Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

Volumn 2015-December, Issue , 2015, Pages 250-269

  Lee, Yin Tat ^a   Sun, He ^b  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

algorithmic spectral graph theory; spectral sparsification

Indexed keywords

ALGORITHMS; CLUSTERING ALGORITHMS;

BARRIER FUNCTIONS; EFFECTIVE RESISTANCES; LINEAR TIME; LINEAR-TIME ALGORITHMS; RANDOM SAMPLING; SPARSIFICATION; SPECTRAL GRAPH THEORY;

GRAPH THEORY;

EID: 84960459284     PISSN: 02725428     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/FOCS.2015.24     Document Type: Conference Paper

References (27)

1. J. D. Batson, D. A. Spielman, and N. Srivastava, "Twice-Ramanujan sparsifiers, " SIAM Journal on Computing, vol. 41, no. 6, pp. 1704-1721, 2012.
2. A. Zouzias, "A matrix hyperbolic cosine algorithm and applications, " in 39th International Colloquium on Automata, Languages, and Programming (ICALP'12), 2012, pp. 846-858.
3. Z. Allen-Zhu, Z. Liao, and L. Orecchia, "Spectral sparsification and regret minimization beyond multiplicative updates, " in 47th Annual ACM Symposium on Theory of Computing (STOC'15), 2015, pp. 237-245.
4. D. A. Spielman and N. Srivastava, "Graph sparsification by effective resistances, " SIAM Journal on Computing, vol. 40, no. 6, pp. 1913-1926, 2011.
5. L. P. Chew, "There are planar graphs almost as good as the complete graph, " Journal of Computer and System Sciences, vol. 39, no. 2, pp. 205-219, 1989.
6. A. Benczúr and D. Karger, "Approximating s-t minimum cuts in -O(n2) time, " in 28th Annual ACM Symposium on Theory of Computing (STOC'96), 1996, pp. 47-55.
7. D. A. Spielman and S.-H. Teng, "Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems, " in 36th Annual IEEE Symposium on Foundations of Computer Science (FOCS'04), 2004, pp. 81-90.
8. I. Koutis, G. L. Miller, and R. Peng, "Approaching optimality for solving SDD linear systems, " in 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS'10), 2010, pp. 235-244.
9. -, "A nearly-mlog n time solver for SDD linear systems, " in 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS'11), 2011, pp. 590-598.
10. I. Koutis, A. Levin, and R. Peng, "Improved spectral sparsification and numerical algorithms for SDD matrices, " in 29th International Symposium on Theoretical Aspects of Computer Science (STACS'12), 2012, pp. 266-277.
11. R. Peng and D. A. Spielman, "An efficient parallel solver for SDD linear systems, " in 46th Annual ACM Symposium on Theory of Computing (STOC'14), 2014, pp. 333-342.
12. Y. T. Lee, R. Peng, and D. A. Spielman, "Sparsified Cholesky solvers for SDD linear systems, " arXiv:1506. 08204, 2015.
13. J. A. Kelner, Y. T. Lee, L. Orecchia, and A. Sidford, "An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations, " in 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'14), 2014, pp. 217-226.
14. J. Sherman, "Nearly maximum flows in nearly linear time, " in 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS'13), 2013, pp. 263-269.
15. J. A. Kelner and A. Levin, "Spectral sparsification in the semi-streaming setting, " Theory of Computing Systems, vol. 53, no. 2, pp. 243-262, 2013.
16. M. Kapralov, Y. T. Lee, C. Musco, C. Musco, and A. Sidford, "Single pass spectral sparsification in dynamic streams, " in 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS'14), 2014, pp. 561-570.
17. M. W. Mahoney, "Randomized algorithms for matrices and data, " Foundations and Trends in Machine Learning, vol. 3, no. 2, pp. 123-224, 2011.
18. M. Li, G. L. Miller, and R. Peng, "Iterative row sampling, " in 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS'13), 2013, pp. 127-136.
19. M. B. Cohen, Y. T. Lee, C. Musco, C. Musco, R. Peng, and A. Sidford, "Uniform sampling for matrix approximation, " in 6th Innovations in Theoretical Computer Science (ITCS'15), 2015, pp. 181-190.
20. Y. T. Lee and A. Sidford, "Path finding methods for linear programming: Solving linear programs in -O(rank) iterations and faster algorithms for maximum flow, " in 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS'14), 2014, pp. 424-433.
21. D. A. Spielman and N. Srivastava, "An elementary proof of the restricted invertibility theorem, " Israel Journal of Mathematics, vol. 190, no. 1, pp. 83-91, 2012.
22. N. Srivastava, "On contact points of convex bodies, " in Geometric Aspects of Functional Analysis, 2012, pp. 393-412.
23. A. Marcus, D. A. Spielman, and N. Srivastava, "Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem, " arXiv:1306. 3969, 2013.
24. A. Barvinok, "Thrifty approximations of convex bodies by polytopes, " International Mathematics Research Notices, vol. 2014, no. 16, pp. 4341-4356, 2014.
25. D. A. Spielman and S.-H. Teng, "Spectral sparsification of graphs, " SIAM Journal on Computing, vol. 40, no. 4, pp. 981-1025, 2011.
26. J. A. Tropp, "User-friendly tail bounds for sums of random matrices, " Foundations of Computational Mathematics, vol. 12, no. 4, pp. 389-434, 2012.
27. E. Lieb and W. Thirring, "Inequalities for the moments of the eigenvalues of the Schrödinger equation and their relation to Sobolev inequalities, " Studies in Mathematical Physics: Essays in honor of Valentine Bargman, Lieb, E., Simon, B., Wightman, AS (eds.), pp. 269-303, 1976.