Improved bounds for randomly colouring simple hypergraphs

TL;DR

This paper presents improved bounds for efficiently sampling nearly uniform proper q-colorings in simple hypergraphs, reducing the number of colors needed compared to previous methods and achieving faster algorithms.

Contribution

The authors develop a more efficient sampling algorithm for hypergraph colorings that requires fewer colors and has improved running time bounds over prior work.

Findings

Requires fewer colors than previous bounds

Achieves near-linear time complexity in the number of vertices

Applicable to hypergraphs with bounded degree and uniformity

Abstract

We study the problem of sampling almost uniform proper $q$-colourings in $k$-uniform simple hypergraphs with maximum degree $\Delta$. For any $\delta > 0$, if $k \geq\frac{20(1+\delta)}{\delta}$ and $q \geq 100\Delta^{\frac{2+\delta}{k-4/\delta-4}}$, the running time of our algorithm is $\tilde{O}(\mathrm{poly}(\Delta k)\cdot n^{1.01})$, where $n$ is the number of vertices. Our result requires fewer colours than previous results for general hypergraphs (Jain, Pham, and Voung, 2021; He, Sun, and Wu, 2021), and does not require $\Omega(\log n)$ colours unlike the work of Frieze and Anastos (2017).