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 $\mathrm{\Delta }$, where the truncation is by restricting each variable to the interval $[0,\frac{1+\delta }{\mathrm{\Delta }}]$, and $\delta \le \frac{C}{\mathrm{\Delta }}$ for some constant $C>0$. We also generalise our result to the fractional matching polytope for hypergraphs of maximum degree $\mathrm{\Delta }$ and maximum hyperedge size $k$, truncated by $[0,\frac{1+\delta }{\mathrm{\Delta }}]$ as well, where $\delta \le C{\mathrm{\Delta }}^{-\frac{2k-3}{k-1}}{k}^{-1}$ 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 $\mathrm{\Delta }=2$). Our approach is based on the cluster expansion technique.