Skip to main content
SpringerLink
Log in

Sampling Edge Covers in 3-Regular Graphs

  • Conference paper
Mathematical Foundations of Computer Science 2009 (MFCS 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5734))

  • 815 Accesses

Abstract

An edge cover C of an undirected graph is a set of edges such that every vertex has an adjacent edge in C. We show that a Glauber dynamics Markov chain for edge covers mixes rapidly for graphs with degrees at most three. Glauber dynamics have been studied extensively in the statistical physics community, with special emphasis on lattice graphs. Our results apply, for example, to the hexagonal lattice. Our proof of rapid mixing introduces a new cycle/path decomposition for the canonical flow argument.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Cooper, C., Dyer, M., Greenhill, C.: Sampling regular graphs and a peer- to-peer network. Combinatorics, Probability and Computing 16, 557–593 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Diaconis, P., Saloff-Coste, L.: Comparison Theorems for Reversible Markov Chains. The Annals of Applied Probability 3(3), 696–730 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Diaconis, P., Stroock, D.: Geometric Bounds for Eigenvalues of Markov Chains. The Annals of Applied Probability 1(1), 36–61 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM Journal on Computing 18, 1149–1178 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jerrum, M., Sinclair, A.: Approximate counting, uniform generation and rapidly mixing Markov chains. Information and Computation 82, 93–133 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. J. ACM 51(4), 671–697 (2004)

    Article  MATH  Google Scholar 

  7. Kannan, R., Tetali, P., Vempala, S.: Simple Markov-chain algorithms for generating bipartite graphs and tournaments. Random Structures and Algorithms (RSA) 14(4), 293–308 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Luby, M., Randall, D., Sinclair, A.: Markov chain algorithms for planar lattice structures. SIAM Journal on Computing 31, 167–192 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Randall, D., Tetali, P.: Analyzing Glauber dynamics by comparison of Markov chains. Journal of Mathematical Physics 41, 1598–1615 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sinclair, A.: Improved Bounds for Mixing Rates of Marked Chains and Multicommodity Flow. Combinatorics, Probability and Computing 1, 351–370 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8, 189–201 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Vigoda, E.: A note on the Glauber dynamics for sampling independent sets. Electronic Journal of Combinatorics 8(1) (2001); Research paper 8

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rochester Institute of Technology, Rochester, NY, USA

    Ivona Bezáková & William A. Rummler

Authors
  1. Ivona Bezáková
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. William A. Rummler
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Faculty of Mathematics, Physics and Informatics, Comenius University, Research partially funded by grant APVV 0433-06, Bratislava, Slovakia

    Rastislav Královič

  2. Members of EACSL Jury for the Ackermann Award, Poland

    Damian Niwiński

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bezáková, I., Rummler, W.A. (2009). Sampling Edge Covers in 3-Regular Graphs. In: Královič, R., Niwiński, D. (eds) Mathematical Foundations of Computer Science 2009. MFCS 2009. Lecture Notes in Computer Science, vol 5734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03816-7_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-03816-7_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03815-0

  • Online ISBN: 978-3-642-03816-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation