Advertisement

Directed Lovász local lemma and Shearer’s lemma

  • Lefteris KirousisEmail author
  • John Livieratos
  • Kostas I. Psaromiligkos
Article
  • 1 Downloads

Abstract

Moser and Tardos (J. ACM (JACM) 57(2), 11 2010) gave an algorithmic proof of the lopsided Lovász local lemma (LLL) in the variable framework, where each of the undesirable events is assumed to depend on a subset of a collection of independent random variables. For the proof, they define a notion of a lopsided dependency between the events suitable for this framework. In this work, we strengthen this notion, defining a novel directed notion of dependency and prove the LLL for the corresponding graph. We show that this graph can be strictly sparser (thus the sufficient condition for the LLL weaker) compared with graphs that correspond to other extant lopsided versions of dependency. Thus, in a sense, we address the problem “find other simple local conditions for the constraints (in the variable framework) that advantageously translate to some abstract lopsided condition” posed by Szegedy (2013). We also give an example where our notion of dependency graph gives better results than the classical Shearer lemma. Finally, we prove Shearer’s lemma for the dependency graph we define. For the proofs, we perform a direct probabilistic analysis that yields an exponentially small upper bound for the probability of the algorithm that searches for the desired assignment to the variables not to return a correct answer within n steps. In contrast, the method of proof that became known as the entropic method, gives an estimate of only the expectation of the number of steps until the algorithm returns a correct answer, unless the probabilities are tinkered with.

Keywords

Lovász local lemma Shearer’s lemma Lopsidependency 

Mathematics Subject Classification (2010)

68W20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

We are truly grateful to Ioannis Giotis and Dimitrios Thilikos for their substantial contribution to earlier versions of this work (see [7] and [8]).

References

  1. 1.
    Achlioptas, D., Iliopoulos, F.: Random walks that find perfect objects and the Lovász local lemma. In: Proceedings 55th Annual Symposium on Foundations of Computer Science (FOCS), pp 494–503. IEEE (2014)Google Scholar
  2. 2.
    Achlioptas, D., Iliopoulos, F.: Random walks that find perfect objects and the Lovász local lemma. J. ACM (JACM) 63(3), 22 (2016)CrossRefGoogle Scholar
  3. 3.
    Bender, E.A., Bruce Richmond, L: A multivariate Lagrange inversion formula for asymptotic calculations. Electron. J. Comb. 5(1), 33 (1998)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Erdős, P., Lovász, L.: Problems and results on 3-chromatic hypergraphs and some related questions. Infinite Finite Sets 10, 609–627 (1975)MathSciNetGoogle Scholar
  5. 5.
    Erdős, P., Spencer, J.: Lopsided Lovász local lemma and Latin transversals. Discret. Appl. Math. 30(2–3), 151–154 (1991)CrossRefGoogle Scholar
  6. 6.
    Giotis, I., Kirousis, L., Psaromiligkos, K.I., Thilikos, D.M.: On the algorithmic Lovász local lemma and acyclic edge coloring. In: Proceedings of the twelfth workshop on analytic algorithmics and combinatorics. Society for Industrial and Applied Mathematics (2015) Available: http://epubs.siam.org/doi/pdf/10.1137/1.9781611973761.2
  7. 7.
    Giotis, I., Kirousis, L.M., Livieratos, J., Psaromiligkos, K.I., Thilikos, D.M.: Alternative proofs of the asymmetric Lovász local lemma and Shearer’s lemma. In: Proceedings of the 11th International Conference on Random and Exhaustive Generation of Combinatorial Structures, GASCom (2018) Available: http://ceur-ws.org/Vol-2113/paper15.pdf
  8. 8.
    Giotis, I., Kirousis, L.M., Psaromiligkos, K.I., Thilikos, D.: An alternative proof for the constructive asymmetric Lovász local lemma. In: 13th Cologne Twente Workshop on Graphs and Combinatorial Optimization (2015)Google Scholar
  9. 9.
    Harris, D.G.: Lopsidependency in the Moser-Tardos framework: Beyond the lopsided Lovász local lemma. In: Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1792–1808. SIAM (2015)Google Scholar
  10. 10.
    Harris, D.G, Srinivasan, A.: A constructive algorithm for the Lovász local lemma on permutations. In: Proceedings 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 907–925. SIAM (2014)Google Scholar
  11. 11.
    Harvey, N.J.A., Vondrák, J.: An algorithmic proof of the Lovász local lemma via resampling oracles. In: Proceedings 56th Annual Symposium on Foundations of Computer Science (FOCS), pp. 1327–1346. IEEE (2015)Google Scholar
  12. 12.
    He, K., Li, L., Liu, X., Wang, Y., Xia, M.: Variable-version Lovász local: Beyond shearer’s bound. In: 58th Annual Symposium on Foundations of Computer Science (FOCS), pp. 451–462. IEEE (2017)Google Scholar
  13. 13.
    Horn, R.A, Johnson, C.R: Matrix Analysis. Cambridge University Press (1990)Google Scholar
  14. 14.
    Kolipaka, K., Rao, B., Szegedy, M.: Moser and Tardos meet Lovász. In: Proceedings 43rd Annual ACM Symposium on Theory of Computing (STOC), pp. 235–244. ACM (2011)Google Scholar
  15. 15.
    Moser, R.A.: A constructive proof of the Lovász local lemma. In: Proceedings 41st Annual ACM Symposium on Theory of Computing (STOC), pp. 343–350. ACM (2009)Google Scholar
  16. 16.
    Moser, R.A., Tardos, G.: A constructive proof of the general Lovász local lemma. J. ACM (JACM) 57(2), 11 (2010)CrossRefGoogle Scholar
  17. 17.
    Sarkar, K., Colbourn, C.J: Upper bounds on the size of covering arrays. SIAM J. Discret. Math. 31(2), 1277–1293 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Shearer, J.B.: On a problem of Spencer. Combinatorica 5(3), 241–245 (1985)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Szegedy, M.: The Lovász local lemma–a survey. In: International Computer Science Symposium in Russia, pp. 1–11. Springer (2013)Google Scholar
  20. 20.
    Tao, T.: Moser’s entropy compression argument (2009) Available: https://terrytao.wordpress.com/2009/08/05/mosers-entropy-compression-argument/

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA

Personalised recommendations