Distributed Symmetry Breaking in Sampling (Optimal Distributed Randomly Coloring with Fewer Colors)

TL;DR

This paper introduces a distributed sampling algorithm that achieves optimal parallelization for graph coloring, significantly improving the efficiency of sampling proper colorings in graphs with bounded degree.

Contribution

It presents the Lazy Local Metropolis Algorithm, a distributed symmetry breaking method that attains optimal $O( ext{log } n)$ mixing time for sampling proper colorings under certain conditions.

Findings

Achieves $O( ext{log } n)$ mixing time for $q extgreater (2+ ext{delta})\Delta$ on general graphs.

Achieves $O( ext{log } n)$ mixing time for $q extgreater ( ext{alpha}^*+ ext{delta})\Delta$ on large girth graphs.

Improves previous algorithms by incorporating distributed symmetry breaking for faster convergence.

Abstract

We examine the problem of almost-uniform sampling proper $q$-colorings of a graph whose maximum degree is $\Delta$. A famous result, discovered independently by Jerrum(1995) and Salas and Sokal(1997), is that, assuming $q > (2+\delta) \Delta$, the Glauber dynamics (a.k.a. single-site dynamics) for this problem has mixing time $O(n \log n)$, where $n$ is the number of vertices, and thus provides a nearly linear time sampling algorithm for this problem. A natural question is the extent to which this algorithm can be parallelized. Previous work Feng, Sun and Yin [PODC'17] has shown that a $O(\Delta \log n)$ time parallelized algorithm is possible, and that $\Omega(\log n)$ time is necessary. We give a distributed sampling algorithm, which we call the Lazy Local Metropolis Algorithm, that achieves an optimal parallelization of this classic algorithm. It improves its predecessor, the Local Metropolis algorithm of Feng, Sun and Yin [PODC'17], by introducing a step of distributed symmetry breaking that helps the mixing of the distributed sampling algorithm. For sampling almost-uniform proper $q$-colorings of graphs $G$ on $n$ vertices, we show that the Lazy Local Metropolis algorithm achieves an optimal $O(\log n)$ mixing time if either of the following conditions is true for an arbitrary constant $\delta>0$: $\bullet$ $q\ge(2+\delta)\Delta$, on general graphs with maximum degree $\Delta$; $\bullet$ $q \geq (\alpha^* + \delta)\Delta$, where $\alpha^* \approx 1.763$ satisfies $\alpha^* = \mathrm{e}^{1/\alpha^*}$, on graphs with sufficiently large maximum degree $\Delta\ge \Delta_0(\delta)$ and girth at least $9$.