Local Algorithms, Regular Graphs of Large Girth, and Random Regular Graphs

Article

Abstract

We introduce a general class of algorithms and analyse their application to regular graphs of large girth. In particular, we can transfer several results proved for random regular graphs into (deterministic) results about all regular graphs with sufficiently large girth. This reverses the usual direction, which is from the deterministic setting to the random one. In particular, this approach enables, for the first time, the achievement of results equivalent to those obtained on random regular graphs by a powerful class of algorithms which contain prioritised actions. As a result, we obtain new upper or lower bounds on the size of maximum independent sets, minimum dominating sets, maximum k-independent sets, minimum k-dominating sets and maximum k-separated matchings in r-regular graphs with large girth.

Mathematics Subject Classification (2000)

05C35 05C80 05C69 05C85 

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References

  1. [1]
    M. Beis, W. Duckworth and M. Zito: Packing vertices and edges in random regular graphs, Random Structures & Algorithms 32 (2008), 20–37.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    E. A. Bender and E. R. Canfield: The asymptotic number of labeled graphs with given degree sequences, J. Combinatorial Theory Ser. A 24 (1978), 296–307.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    B. Bollobás: Random graphs, Academic Press, London, 1985.MATHGoogle Scholar
  4. [4]
    E. Csóka, B. Gerencsér, V. Harangi and B. Virág: Invariant Gaussian processes and independent sets on regular graphs of large girth, Random Structures and Algorithms 47 (2015), 284–303.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    W. Duckworth and B. Mans: Randomized greedy algorithms for finding small k-dominating sets of random regular graphs, Random Structures and Algorithms 27 (2005), 401–412.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    W. Duckworth and N. C. Wormald: On the independent domination number of random regular graphs, Combinatorics, Probability and Computing 15 (2006), 513–522.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    W. Duckworth and M. Zito: Large 2-independent sets of regular graphs, Electronic Notes in Theoretical Computer Science 78 (2003), 1–13.CrossRefMATHGoogle Scholar
  8. [8]
    W. Duckworth and M. Zito: Large independent sets in random regular graphs, Theoretical Computer Science 410 (2009), 5236–5243.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    P. Erdős: Graph theory and probability, Canadian Journal of Mathematics 11 (1959), 34–38.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    D. Gamarnik and D. A. Goldberg: Randomized greedy algorithms for independent sets and matchings in regular graphs: exact results and finite girth corrections, Combin. Probab. Comput. 19 (2010), 61–85.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    D. Gamarnik and M. Sudan: Limits of local algorithms over sparse random graphs, http://arxiv.org/abs/1304.1831. Proceedings of the 5th conference on Innovations in theoretical computer science (2014), 369–376.CrossRefGoogle Scholar
  12. [12]
    H. Hatami, L. Lovász and B. Szegedy: Limits of locally-globally convergent graph sequences, Geom. Funct. Anal. 24 (2014), 269–296.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    G. Hopkins and W. Staton, Girth and independence ratio, Canadian Mathematical Bulletin 25 (1982), 179–186.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    C. Hoppen: Properties of graphs with large girth, Doctoral thesis, University of Waterloo, 2008.MATHGoogle Scholar
  15. [15]
    C. Hoppen and N. Wormald: Local algorithms, regular graphs of large girth, and random regular graphs, http://arxiv.org/abs/1308.0266v2.
  16. [16]
    C. Hoppen and N. Wormald: Properties of regular graphs with large girth via local algorithms Journal of Combinatorial Theory, Series B 121 (2016), 367–397.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    F. Kardoš, D. Král and J. Volec: Fractional colorings of cubic graphs with large girth, SIAM J. Discrete Math. 25 (2011), 1454–1476.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    F. Kardoš, D. Král and J. Volec: Maximum edge-cuts in cubic graphs with large girth and in random cubic graphs, Random Structures & Algorithms 41 (2012), 506–520.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    K. Kawarabayashi, M. D. Plummer and A. Saito: Domination in a graph with a 2 factor, J. Graph Theory 52 (2006), 1–6.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    D. Král, P. Škoda and J. Volec: Domination number of cubic graphs with large girth, J. Graph Theory 69 (2012), 131–142.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    J. Lauer and N. Wormald: Large independent sets in random graphs with large girth, Journal of Combinatorial Theory, Series B 97 (2007), 999–1009.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    L. Lovász: Large networks and graph limits, American Mathematical Society Colloquium Publications, 60, American Mathematical Society, Providence (2012).MATHGoogle Scholar
  23. [23]
    B. D. McKay: Independent sets in regular graphs of high girth, Ars Combinatoria 23A (1987), 179–185.MathSciNetMATHGoogle Scholar
  24. [24]
    M. Rahman and B. Virág: Local algorithms for independent sets are half-optimal, to appear in Annals of Probability, http://arxiv.org/abs/1402.0485.
  25. [25]
    B. Reed: Paths, stars and the number three, Combin. Probab. Comput. 5 (1996), 277–295.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    J. B. Shearer: A note on the independence number of triangle-free graphs, II, Journal of Combinatorial Theory, Series B 53 (1991), 300–307.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    N. C. Wormald: Differential equations for random processes and random graphs, Annals of Applied Probability 5 (1995), 1217–1235.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    N. C. Wormald: The differential equation method for random graph processes and greedy algorithms, in: Lectures on Approximation and Randomized Algorithms, M. Karoński and H.J. Prömel (eds), 73–155, PWN, Warsaw, 1999.Google Scholar
  29. [29]
    N. C. Wormald: Models of random regular graphs, Surveys in Combinatorics, 1999, London Mathematical Society Lecture Note Series 267 (J.D. Lamb and D.A. Preece, eds), Cambridge University Press, Cambridge, 239–298, 1999.CrossRefGoogle Scholar
  30. [30]
    N. C. Wormald: Analysis of greedy algorithms on graphs with bounded degrees, Discrete Mathematics 273 (2003), 235–260.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    N.C. Wormald: Random graphs and asymptotics, Section 8.2 in Handbook of Graph Theory (J. L. Gross and J. Yellen eds), 817–836, CRC, Boca Raton, 2004.Google Scholar
  32. [32]
    M. Zito: Greedy algorithms for minimisation problems in random regular graphs, Lecture Notes in Computer Science 2161 524–536, Springer-Verlag, 2001.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUniversidade Federal do Rio Grande do SulPorto AlegreBrazil
  2. 2.School of Mathematical SciencesMonash UniversityMelbourneAustralia

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