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New Abilities and Limitations of Spectral Graph Bisection

Authors Martin R. Schuster, Maciej Liskiewicz

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Martin R. Schuster
Maciej Liskiewicz

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Martin R. Schuster and Maciej Liskiewicz. New Abilities and Limitations of Spectral Graph Bisection. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ESA.2017.66

Abstract

Spectral based heuristics belong to well-known commonly used methods which determines provably minimal graph bisection or outputs "fail" when the optimality cannot be certified. In this paper we focus on Boppana's algorithm which belongs to one of the most prominent methods of this type. It is well known that the algorithm works well in the random planted bisection model - the standard class of graphs for analysis minimum bisection and relevant problems. In 2001 Feige and Kilian posed the question if Boppana's algorithm works well in the semirandom model by Blum and Spencer. In our paper we answer this question affirmatively. We show also that the algorithm achieves similar performance on graph classes which extend the semirandom model. Since the behavior of Boppana's algorithm on the semirandom graphs remained unknown, Feige and Kilian proposed a new semidefinite programming (SDP) based approach and proved that it works on this model. The relationship between the performance of the SDP based algorithm and Boppana's approach was left as an open problem. In this paper we solve the problem in a complete way by proving that the bisection algorithm of Feige and Kilian provides exactly the same results as Boppana's algorithm. As a consequence we get that Boppana's algorithm achieves the optimal threshold for exact cluster recovery in the stochastic block model. On the other hand we prove some limitations of Boppana's approach: we show that if the density difference on the parameters of the planted bisection model is too small then the algorithm fails with high probability in the model.
Keywords
  • Minimum Graph Bisection
  • Spectral Methods
  • Convex Programming

