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Improved Distributed Algorithms for Random Colorings

Authors Charlie Carlson, Daniel Frishberg, Eric Vigoda

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Author Details

Charlie Carlson
  • Department of Computer Science, University of California, Santa Barbara, CA, USA
Daniel Frishberg
  • Department of Computer Science and Software Engineering, California Polytechnic State University, San Luis Obispo, CA, USA
Eric Vigoda
  • Department of Computer Science, University of California, Santa Barbara, CA, USA

Funding

  • Vigoda, Eric: Research supported in part by NSF grant CCF-2147094.

Cite AsGet BibTex

Charlie Carlson, Daniel Frishberg, and Eric Vigoda. Improved Distributed Algorithms for Random Colorings. In 27th International Conference on Principles of Distributed Systems (OPODIS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 286, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.OPODIS.2023.13

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 Δ and an integer k ≥ Δ+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 > 2Δ. 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+ε)Δ for any ε > 0. We improve this result to k > (11/6-δ)Δ for a fixed δ > 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Random walks and Markov chains
Keywords
  • Distributed Graph Algorithms
  • Local Algorithms
  • Coloring
  • Glauber Dynamics
  • Sampling
  • Markov Chains

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