Flip Dynamics for Sampling Colorings: Improving $(11/6-\epsilon)$ Using a Simple Metric

TL;DR

This paper improves the bounds for sampling proper colorings of graphs using flip dynamics, achieving optimal mixing times for a broader range of color counts relative to maximum degree, advancing understanding of Markov chain convergence.

Contribution

It establishes the first significant improvement over previous bounds, proving optimal mixing for flip dynamics when the number of colors is at least 1.809 times the maximum degree.

Findings

Achieves $O(n ext{log}n)$ mixing time for flip dynamics at $k ext{ } ext{geq} ext{ } 1.809 imes ext{ } ext{max degree}$

Extends spectral independence results to Glauber dynamics for bounded degree graphs

Utilizes path coupling with a weighted Hamming distance for analysis

Abstract

We present improved bounds for randomly sampling $k$-colorings of graphs with maximum degree $\Delta$; our results hold without any further assumptions on the graph. The Glauber dynamics is a simple single-site update Markov chain. Jerrum (1995) proved an optimal $O(n\log{n})$ mixing time bound for Glauber dynamics whenever $k>2\Delta$ where $\Delta$ is the maximum degree of the input graph. This bound was improved by Vigoda (1999) to $k > (11/6)\Delta$ using a "flip" dynamics which recolors (small) maximal 2-colored components in each step. Vigoda's result was the best known for general graphs for 20 years until Chen et al. (2019) established optimal mixing of the flip dynamics for $k > (11/6 - \epsilon ) \Delta$ where $\epsilon \approx 10^{-5}$. We present the first substantial improvement over these results. We prove an optimal mixing time bound of $O(n\log{n})$ for the flip dynamics when $k \geq 1.809 \Delta$. This yields, through recent spectral independence results, an optimal $O(n\log{n})$ mixing time for the Glauber dynamics for the same range of $k/\Delta$ when $\Delta=O(1)$. Our proof utilizes path coupling with a simple weighted Hamming distance for "unblocked" neighbors.