Deterministic approximation for the volume of the truncated fractional matching polytope
Heng Guo, Vishvajeet N
TL;DR
This work develops a deterministic FPTAS for the volume of the truncated fractional matching polytope on graphs of maximum degree $\Delta$ (with $\delta\le C/\Delta$) and extends the result to hypergraphs with maximum edge size $k$ (with $\delta\le C\Delta^{-(2k-3)/(k-1)}k^{-1}$). The core technique is an algorithmic cluster expansion: the volume is expressed as a polymer partition function, the Kotecký–Preiss criterion is verified to ensure convergence, and weights are computed via polynomial-time polytope-volume calculations. The method generalises prior work on the truncated independence polytope and BR24, and relies on a careful polymer formulation and a hypergraph-specific MCS-based model to achieve the necessary decay. The results provide explicit truncation-parameter bounds under which a deterministic approximation is feasible, while leaving open the challenge of reducing these bounds further or removing truncation altogether.
Abstract
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.