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References

  1. Emmanuel Abbe. Community detection and stochastic block models: recent developments. arXiv preprint arXiv:1703.10146, 2017. Google Scholar
  2. Emmanuel Abbe, Afonso S. Bandeira, and Georgina Hall. Exact recovery in the stochastic block model. IEEE Transactions on Information Theory, 62(1):471-487, 2016. Google Scholar
  3. Farid Alizadeh. Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM Journal on Optimization, 5(1):13-51, 1995. Google Scholar
  4. Sanjeev Arora, David Karger, and Marek Karpinski. Polynomial time approximation schemes for dense instances of NP-hard problems. In Proc. of the 27th Annual ACM Symposium on Theory of Computing (STOC), pages 284-293. ACM, 1995. Google Scholar
  5. Sandeep N Bhatt and Frank Thomson Leighton. A framework for solving VLSI graph layout problems. Journal of Computer and System Sciences, 28(2):300-343, 1984. Google Scholar
  6. Avrim Blum and Joel Spencer. Coloring random and semi-random k-colorable graphs. Journal of Algorithms, 19(2):204-234, 1995. Google Scholar
  7. Robert D. Blumofe. Spectral methods for bisecting graphs. Unpublished Manuscript, 1993. Google Scholar
  8. Béla Bollobás and Alex D. Scott. Max cut for random graphs with a planted partition. Combinatorics, Probability and Computing, 13(4-5):451-474, 2004. Google Scholar
  9. Ravi B. Boppana. Eigenvalues and graph bisection: An average-case analysis. In Proc. of the 28th Annual Symposium on Foundations of Computer Science (FOCS), pages 280-285. IEEE Computer Society, 1987. URL: http://dx.doi.org/10.1109/SFCS.1987.22.
  10. T. N. Bui, S. Chaudhuri, F. T. Leighton, and M. Sipser. Graph bisection algorithms with good average case behavior. Combinatorica, 7(2):171-191, 1987. URL: http://dx.doi.org/10.1007/BF02579448.
  11. Ted Carson and Russell Impagliazzo. Hill-climbing finds random planted bisections. In Proc. of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 903-909. SIAM, 2001. Google Scholar
  12. Amin Coja-Oghlan. A spectral heuristic for bisecting random graphs. In Proc. of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 850-859. SIAM, 2005. URL: http://dl.acm.org/citation.cfm?id=1070432.1070552.
  13. Amin Coja-Oghlan, Charilaos Efthymiou, and Nor Jaafari. Local convergence of random graph colorings. In Proc. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, (APPROX/RANDOM), volume 40 of LIPIcs, pages 726-737, 2015. URL: http://arxiv.org/abs/1505.02985.
  14. Anne Condon and Richard M. Karp. Algorithms for graph partitioning on the planted partition model. Random Structures and Algorithms, 18(2):116-140, 2001. Google Scholar
  15. Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Minimum bisection is fixed parameter tractable. In Proc. of the 46th Annual ACM Symposium on Theory of Computing (STOC), pages 323-332. ACM, 2014. URL: http://dx.doi.org/10.1145/2591796.2591852.
  16. Martin E. Dyer and Alan M. Frieze. The solution of some random NP-hard problems in polynomial expected time. Journal of Algorithms, 10(4):451-489, 1989. Google Scholar
  17. Julie Falkner, Franz Rendl, and Henry Wolkowicz. A computational study of graph partitioning. Mathematical Programming, 66(1-3):211-239, 1994. URL: http://dx.doi.org/10.1007/BF01581147.
  18. Uriel Feige and Joe Kilian. Heuristics for semirandom graph problems. Journal of Computer and System Sciences, 63(4):639-671, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1773.
  19. Uriel Feige and Robert Krauthgamer. A polylogarithmic approximation of the minimum bisection. SIAM Journal on Computing, 31(4):1090-1118, April 2002. URL: http://dx.doi.org/10.1137/S0097539701387660.
  20. Uriel Feige, Robert Krauthgamer, and Kobbi Nissim. Approximating the minimum bisection size. In Proc. of the 32nd Annual ACM Symposium on Theory of Computing (STOC), pages 530-536. ACM, 2000. Google Scholar
  21. Miroslav Fiedler. A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czechoslovak Mathematical Journal, 25(4):619-633, 1975. Google Scholar
  22. Michael R. Garey, David S. Johnson, and Larry Stockmeyer. Some simplified np-complete graph problems. Theoretical Computer Science, 1(3):237-267, 1976. Google Scholar
  23. M. Grötschel, L. Lovász, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1(2):169-197, 1981. URL: http://dx.doi.org/10.1007/BF02579273.
  24. Stephen Guattery and Gary L. Miller. On the quality of spectral separators. SIAM Journal on Matrix Analysis and Applications, 19(3):701-719, July 1998. URL: http://dx.doi.org/10.1137/S0895479896312262.
  25. Bruce Hajek, Yihong Wu, and Jiaming Xu. Achieving exact cluster recovery threshold via semidefinite programming. IEEE Transactions on Information Theory, 62(5):2788-2797, 2016. Google Scholar
  26. Frank Harary, John P. Hayes, and Horng-Jyh Wu. A survey of the theory of hypercube graphs. Computers and Mathematics with Applications, 15(4):277-289, 1988. Google Scholar
  27. Paul W Holland, Kathryn Blackmond Laskey, and Samuel Leinhardt. Stochastic blockmodels: First steps. Social networks, 5(2):109-137, 1983. Google Scholar
  28. Subhash Khot. Ruling out PTAS for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM Journal on Computing, 36(4):1025-1071, 2006. Google Scholar
  29. Vivek Kwatra, Arno Schödl, Irfan Essa, Greg Turk, and Aaron Bobick. Graphcut textures: image and video synthesis using graph cuts. ACM Transactions on Graphics (ToG), 22(3):277-286, 2003. Google Scholar
  30. Thomas Lengauer. Combinatorial algorithms for integrated circuit layout. Springer Science &Business Media, 2012. Google Scholar
  31. Richard J. Lipton and Robert Endre Tarjan. Applications of a planar separator theorem. SIAM Journal on Computing, 9(3):615-627, 1980. Google Scholar
  32. Konstantin Makarychev, Yury Makarychev, and Aravindan Vijayaraghavan. Approximation algorithms for semi-random partitioning problems. In Proc. of the 44th Annual ACM Symposium on Theory of Computing (STOC), pages 367-384. ACM, 2012. Google Scholar
  33. Dániel Marx. Parameterized graph separation problems. Theoretical Computer Science, 351(3):394-406, 2006. Google Scholar
  34. Frank McSherry. Spectral partitioning of random graphs. In Proc. of the 42nd Annual Symposium on Foundations of Computer Science (FOCS), pages 529-537. IEEE, 2001. Google Scholar
  35. Cristopher Moore. The computer science and physics of community detection: Landscapes, phase transitions, and hardness. arXiv preprint arXiv:1702.00467, 2017. Google Scholar
  36. Elchanan Mossel, Joe Neeman, and Allan Sly. Consistency thresholds for the planted bisection model. In Proc. of the 47th ACM Symposium on Theory of Computing (STOC), pages 69-75. ACM, 2015. Google Scholar
  37. Huzur Saran and Vijay V. Vazirani. Finding k cuts within twice the optimal. SIAM Journal on Computing, 24(1):101-108, 1995. Google Scholar
  38. Kirk Schloegel, George Karypis, and Vipin Kumar. Graph partitioning for high performance scientific simulations. Army High Performance Computing Research Center, 2000. Google Scholar
  39. Martin R. Schuster and Maciej Liśkiewicz. New abilities and limitations of spectral graph bisection. CoRR, 2017. URL: https://arxiv.org/abs/1701.01337.
  40. Daniel A. Spielman and Shang-Hua Teng. Spectral partitioning works: Planar graphs and finite element meshes. In Proc. of the 37th Annual Symposium on Foundations of Computer Science (FOCS), pages 96-105. IEEE Computer Society, 1996. URL: http://dl.acm.org/citation.cfm?id=874062.875505.
  41. Daniel A. Spielman and Shang-Hua Teng. Spectral partitioning works: Planar graphs and finite element meshes. Linear Algebra and its Applications, 421(2):284-305, 2007. URL: http://dx.doi.org/10.1016/j.laa.2006.07.020.
  42. Chih-Chien Tu and Hsuanjen Cheng. Spectral methods for graph bisection problems. Computers &operations research, 25(7):519-530, 1998. URL: http://dx.doi.org/10.1016/S0305-0548(98)00021-5.
  43. Chih-Chien Tu, Ce-Kuen Shieh, and Hsuanjen Cheng. Algorithms for graph partitioning problems by means of eigenspace relaxations. European Journal of Operational Research, 123(1):86-104, 2000. URL: http://dx.doi.org/10.1016/S0377-2217(99)00060-0.
  44. René van Bevern, Andreas Emil Feldmann, Manuel Sorge, and Ondřej Suchỳ. On the parameterized complexity of computing graph bisections. In Proc. International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pages 76-87. Springer, 2013. Google Scholar
  45. Lieven Vandenberghe and Stephen Boyd. Semidefinite programming. SIAM Review, 38(1):49-95, March 1996. URL: http://dx.doi.org/10.1137/1038003.
  46. Zhenyu Wu and Richard Leahy. An optimal graph theoretic approach to data clustering: Theory and its application to image segmentation. IEEE transactions on pattern analysis and machine intelligence, 15(11):1101-1113, 1993. Google Scholar
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