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

We examine the problem of almost-uniform sampling proper q-colorings of a graph whose maximum degree is ∆. A famous result, discovered independently by Jerrum [30] and Salas and Sokal [38], is that, assuming q > (2+δ)∆, the Glauber dynamics (a.k.a. single-site dynamics) for this problem has mixing time O(n logn), 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 [15] has shown that a O(∆ log n) time parallelized algorithm is possible, and that Ω(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 δ > 0: • q ≥ (2 + δ)∆, on general graphs with maximum degree ∆; • q ≥ (α + δ)∆, where α ≈ 1.763 satisfies α = e1/α∗ , on graphs with sufficiently large maximum degree ∆ ≥ ∆0(δ) and girth at least 9. Department of Computer Science and Technology, Nanjing University. Email: [email protected]. Department of Computer Science, University of New Mexico. Email:[email protected]. Partially supported by NSF CAREER award CCF-1150281. State Key Laboratory for Novel Software Technology, Nanjing University. Email: [email protected]. Supported by the National Science Foundation of China under Grant No. 61672275 and No. 61722207.