Improved cheeger's inequality: Analysis of spectral partitioning algorithms through higher order spectral gap

Proceedings of the Annual ACM Symposium on Theory of Computing

Volumn , Issue , 2013, Pages 11-20

  Kwok, Tsz Chiu ^a   Lau, Lap Chi ^a   Lee, Yin Tat ^a   Gharan, Shayan Oveis ^b   Trevisan, Luca ^b  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^b STANFORD UNIVERSITY   (United States)

Author keywords

Graph partitioning; Spectral algorithm

Indexed keywords

CHEEGER'S INEQUALITIES; CONSTANT-FACTOR APPROXIMATION ALGORITHMS; EMPIRICAL PERFORMANCE; GRAPH PARTITIONING; PERFORMANCE GUARANTEES; SEGMENTATION AND CLUSTERING; SPECTRAL ALGORITHM; SPECTRAL PARTITIONING;

APPROXIMATION ALGORITHMS; AXIAL FLOW; EIGENVALUES AND EIGENFUNCTIONS; GRAPH THEORY; MATRIX ALGEBRA;

COMPUTATION THEORY;

EID: 84879815963     PISSN: 07378017     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/2488608.2488611     Document Type: Conference Paper

References (46)

1. N. Alon. Eigenvalues and expanders. Combinatorica, 6, 83-96, 1986.
2. N. Alon, V. Milman. λ1, isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B, 38(1), 73-88, 1985.
3. N. Alon, M. Krivelevich, B. Sudakov. Finding a large hidden clique in a random graph. Random Structures and Algorithms 13(3-4), 457-466, 1998.
4. S. Arora, B. Barak, D. Steurer. Subexponential algorithms for unique games and related problems. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), 563-572, 2010.
5. S. Arora, S. Rao, U. Vazirani. Expander ows, geometric embeddings and graph partitioning. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), 222-231, 2004.
6. P. Awasthi, A. Blum, O. Sheffet. Stability yields a PTAS for k-median and k-means clustering. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), 309-318, 2010.
7. M.-F. Balcan, A. Blum, A. Gupta. Approximate clustering without the approximation. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 1068-1077, 2009.
8. Y. Bilu, N. Linial. Are stable instances easy? In Proceedings of Innovations in Computer Science (ICS), 332-341, 2010.
9. Y. Bilu, A. Daniely, N. Linial, M. Saks. On the practically interesting instances of MAXCUT. In arXiv:1205.4893, 2012.
10. P. Biswal, J.R. Lee, S. Rao. Eigenvalue bounds, spectral partitioning, and metrical deformations via ows. Journal of the ACM 57(3), 2010.
11. R. Boppana. Eigenvalues and graph bisection: An average-case analysis. In Proceedings of the 28th Annual Symposium on Foundations of Computer Science (FOCS), 280-285, 1987.
12. J. Cheeger. A lower bound for the smallest eigenvalue of the Laplacian. Problems in Analysis, Princeton University Press, 195-199, 1970.
13. F. R. K. Chung. Laplacians of graphs and Cheeger's inequalities. Combinatorics, Paul Erdös is eighty, Vol. 2 (Keszthely, 1993), volume 2 of Bolyai Soc. Math. Stud., pages 157-172. János Bolyai Math. Soc., Budapest, 1996.
14. Fan R. K. Chung. Spectral graph theory. volume 92 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997. 9.
15. U. Feige, R. Krauthgamer. A polylogarithmic approximation of the minimum bisection. SIAM Journal on Computing 31(3), 1090-1118, 2002.
16. K. Fujiwara. Eigenvalues of Laplacians on a closed Riemannian manifold and its nets. Proceedings of the American Mathematical Society 123(8), 1995.
17. M.X. Goemans, D.P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42(6), 1115-1145, 1995.
18. S. Guattery, G.L. Miller. On the quality of spectral separators. SIAM J. Matrix Anal. Appl. 19(3), 701-719, 1998.
19. V. Guruswami, A.K. Sinop. Faster SDP hierarchy solvers for local rounding algorithms. In Proceedings of the 53rd IEEE Symposium on Foundations of Computer Science (FOCS), 197-206, 2012.
20. V. Guruswami, A.K. Sinop. Approximating non-uniform sparsest cut via generalized spectra. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2013.
