Deterministic approximation for the volume of the truncated fractional matching polytope

TL;DR

This paper presents a deterministic polynomial-time approximation scheme for calculating the volume of truncated fractional matching polytopes in graphs and hypergraphs, extending previous results and employing cluster expansion techniques.

Contribution

It introduces a novel FPTAS for the volume of truncated fractional matching polytopes in graphs and hypergraphs, generalizing prior work and applying cluster expansion methods.

Findings

Provides a polynomial-time approximation scheme for graphs of maximum degree Δ.

Extends the approximation scheme to hypergraphs with maximum degree Δ and hyperedge size k.

Generalizes previous results on the truncated independence polytope.

Abstract

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