Document Open Access Logo

The Minimum Bisection in the Planted Bisection Model

Authors Amin Coja-Oghlan, Oliver Cooley, Mihyun Kang, Kathrin Skubch

PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2015.710.pdf
  • Filesize: 0.49 MB
  • 16 pages

Document Identifiers

Author Details

Amin Coja-Oghlan
Oliver Cooley
Mihyun Kang
Kathrin Skubch

Cite As Get BibTex

Amin Coja-Oghlan, Oliver Cooley, Mihyun Kang, and Kathrin Skubch. The Minimum Bisection in the Planted Bisection Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 710-725, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.710

Abstract

In the planted bisection model a random graph G(n,p_+,p_-) with n vertices is created by partitioning the vertices randomly into two classes of equal size (up to plus or minus 1). Any two vertices that belong to the same class are linked by an edge with probability p_+ and any two that belong to different classes with probability (p_-) <(p_+) independently. The planted bisection model has been used extensively to benchmark graph partitioning algorithms. If (p_+)=2(d_+)/n and (p_-)=2(d_-)/n for numbers 0 <= (d_-) <(d_+) that remain fixed as n tends to infinity, then with high probability the "planted" bisection (the one used to construct the graph) will not be a minimum bisection. In this paper we derive an asymptotic formula for the minimum bisection width under the assumption that (d_+)-(d_-) > c * sqrt((d_+)ln(d_+)) for a certain constant c>0.

Subject Classification

Keywords
  • Random graphs
  • minimum bisection
  • planted bisection
  • belief propagation.

