Spectral sparsification of graphs

SIAM Journal on Computing

Volumn 40, Issue 4, 2011, Pages 981-1025

  Spielman, Daniel A ^a   Teng, Shang Hua ^b  

^a Yale University   (United States)

^b University of Southern California ^*  (United States)

Author keywords

Graph Laplacian; Graph partitioning; Sparsification

Indexed keywords

DIAGONALLY DOMINANT; GRAPH LAPLACIAN; GRAPH LAPLACIANS; GRAPH PARTITIONING; GRAPH SPARSIFICATION; KEY COMPONENT; LAPLACIANS; LINEAR-TIME ALGORITHMS; PRECONDITIONERS; QUADRATIC FORM; SPARSIFICATION; SPECTRAL SIMILARITY; SUBGRAPHS;

GRAPH THEORY; LAPLACE TRANSFORMS; NUMBER THEORY;

ALGORITHMS;

EID: 80053214357     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/08074489X     Document Type: Article

References (31)

1. R. Andersen, F. Chung, and K. Lang, Local graph partitioning using pagerank vectors, in Proceedings of the 47th IEEE Annual Symposium on Foundations of Computer Science, IEEE Press, Piscataway, NJ, 2006, pp. 475-486.
2. D. Achlioptas and F. McSherry, Fast computation of low rank matrix approximations, in Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, ACM, New York, 2001, pp. 611-618.
3. R. Andersen and Y. Peres, Finding sparse cuts locally using evolving sets, in Proceedings of the 41st Annual ACM Symposium on Theory of Computing, ACM, New York, 2009, pp. 235-244.
4. O. Axelsson, A survey of preconditioned iterative methods for linear systems of algebraic equations, BIT, 25 (1985), pp. 165-187.
5. M. Bern, J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo, Supportgraph preconditioners, SIAM J. Matrix Anal. Appl., 27 (2006), pp. 930-951.
6. E. G. Boman and B. Hendrickson, Support theory for preconditioning, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 694-717.
7. 2) time, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, ACM, New York, 1996, pp. 47-55.
8. B. Bollobás, Modern Graph Theory, Springer-Verlag, New York, 1998.
9. J. D. Batson, D. A. Spielman, and N. Srivastava, Twice-Ramanujan sparsifiers, in Proceedings of the 41st Annual ACM Symposium on Theory of Computing, ACM, New York, 2009, pp. 255-262.
10. J. Cheeger, A lower bound for smallest eigenvalue of Laplacian, in Problems in Analysis, R. C. Gunning, ed., Princeton University Press, Princeton, NJ, 1970, pp. 195-199.
11. P. Chew, There are planar graphs almost as good as the complete graph, J. Comput. Syst. Sci., 39 (1989), pp. 205-219.
12. F. R. K. Chung, Spectral Graph Theory, CBMS Reg. Conf. Ser. Math., AMS, New York, 1997.
13. D. Dubhashi and D. Ranjan, Balls and bins: A study in negative dependence, Random Structures Algorithms, 13 (1998), pp. 99-124.
14. P. Diaconis and D. Stroock, Geometric bounds for eigenvalues of Markov chains, Ann. Appl. Probab., 1 (1991), pp. 36-61.
15. Z. Füredi and J. Komlós, The eigenvalues of random symmetric matrices, Combinatorica, 1 (1981), pp. 233-241.
16. C. Godsil and G. Royle, Algebraic Graph Theory, Grad. Texts in Math., Springer, Berlin, New York, 2001.
17. I. Koutis, G. L. Miller, and R. Peng, Approaching Optimality for Solving SDD Systems, preprint, 2010; available online at http://arxiv.org/abs/1003. 2958v1.
18. R. Kannan, S. Vempala, and A. Vetta, On clusterings: Good, bad and spectral, J. ACM, 51 (2004), pp. 497-515.
19. A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica, 8 (1988), pp. 261-277.
20. G. A. Margulis, Explicit group theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators, Prob. Inform. Trans., 24 (1988), pp. 39-46.
21. B. Mohar, The Laplacian spectrum of graphs, in Graph Theory, Combinatorics, and Applications, Wiley, New York, 1991, pp. 871-898.
22. P. Raghavan, Probabilistic construction of deterministic algorithms: Approximating packing integer programs, J. Comput. Syst. Sci., 37 (1988), pp. 130-143.
23. A. Sinclair and M. Jerrum, Approximate counting, uniform generation and rapidly mixing Markov chains, Inform. Comput., 82 (1989), pp. 93-133.
24. D. A. Spielman and N. Srivastava, Graph sparsification by effective resistances, in Proceedings of the 40th Annual ACM Symposium on Theory of Computing, ACM, New York, 2008, pp. 563-568.
25. D. A. Spielman and S.-H. Teng, Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems, in Proceedings of the 36th Annual ACM Symposium on Theory of Computing, ACM, New York, 2004, pp. 81-90. Full version available online at http://arxiv.org/abs/cs.DS/0310051.
26. D. A. Spielman and S.-H. Teng, A Local Clustering Algorithm for Massive Graphs and Its Application to Nearly-Linear Time Graph Partitioning, preprint, 2008. Available online at http://arxiv.org/abs/0809.3232. Submitted to SIAM J. Comput.
27. D. A. Spielman and S.-H. Teng, Nearly-Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems, preprint, 2008. Available online at http://www.arxiv.org/abs/cs.NA/0607105. Submitted to SIAM J. Matrix Anal. Appl.
28. R. E. Tarjan, Efficiency of a good but not linear set union algorithm, J. ACM, 22 (1975), pp. 448-501.
29. L. N. Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM, Philadelphia, 1997.
30. L. Trevisan, Approximation algorithms for unique games, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE Press, Piscataway, NJ, 2005, pp. 197-205.
31. V. Vu, Spectral norm of random matrices, Combinatorica, 27 (2007), pp. 721-736.