Lp metrics on the Heisenberg group and the Goemans-Linial conjecture

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

Volumn , Issue , 2006, Pages 99-108

  Lee, James R ^a   Naor, Assaf ^b  

^a INSTITUTE FOR ADVANCED STUDY   (United States)

^b MICROSOFT RESEARCH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

FOURIER ANALYSIS; MATHEMATICAL MODELS; METRIC SYSTEM; PROBLEM SOLVING;

DACUNHA-CASTELLE; GOEMANS-LINIAL CONJECTURE; LIPSCHITZ FUNCTIONS; SPARSEST CUT PROBLEM;

FUNCTION EVALUATION;

EID: 38749129134     PISSN: 02725428     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/FOCS.2006.47     Document Type: Conference Paper

References (44)

1. A. Agrawal, P. Klein, R. Ravi, and S. Rao. Approximation through multicommodity flow. In 31st Annual Symposium on Foundations of Computer Science, pages 726-737. IEEE Computer Soc., Los Alamitos, CA, 1990.
2. S. Arora, J. R. Lee, and A. Naor. Euclidean distortion and the sparsest cut [extended abstract], In STOC'05: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pages 553-562, New York, 2005. ACM.
3. S. Arora, S. Rao, and U. Vazirani. Expander flows, geometric embeddings and graph partitioning. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pages 222-231 (electronic), New York, 2004. ACM.
4. n. Bull. Soc. Math. France, 111(4):429-448, 1983.
5. Y. Aumann and Y Rabani. An O(log k) approximate mincut max-flow theorem and approximation algorithm. SIAM J. Comput., 27(1):291-301, 1998.
6. Y. Benyamini and J. Lindenstrauss. Geometric nonlinear functional analysis. Vol. 1, volume 48 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2000.
7. p. Ann. Inst. H. Poincaré Sect. B (N.S.), 2:231-259, 1965/1966.
8. S. Chawla, R. Krauthgamer, R. Kumar, Y. Rabani, and D. Sivakumar. On the hardness of approximating multicut and sparsest-cut. In 20th Annual IEEE Conference on Computational Complexity, pages 144-153, 2005.
9. 1 and the geometry of BV functions. Preprint, 2006.
10. J. Cheeger, B. Kleiner, and A. Naor. Quantitative bounds on the rate of central collapse of Lipschitz maps on the Heisenberg group. In preparation, 2006.
11. J. Cygan. Subadditivity of homogeneous norms on certain nilpotent Lie groups. Proc. Amer. Math. Soc., 83(1):69-70, 1981.
12. N. Devanur, S. Khot, R. Saket, and N. Vishnoi. Integrality gaps for sparsest cut and minimum linear arrangement problems. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pages 537-546, 2006.
13. M. M. Deza and M. Laurent. Geometry of cuts and metrics, volume 15 of Algorithms and Combinatorics. Springer-Verlag, Berlin, 1997.
14. R. Durrett. Probability: theory and examples. Duxbury Press, Belmont, CA, second edition, 1996.
15. M. X. Goemans. Semidefinite programming in combinatorial optimization. Mathematical Programming, 79:143-161, 1997.
16. M. Gromov. Carnot-Carathéodory spaces seen from within. In Sub-Riemannian geometry, volume 144 of Progr. Math, pages 79-323. Birkhauser, Basel, 1996.
17. A. Gupta, R. Krauthgamer, and J. R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In 44th Symposium on Foundations of Computer Science, pages 534-543, 2003.
18. 1 -embeddings of graphs. Combinatorica, 24(2):233-269, 2004.
19. J. Heinonen. Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001.
20. S. Heinrich. Ultraproducts in Banach space theory. J. Reine Angew. Math., 313:72-104, 1980.
21. 1. In 46th Annual Symposium on Foundations of Computer Science, pages 53-62. IEEE Computer Soc., Los Alamitos, CA, 2005.
22. P. Klein, S. A. Plotkin, and S. Rao. Excluded minors, network decomposition, and multicommodity flow. In 25th Annual ACM Symposium on Theory of Computing, pages 682-690, May 1993.
23. R. Krauthgamer, J. R. Lee, M. Mendel, and A. Naor. Measured descent: A new embedding method for finite metrics. Geom. Funct. Anal., 15(4):839-858, 2005.
24. R. Krauthgamer and Y Rabani. Improved lower bounds for embeddings into L1. In SODA '06: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pages 1010-1017, New York, NY, USA, 2006. ACM Press.
25. J. R. Lee. Improved distortion estimates for negative type metrics. Preprint, 2006.
26. 1: dimension, snowflakes, and average distortion. European J. Combin., 26(8): 1180-1190, 2005.
27. J. R. Lee and A. Naor. Extending Lipschitz functions via random metric partitions. Invent. Math., 160(1):59-95, 2005.
28. T. Leighton and S. Rao. Multicommodity max-flow mincut theorems and their use in designing approximation algorithms. J. ACM, 46(6):787-832, 1999.
29. J. Lindenstrauss and L. Tzafriri. Classical Banach spaces. I. Springer-Verlag, Berlin, 1977. Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92.
30. N. Linial. Finite metric spaces - combinatorics, geometry and algorithms. In Proceedings of the International Congress of Mathematicians III, pages 573-586, 2002.
31. 1. In J. Matoušek, editor, Open Problems, Workshop on Discrete Metric Spaces and their Algorithmic Applications. Haifa, March 2002.
32. 1. In J. Matoušek, editor, Open Problems, Workshop on Discrete Metric Spaces and their Algorithmic Applications. Haifa, March 2002.
33. N. Linial, E. London, and Y Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215-245, 1995.
34. D. W. Matula and F. Shahrokhi. The maximum concurrent flow problem. J Assoc. Comput. Mach., 37(2):318-334, 1990.
35. R. Montgomery. A tour of subriemannian geometries, their geodesies and applications, volume 91 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2002.
36. P. Pansu. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. of Math. (2), 129(1):1-60, 1989.
37. D. Preiss and L. Zajíček. Sigma-porous sets in products of metric spaces and sigma-directionally porous sets in Banach spaces. Real Anal. Exchange, 24(1):295-313, 1998/99.
38. Y. Rabinovich. On average distortion of embedding metrics into the line and into I1. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pages 456-462 (electronic), New York, 2003. ACM.
39. I. J. Schoenberg. Metric spaces and positive definite functions. Trans. Amer. Math. Soc., 44(3):522-536, 1938.
40. ∞ -weights. Rev. Mat. Iberoamericana, 12(2):337-410, 1996.
41. D. B. Shmoys. Cut problems and their application to divide-and-conquer. In Approximation Algorithms for NP-hard Problems, (D.S. Hochbaum, ed.), pages 192-235. PWS, 1997.
42. E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.
43. K. Talwar. Decomposability of metrics. In J. Matoušek, editor, Open Problems, Workshop on Discrete Metric Spaces and their Algorithmic Applications. Haifa, March 2002.
44. J. H. Wells and L. R. Williams. Embeddings and extensions in analysis. Springer-Verlag, New York, 1975. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84.