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. E. Abbe, A. Bandeira, and G. Hall. Exact recovery in the stochastic block model. arXiv preprint arXiv:1405.3267, 2014. Google Scholar
  2. D. Aldous and J. Steele. The objective method: probabilistic combinatorial optimization and local weak convergence. In Probability on discrete structures, pages 1-72. Springer, 2004. Google Scholar
  3. S. Arora, S. Rao, and U. Vazirani. Expander flows, geometric embeddings and graph partitioning. Journal of the ACM (JACM), 56:5, 2009. Google Scholar
  4. V. Bapst, A. Coja-Oghlan, S. Hetterich, F. Rassmann, and D. Vilenchik. The condensation phase transition in random graph coloring. arXiv preprint arXiv:1404.5513, 2014. Google Scholar
  5. B. Bollobás and A. Scott. Max cut for random graphs with a planted partition. Combinatorics, Probability and Computing, 13:451-474, 2004. Google Scholar
  6. R. Boppana. Eigenvalues and graph bisection: An average-case analysis. In Proc. 28th Foundations of Computer Science, pages 280-285. IEEE, 1987. Google Scholar
  7. T. Bui, S. Chaudhuri, T. Leighton, and M. Sipser. Graph bisection algorithms with good average case behavior. Combinatorica, 7:171-191, 1987. Google Scholar
  8. T. Carson and R. Impagliazzo. Hill-climbing finds random planted bisections. In Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, pages 903-909. Society for Industrial and Applied Mathematics, 2001. Google Scholar
  9. A. Coja-Oghlan. A spectral heuristic for bisecting random graphs. In Random Structures and Algorithms, pages 351-398, 2006. Google Scholar
  10. A. Coja-Oghlan, O. Cooley, M. Kang, and K. Skubch. How does the core sit inside the mantle? arXiv preprint arXiv:1503.09030, 2015. Google Scholar
  11. A. Condon and R. Karp. Algorithms for graph partitioning on the planted partition model. Random Structures and Algorithms, 18:116-140, 2001. Google Scholar
  12. A. Decelle, F. Krzakala, C. Moore, and L. Zdeborová. Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Physical Review E, 84:066106, 2011. Google Scholar
  13. A. Dembo, A. Montanari, and S. Sen. Extremal cuts of sparse random graphs. arXiv preprint arXiv:1503.03923, 2015. Google Scholar
  14. T. Dimitriou and R. Impagliazzo. Go with the winners for graph bisection. In Proc. 9th SODA, pages 510-520. ACM/SIAM, 1998. Google Scholar
  15. M. Dyer and A. Frieze. The solution of some random np-hard problems in polynomial expected time. Journal of Algorithms, 10:451-489, 1989. Google Scholar
  16. U. Feige and J. Kilian. Heuristics for semirandom graph problems. Journal of Computer and System Sciences, 63:639-671, 2001. Google Scholar
  17. U. Feige and R. Krauthgamer. A polylogarithmic approximation of the minimum bisection. SIAM Journal on Computing, 31:1090-1118, 2002. Google Scholar
  18. M. Garey, D. Johnson, and L. Stockmeyer. Some simplified np-complete graph problems. Theoretical computer science, 1:237-267, 1976. Google Scholar
  19. M. Goemans and D. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM (JACM), 42:1115-1145, 1995. Google Scholar
  20. P. Holland, K. Laskey, and S. Leinhardt. Stochastic blockmodels: First steps. Social networks, 5:109-137, 1983. Google Scholar
  21. S. Janson, T. Łuczak, and A. Ruciński. Random graphs. John Wiley & Sons, 2000. Google Scholar
  22. M. Jerrum and G. Sorkin. The metropolis algorithm for graph bisection. Discrete Applied Mathematics, 82:155-175, 1998. Google Scholar
  23. A. Juels. Topics in black-box combinatorial function optimization. PhD thesis, UC Berkeley, 1996. Google Scholar
  24. R. Karp. Reducibility among combinatorial problems. Springer, 1972. Google Scholar
  25. M. Karpinski. Approximability of the minimum bisection problem: An algorithmic challenge. In Proc. 27th Mathematical Foundations of Computer Science, pages 59-67. Springer, 2002. Google Scholar
  26. S. Khot. Ruling out ptas for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM Journal on Computing, 36:1025-1071, 2006. Google Scholar
  27. L. Kučera. Expected complexity of graph partitioning problems. Discrete Applied Mathematics, 57:193-212, 1995. Google Scholar
  28. M. Luczak and C. McDiarmid. Bisecting sparse random graphs. Random Structures and Algorithms, 18:31-38, 2001. Google Scholar
  29. K. Makarychev, Y. Makarychev, and A. Vijayaraghavan. Approximation algorithms for semi-random partitioning problems. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 367-384. ACM, 2012. Google Scholar
  30. L. Massoulié. Community detection thresholds and the weak ramanujan property. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 694-703. ACM, 2014. Google Scholar
  31. F. McSherry. Spectral partitioning of random graphs. In Proc. 42nd Foundations of Computer Science, pages 529-537. IEEE, 2001. Google Scholar
  32. M. Mézard and A. Montanari. Information, physics, and computation. Oxford University Press, 2009. Google Scholar
  33. E. Mossel, J. Neeman, and A. Sly. Stochastic block models and reconstruction. arXiv preprint arXiv:1202.1499, 2012. Google Scholar
  34. E. Mossel, J. Neeman, and A. Sly. Belief propagation, robust reconstruction, and optimal recovery of block models. arXiv preprint arXiv:1309.1380, 2013. Google Scholar
  35. E. Mossel, J. Neeman, and A. Sly. A proof of the block model threshold conjecture. arXiv preprint arXiv:1311.4115, 2013. Google Scholar
  36. E. Mossel, J. Neeman, and A. Sly. Consistency thresholds for the planted bisection model. arXiv preprint arXiv:1407.1591 v2, 2014. Google Scholar
  37. R. Neininger and L. Rüschendorf. A general limit theorem for recursive algorithms and combinatorial structures. The Annals of Applied Probability, 14:378-418, 2004. Google Scholar
  38. H. Räcke. Optimal hierarchical decompositions for congestion minimization in networks. In Proc. 40th ACM symposium on Theory of computing, pages 255-264. ACM, 2008. Google Scholar
  39. M. Talagrand. The parisi formula. Annals of Mathematics, 163:221-263, 2006. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact