O(√log n) approximation to sparsest cut in õ(n2) time

SIAM Journal on Computing

Volumn 39, Issue 5, 2010, Pages 1748-1771

  Arora, Sanjeev ^a   Hazan, Elad ^b   Kale, Satyen ^c  

^a Princeton University   (United States)

^b IBM ALMADEN RESEARCH CENTER   (United States)

^c YAHOO RESEARCH   (United States)

Author keywords

Expander flows; Graph partitioning; Multiplicative weights

Indexed keywords

GRAPH PARTITIONING; SEMI-DEFINITE PROGRAMMING; SEMIDEFINITE PROGRAMS; SPARSEST CUT;

GRAPH THEORY; MATHEMATICAL PROGRAMMING;

COMPUTATION THEORY;

EID: 77049125312     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/080731049     Document Type: Article

References (31)

1. F. ALIZADEH, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim., 5 (1995), pp. 13-51.
2. N. Alon, Eigenvalues and expanders, Combinatorica, 6 (1986), pp. 83-96.
3. 1, isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B, 38 (1985), pp. 73-88.
4. S. Arora, E. HAZAN, AND S. Kale, The multiplicative weights method: A meta-algorithm and some applications, Theory Comput., submitted.
5. S. Arora and S. Kale, A combinatorial, primal-dual approach to semidefinite programs, in Proceedings of the 39th Annual ACM Symposium on Theory of Computing, 2007, pp. 227-236.
6. S. Arora, S. Rao, AND U. V. Vazirani, Expander flows, geometric embeddings and graph partitioning, in Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 222-231.
7. Y. Aumann AND Y. Rabani, An O(log k) approximate min-cut max-flow theorem and approximation algorithm, SIAM J. Comput., 27 (1998), pp. 291-301.
8. 2) time, in Proceedings of the 28th Annual ACM Symposium on Theory of Computing, 1996, pp. 47-55.
9. M. Blum, R. M. Karp, O. Vornberger, C. H. Papadimitriou, and M. Yannakakis, The complexity of testing whether a graph is a superconcentrator, Inform. Process. Lett., 13 (1981), pp. 164-167.
10. S. Boyd AND L. VANDENBERGHE, Convex Optimization, Cambridge University Press, Cambridge, UK, 2004.
11. S. CHAWLA, A. Gupta, AND H. Räcke, Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut, in Proceedings of the Sixteenth Annual ACM- SIAM Symposium on Discrete Algorithms, 2005, pp. 102-111.
12. J. CHEEGER, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, Princeton University Press, Princeton, NJ, 1970, pp. 195-199.
13. F. CHUNG, Spectral Graph Theory, CBMS Regional Conf. Ser. in Math. 92, AMS, Providence, RI, 1997.
14. L. K. FLEISCHER, Approximating fractional multicommodity flow independent of the number of commodities, SIAM J. Discrete Math., 13 (2000), pp. 505-520.
15. Y. FREUND AND R. E. SCHAPIRE, Adaptive game playing using multiplicative weights, Games Econom. Behav., 29 (1999), pp. 79-103.
16. N. Garg AND J. KÖNEMANN, Faster and simpler algorithms for multicommodity flow and other fractional packing problems, in Proceedings of the 39th Annual Symposium on Foundations of Computer Science, 1998, pp. 300-309.
17. M. X. Goemans AND D. P. WILLIAMSON, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM, 42 (1995), pp. 1115-1145.
18. A. V. GOLDBERG AND R. E. TARJAN, A new approach to the maximum-flow problem, J. ACM, 35 (1988), pp. 921-940.
19. R. Khandekar, S. Rao, AND U. VAZIRANI, Graph partitioning using single commodity flows, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 385-390.
20. P. Klein, S. Plotkin, C. Stein, AND É. TARDOS, Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts, SIAM J. Comput., 23 (1994), pp. 466-487.
21. J. Kuczyński AND H. WOŹNIAKOWSKI, Estimating the largest eigenvalue by the power and Lanczos algorithms with a random start, SIAM J. Mat rix Anal. Appl., 13 (1992), pp. 1094-1122.
22. F. T. LEIGHTON AND S. Rao, Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms, J. ACM, 46 (199 9), pp. 787-832.
23. N. LINIAL, E. LONDON, AND Y. RABINOVICH, The geometry of graphs and some of its algorithmic applications, Combinatorica, 15 (1995), pp. 215-245.
24. D. W. MATULA, A linear time 2 + ε approximation algorithm for edge connectivity, 'm Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, 1993, pp. 500-504.
25. r. motwani and p. raghavan, Randomized Algorithms, Cambridge University Press, Cambridge, UK, 1995.
26. l. Orecchia, l. j. Schulman, u. v. Vazirani, and n. k. Vishnoi, On partitioning graphs via single commodity flows, in Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2008, pp. 461-470.
27. s. a. plotkin, d. b. Shmoys, and É. Tardos, Fast approximation algorithms for fractional packing and covering problems, in Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, 1991, pp. 495-504.
28. f. shahrokhi and d. w. Matula, The maximum concurrent flow problem, J. ACM, 37 (1990), pp. 318-334.
29. d. s. Shmoys, Cut problems and their application to divide and conquer, in Approximation Algorithms for NP-hard Problems, PWS, Boston, 1995.
30. v. vazirani, Approximation Algorithms, Springer-Verlag, Berlin, 2002.
31. n. e. young, Randomized rounding without solving the linear program, in Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 1995, pp. 170-178.