Improved Distributed Algorithms for Random Colorings

TL;DR

This paper introduces a new distributed flip dynamics algorithm for generating random graph colorings, achieving faster mixing times than previous methods for certain parameters, and extends the understanding of distributed MCMC algorithms.

Contribution

It presents a novel distributed flip dynamics algorithm with proven $O(n ext{log}n)$ mixing time for $k>(11/6- ext{delta}) imes ext{max degree}$, improving upon prior distributed algorithms.

Findings

Achieves $O(n ext{log}n)$ mixing time for specific $k$ and $ ext{max degree}$.

Generalizes distributed Glauber dynamics to flip dynamics.

Provides analysis of stationary distribution for the new dynamics.

Abstract

We study distributed versions of Markov Chain Monte Carlo (MCMC) algorithms for generating random $k$-colorings of an input graph with maximum degree $\Delta$. In the sequential setting, the Glauber dynamics is the simple MCMC algorithm which updates the color at a randomly chosen vertex in each step. 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 present the distributed flip dynamics and prove $O(n\log{n})$ mixing for $k>(11/6-\delta)\Delta$ for a fixed $\delta>0$. Our new Markov chain is a generalization of the distributed Glauber dynamics previously analyzed, and is a parallel and distributed version of the more general flip dynamics considered in the sequential setting which recolors local maximal two-colored components in each step. While the distributed Glauber dynamics and the sequential flip dynamics are symmetric Markov chains, and hence their stationary distribution is uniformly distributed over colorings, our distributed flip dynamics is not symmetric and hence the stationary distribution is unclear.