Counting hypergraph colorings in the local lemma regime

TL;DR

This paper presents the first efficient algorithms for approximately counting and sampling hypergraph colorings in the local lemma regime, extending previous work by removing dependencies on hypergraph parameters.

Contribution

It introduces an FPTAS and an almost uniform sampler for hypergraph colorings, removing the dependency on hypergraph parameters in prior approaches.

Findings

First FPTAS for hypergraph colorings in the local lemma regime

First polynomial-time sampler in the regime q << Δ

Removes dependency of parameters k and Δ in previous methods

Abstract

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of $q$-colorings for $k$-uniform hypergraphs with maximum degree $\Delta$ if $k\ge 28$ and $q >357\Delta^{\frac{14}{k-14}}$ . We also obtain a polynomial-time almost uniform sampler if $q>931\Delta^{\frac{16}{k-16/3}}$. These are the first approximate counting and sampling algorithms in the regime $q\ll\Delta$ (for large $\Delta$ and $k$) without any additional assumptions. Our method is based on the recent work of Moitra (STOC, 2017). One important contribution of ours is to remove the dependency of $k$ and $\Delta$ in Moitra's approach.