum degree $Δ$, where the truncation is by restricting each variable to the interval $[0,\frac{1+δ}Δ]$, and $δ\le \frac{C}Δ$ for some constant $C>0$. We also generalise our result to the fractional matching polytope for hypergraphs of maximum degree $Δ$ and maximum hyperedge size $k$, truncated by $[0,\frac{1+δ}Δ]$ as well, where $δ\le CΔ^{-\frac{2k-3}{k-1}}k^{-1}$ for some constant $C>0$. The latter result generalises both the first result for graphs (when $k=2$), and a result by Bencs and Regts (2024) for the truncated independence polytope (when $Δ=2$). Our approach is based on the cluster expansion technique.