Rapid Mixing of Hypergraph Independent Set

TL;DR

This paper proves that Glauber dynamics rapidly mixes for sampling independent sets in hypergraphs with certain degree bounds, narrowing the gap between algorithmic efficiency and computational hardness.

Contribution

It improves the known bounds on hypergraph degree for rapid mixing of Glauber dynamics, approaching the theoretical hardness threshold.

Findings

Mixing time is $O(n ext{log} n)$ under specified degree conditions.

Improves previous bounds from $ riangle ext{max degree} extless k-2$ to $ riangle extless c 2^{k/2}$.

Bridges the gap between efficient sampling algorithms and NP-hardness results.

Abstract

We prove that the the mixing time of the Glauber dynamics for sampling independent sets on $n$-vertex $k$-uniform hypergraphs is $O(n\log n)$ when the maximum degree $\Delta$ satisfies $\Delta \leq c 2^{k/2}$, improving on the previous bound [BDK06] of $\Delta \leq k-2$. This result brings the algorithmic bound to within a constant factor of the hardness bound of [BGG+16] which showed that it is NP-hard to approximately count independent sets on hypergraphs when $\Delta \geq 5 \cdot 2^{k/2}$.