research-article

Perfectly sampling k ≥ (8/3 + o(1))Δ-colorings in graphs

Authors:

Mehtaab SawhneyAuthors Info & Claims

STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

Pages 1589 - 1600

https://doi.org/10.1145/3406325.3451012

Published: 15 June 2021 Publication History

Abstract

We present a randomized algorithm which takes as input an undirected graph G on n vertices with maximum degree Δ, and a number of colors k ≥ (8/3 + o_Δ(1))Δ, and returns – in expected time Õ(nΔ²logk) – a proper k-coloring of G distributed perfectly uniformly on the set of all proper k-colorings of G. Notably, our sampler breaks the barrier at k = 3Δ encountered in recent work of Bhandari and Chakraborty [STOC 2020]. We also discuss how our methods may be modified to relax the restriction on k to k ≥ (8/3 − є₀)Δ for an absolute constant є₀ > 0.

As in the work of Bhandari and Chakraborty, and the pioneering work of Huber [STOC 1998], our sampler is based on Coupling from the Past [Propp&Wilson, Random Struct. Algorithms, 1995] and the bounding chain method [Huber, STOC 1998; H'aggstr'om& Nelander, Scand. J. Statist., 1999]. Our innovations include a novel bounding chain routine inspired by Jerrum’s analysis of the Glauber dynamics [Random Struct. Algorithms, 1995], as well as a preconditioning routine for bounding chains which uses the algorithmic Lovász Local Lemma [Moser&Tardos, J.ACM, 2010].

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Cited By

Feng WGuo HWang CWang JYin Y(2023)Towards derandomising Markov chain Monte Carlo2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00120(1963-1990)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00120
Feng WGuo HYin Y(2022)Perfect sampling from spatial mixingRandom Structures & Algorithms10.1002/rsa.2107961:4(678-709)Online publication date: 18-Feb-2022
https://doi.org/10.1002/rsa.21079

Index Terms

Perfectly sampling k ≥ (8/3 + o(1))Δ-colorings in graphs
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph coloring

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cover image ACM Conferences

STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

June 2021

1797 pages

ISBN:9781450380539

DOI:10.1145/3406325

General Chair:
Samir Khuller
Northwestern University, USA
,
Program Chair:
Virginia Vassilevska Williams
Massachusetts Institute of Technology, USA

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Published: 15 June 2021

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STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing

June 21 - 25, 2021

Virtual, Italy

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Cited By

Feng WGuo HWang CWang JYin Y(2023)Towards derandomising Markov chain Monte Carlo2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00120(1963-1990)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00120
Feng WGuo HYin Y(2022)Perfect sampling from spatial mixingRandom Structures & Algorithms10.1002/rsa.2107961:4(678-709)Online publication date: 18-Feb-2022
https://doi.org/10.1002/rsa.21079

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