Sampling Colorings Close to the Maximum Degree: Non-Markovian Coupling and Local Uniformity

2026-04-13

AI Summary

This work investigates the problem of Markov Chain Monte Carlo (MCMC) sampling for $k$-colorings of graphs with maximum degree $\Delta$, focusing on the long-standing open question of whether Glauber dynamics exhibits polynomial mixing time when $k$ is close to $\Delta + 2$. For graphs of girth at least 11, the authors introduce a refined local non-Markovian coupling technique combined with a novel local uniformity analysis under Metropolis dynamics. This approach constitutes the first complete application of non-Markovian coupling to the constant-degree setting, overcoming the limitation of traditional methods that suffer from a constant failure probability on low-degree graphs. They prove that when $k \geq (1+\delta)\Delta$ for any fixed $\delta > 0$ and sufficiently large $\Delta$, Glauber dynamics achieves the optimal mixing time of $O(|V| \log |V|)$, thereby establishing a unified analytical framework that bridges the gap between high-degree and constant-degree regimes.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

Sampling graph colorings via local Markov chains is a central problem in approximate counting and Markov chain Monte Carlo (MCMC). We address the problem of sampling a random $k$-coloring of a graph with maximum degree $Δ$. The simplest algorithmic approach is to establish rapid mixing of the single-site update chain known as the Metropolis Glauber dynamics, which at each step chooses a random vertex $v$ and proposes a random color $c$, recoloring $v$ to $c$ if the resulting coloring remains proper. It is a long-standing open problem to prove that the Glauber dynamics has polynomial mixing time on all graphs whenever $k\geqΔ+2$. We prove that for every $δ>0$ and all $Δ\geq Δ_0(δ)$, if $k\ge (1+δ)Δ$ then the Glauber dynamics has optimal mixing time of $O_δ(|V| \log |V|)$ on any graph of girth $\geq 11$ and maximum degree $Δ$. Our approach builds on a non-Markovian coupling introduced by Hayes and Vigoda (2003) for the large-degree regime $Δ=Ω(\log n)$, in which updates at time $t$ may depend on and modify proposed updates at future times. A complete analysis of this framework requires resolving substantial technical obstacles that remain in the original argument, and extending it to the constant-degree regime introduces further difficulties, since non-Markovian updates may fail with constant probability. We overcome these obstacles by developing and analyzing a refined local non-Markovian coupling, and by establishing new local-uniformity results for the Metropolis dynamics, extending prior results for the heat-bath chain due to Hayes (2013). Together, these ingredients provide a complete analysis of the non-Markovian coupling framework in the large-degree regime, while simultaneously strengthening it substantially to obtain optimal mixing all the way down to the constant-degree setting.

Problem

• graph coloring
• Glauber dynamics
• mixing time
• maximum degree
• Markov chain Monte Carlo

Innovation

• non-Markovian coupling
• local uniformity
• Glauber dynamics
• graph coloring
• mixing time