21. S. Hoory, N. Linial, A. Wigderson. Expander graphs and their applications. Bulletin of the American Mathematical Society 43(4), 439-561, 2006.
22. M. Jerrum, A. Sinclair, and E. Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. Journal of the ACM, 51(4), 671-697, 2004.
23. S. Jimbo and A. Maruoka. Expanders obtained from affine transformations. In Proceedings of the 17th Annual Symposiumon Theory of Computing (STOC), 88-97, 1985.
24. R. Kannan, S. Vempala, A. Vetta. On cluterings: good, bad, and spectral. Journal of the ACM 51, 497-515, 2004.
25. J. Kelner. Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus. SIAM Journal on Computing 35(4), 882-902, 2006.
26. J. R. Lee. On expanders from the action of GL(2, Z), manuscript, 2013.
27. J. Kelner, J.R. Lee, G. Price, S.-H. Teng. Metric uniformization and spectral bounds for graphs. Geom. Funct. Anal., 21(5), 1117-1143, 2011.
28. J. Leskovec, K.J. Lang, M.W. Mahoney. Empirical comparison of algorithms for network community detection. In Proceedings of the 19th International Conference on World Wide Web (WWW), 631-640, 2010.
29. J.R. Lee, S. Oveis Gharan, L. Trevisan. Multi-way spectral partitioning and higher-order Cheeger inequalities. In Proceedings of the 44th Annual Symposium on Theory of Computing (STOC), 1117-1130, 2012.
30. F.T. Leighton, S. Rao. Multicommodity max-ow min-cut theorem and their use in designing approximation algorithms. Journal of the ACM 46(6), 787-832, 1999.
31. A. Louis, P. Raghavendra, P. Tetali, S. Vempala. Many sparse cuts via higher eigenvalues. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC), 1131-1140, 2012.
32. U. von Luxburg. A tutorial on spectral clustering. Statistics and Computing 17(4), 395-416, 2007.
33. U. von Luxburg. Clustering stability: an overview. Foundations and Trends in Machine Learning 2(3), 235-274, 2010.
34. F. McSherry. Spectral partitioning of random graphs. In Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science (FOCS), 529-537, 2001.
35. A. Ng, M. Jordan, Y. Weiss. On spectral clustering: Analysis and an algorithm. Advances in Neural Information Processing Systems 14, 849-856, 2001.
36. R. O'Donnell, D. Witmer. Improved small-set expansion from higher eigenvalues. CoRR, abs/1204.4688, 2012.
37. S. Oveis Gharan, L. Trevisan. Approximating the expansion profile and almost optimal local graph clustering. In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2012.
38. Harald Räcke. Optimal hierarchical decompositions for congestion minimization in networks. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC), 255-264, 2008.
39. J. Shi, J. Malik. Normalized cuts and image segmentation. IEEE Pattern Anal. Mach. Intell., 22(8), 888-905, 2000.
40. D.B. Shmoys. Approximation algorithms for cut problems and their applications to divide-and-conquer. In Approximation Algorithms for NP-hard Problems, (D.S. Hochbaum, ed.) PWS, 192-235, 1997.
41. A. Sinclair and M. Jerrum. Approximate counting, uniform generation and rapidly mixing markov chains. Inf. Comput., 82(1), 93-133, July 1989.
42. D.A. Spielman, S.-H. Teng. Spectral partitioning works: Planar graphs and finite element mashes. Linear Algebra and its Applicatiosn 421(2-3), 284-305, 2007.
43. D. Steurer. On the complexity of unique games and graph expansion. Ph.D. thesis, Princeton University, 2010.
44. M. Tanaka. Higher eigenvalues and partitions of graphs. In arXiv:1112.3434, 2012.
45. L. Trevisan. Max cut and the smallest eigenvalue. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), 263-272, 2009.
46. D.A. Tolliver, G.L. Miller. Graph partitioning by spectral rounding: Applications in image segmentation and clustering. In Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), 1053-1060.