Problem
Research questions and friction points this paper is trying to address.
Develop distributed MCMC algorithms for graph coloring
Improve convergence bounds for k-colorings in graphs
Analyze asymmetric Markov chains in distributed settings
Improved Distributed Algorithms for Random Colorings
🤖 AI Summary
This paper studies the problem of sampling random $k$-colorings of graphs with maximum degree $Delta$ in the distributed computing model. For $k > (11/6 - delta)Delta$ with $delta > 0$, we present the first parallel distributed Markov chain algorithm based on Vigoda’s flip dynamics—breaking the prior restriction to Glauber dynamics. Our method generalizes the flip operation to recoloring locally maximal bichromatic connected components and introduces a lightweight message-synchronization protocol. The algorithm achieves rapid mixing within $O(log n)$ communication rounds, substantially improving upon prior distributed results requiring $k > (2 + varepsilon)Delta$. It is the first distributed algorithm to match the current best sequential lower bound on $k$, thereby attaining state-of-the-art theoretical performance.
📝 Abstract
Markov Chain Monte Carlo (MCMC) algorithms are a widely-used algorithmic tool for sampling from high-dimensional distributions, a notable example is the equilibirum distribution of graphical models. The Glauber dynamics, also known as the Gibbs sampler, is the simplest example of an MCMC algorithm; the transitions of the chain update the configuration at a randomly chosen coordinate at each step. Several works have studied distributed versions of the Glauber dynamics and we extend these efforts to a more general family of Markov chains. An important combinatorial problem in the study of MCMC algorithms is random colorings. Given a graph $G$ of maximum degree $Delta$ and an integer $kgeqDelta+1$, the goal is to generate a random proper vertex $k$-coloring of $G$. Jerrum (1995) proved that the Glauber dynamics has $O(nlog{n})$ mixing time when $k>2Delta$. Fischer and Ghaffari (2018), and independently Feng, Hayes, and Yin (2018), presented a parallel and distributed version of the Glauber dynamics which converges in $O(log{n})$ rounds for $k>(2+varepsilon)Delta$ for any $varepsilon>0$. We improve this result to $k>(11/6-delta)Delta$ for a fixed $delta>0$. This matches the state of the art for randomly sampling colorings of general graphs in the sequential setting. Whereas previous works focused on distributed variants of the Glauber dynamics, our work presents a parallel and distributed version of the more general flip dynamics presented by Vigoda (2000) (and refined by Chen, Delcourt, Moitra, Perarnau, and Postle (2019)), which recolors local maximal two-colored components in each step.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Distributed flip dynamics for colorings
Parallel MCMC with local component updates
Non-symmetric Markov chain for efficiency
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