A Perfect Sampler for Hypergraph Independent Sets

TL;DR

This paper introduces an efficient perfect sampling algorithm for hypergraph independent sets that operates under conditions similar to the Lovász Local Lemma, with improved bounds for regular and linear hypergraphs.

Contribution

The paper presents a novel perfect sampler for hypergraph independent sets with expected polynomial runtime under specific degree conditions, extending previous rapid mixing results.

Findings

Expected $O(n ext{log} n)$ runtime for regular hypergraphs

Weakened degree condition for linear hypergraphs

Matching the rapid mixing condition for Glauber dynamics

Abstract

The problem of uniformly sampling hypergraph independent sets is revisited. We design an efficient perfect sampler for the problem under a condition similar to that of the asymmetric Lov\'asz Local Lemma. When applied to $d$-regular $k$-uniform hypergraphs on $n$ vertices, our sampler terminates in expected $O(n\log n)$ time provided $d\le c\cdot 2^{k/2}/k$ for some constant $c>0$. If in addition the hypergraph is linear, the condition can be weaken to $d\le c\cdot 2^{k}/k^2$ for some constant $c>0$, matching the rapid mixing condition for Glauber dynamics in Hermon, Sly and Zhang [HSZ19].