skip to main content
10.1145/3406325.3451012acmconferencesArticle/Chapter ViewAbstractPublication PagesstocConference Proceedingsconference-collections
research-article

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

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 + oΔ(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].

References

[1]
Noga Alon and Joel H. Spencer. 2016. The probabilistic method (fourth ed.). John Wiley & Sons, Inc., Hoboken, NJ. xiv+375 pages. isbn:978-1-119-06195-3
[2]
Siddharth Bhandari and Sayantan Chakraborty. 2020. Improved bounds for perfect sampling of k-colorings in graphs. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing. 631–642.
[3]
Russ Bubley and Martin Dyer. 1997. Path coupling: A technique for proving rapid mixing in Markov chains. In Proceedings 38th Annual Symposium on Foundations of Computer Science. IEEE, 223–231.
[4]
Sitan Chen, Michelle Delcourt, Ankur Moitra, Guillem Perarnau, and Luke Postle. 2019. Improved bounds for randomly sampling colorings via linear programming. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, Philadelphia, PA, 2216–2234.
[5]
Weiming Feng, Heng Guo, Yitong Yin, and Chihao Zhang. 2020. Fast sampling and counting $k$-SAT solutions in the local lemma regime. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing. 854–867.
[6]
Alan Frieze and Eric Vigoda. 2007. A survey on the use of Markov chains to randomly sample colourings. In Combinatorics, complexity, and chance. Oxford Lecture Ser. Math. Appl., Vol. 34. Oxford Univ. Press, Oxford, 53–71.
[7]
David Gamarnik and Dmitriy Katz. 2012. Correlation decay and deterministic FPTAS for counting colorings of a graph. J. Discrete Algorithms 12 (2012), 29–47. issn:1570-8667
[8]
Olle Häggström and Karin Nelander. 1998. Exact sampling from anti-monotone systems. Statistica Neerlandica 52, 3 (1998), 360–380.
[9]
Mark Huber. 1999. Exact sampling and approximate counting techniques. In STOC '98 (Dallas, TX). ACM, New York, 31–40.
[10]
Mark Jerrum. 1995. A very simple algorithm for estimating the number of $k$-colorings of a low-degree graph. Random Structures Algorithms 7, 2 (1995), 157–165. issn:1042-9832
[11]
Mark R. Jerrum, Leslie G. Valiant, and Vijay V. Vazirani. 1986. Random generation of combinatorial structures from a uniform distribution. Theoret. Comput. Sci. 43, 2-3 (1986), 169–188. issn:0304-3975
[12]
Jingcheng Liu, Alistair Sinclair, and Piyush Srivastava. 2019. A Deterministic Algorithm for Counting Colorings with 2-Delta Colors. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 1380–1404.
[13]
Pinyan Lu and Yitong Yin. 2013. Improved FPTAS for multi-spin systems. In Approximation, randomization, and combinatorial optimization. Lecture Notes in Comput. Sci., Vol. 8096. Springer, Heidelberg, 639–654.
[14]
Ankur Moitra. 2019. Approximate counting, the Lovász local lemma, and inference in graphical models. J. ACM 66, 2 (2019), Art. 10, 25. issn:0004-5411
[15]
Robin A. Moser and Gábor Tardos. 2010. A constructive proof of the general Lovász local lemma. J. ACM 57, 2 (2010), Art. 11, 15. issn:0004-5411
[16]
James Gary Propp and David Bruce Wilson. 1996. Exact sampling with coupled Markov chains and applications to statistical mechanics, In Proceedings of the Seventh International Conference on Random Structures and Algorithms (Atlanta, GA, 1995). Random Structures Algorithms 9, 1-2, 223–252. issn:1042-9832
[17]
Eric Vigoda. 2000. Improved bounds for sampling colorings. J. Math. Phys. 41, 3 (2000), 1555–1569. issn:0022-2488 Probabilistic techniques in equilibrium and nonequilibrium statistical physics.

Cited By

View all
  • (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
  • (2022)Perfect sampling from spatial mixingRandom Structures & Algorithms10.1002/rsa.2107961:4(678-709)Online publication date: 18-Feb-2022

Index Terms

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

    Recommendations

    Comments

    Information & Contributors

    Information

    Published In

    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
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

    Sponsors

    Publisher

    Association for Computing Machinery

    New York, NY, United States

    Publication History

    Published: 15 June 2021

    Permissions

    Request permissions for this article.

    Check for updates

    Author Tags

    1. coupling from the past
    2. graph coloring
    3. perfect sampling

    Qualifiers

    • Research-article

    Conference

    STOC '21
    Sponsor:

    Acceptance Rates

    Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

    Upcoming Conference

    STOC '25
    57th Annual ACM Symposium on Theory of Computing (STOC 2025)
    June 23 - 27, 2025
    Prague , Czech Republic

    Contributors

    Other Metrics

    Bibliometrics & Citations

    Bibliometrics

    Article Metrics

    • Downloads (Last 12 months)17
    • Downloads (Last 6 weeks)1
    Reflects downloads up to 28 Feb 2025

    Other Metrics

    Citations

    Cited By

    View all
    • (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
    • (2022)Perfect sampling from spatial mixingRandom Structures & Algorithms10.1002/rsa.2107961:4(678-709)Online publication date: 18-Feb-2022

    View Options

    Login options

    View options

    PDF

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader

    Figures

    Tables

    Media

    Share

    Share

    Share this Publication link

    Share on social media