Pipage rounding, pessimistic estimators and matrix concentration

Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Volumn , Issue , 2014, Pages 926-945

  Harvey, Nicholas J A ^a   Olver, Neil ^b,c  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^b VU UNIVERSITY AMSTERDAM   (Netherlands)

^c CWI   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

FORESTRY; POLYNOMIAL APPROXIMATION; TRAVELING SALESMAN PROBLEM; TREES (MATHEMATICS);

ASYMMETRIC TRAVELING SALESMAN PROBLEM; DIVERSE APPLICATIONS; ISOTROPIC DISTRIBUTIONS; NEGATIVE CORRELATION; ORTHONORMAL BASIS; POLYNOMIAL-TIME ALGORITHMS; SEMIDEFINITE PROGRAMS; SUBMODULAR FUNCTIONS;

APPLICATION PROGRAMS;

FORESTRY; MATHEMATICS; TREES;

EID: 84902105494     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1137/1.9781611973402.69     Document Type: Conference Paper

References (55)

1. A. A. Ageev and M. Sviridenko. Pipage rounding: A new method of constructing algorithms with proven performance guarantee. Journal of Combinatorial Optimization, 8, 2004.
2. R. Ahlswede and A. Winter. Strong converse for identification via quantum channels. IEEE Transactions on Information Theory, 48(3):569-579, Mar. 2002.
3. A. Asadpour, M. X. Goemans, A. Madry, S. Oveis Gharan, and A. Saberi. An O(logn/loglogn)- Approximation algorithm for the asymmetric traveling salesman problem. In SODA, 2010.
4. N. Bansal. Refinements of randomized rounding. Invited lecture at the Landelijk Netwerk Mathematische Beslikunde Lunteren Conference, 2012.
5. J. Batson, D. A. Spielman, and N. Srivastava. Twice- Ramanujan sparsifiers. In STOC, 2009.
6. R. Bhatia. Matrix Analysis. Springer, 1996.
7. R. Bhatia. The logarithmic mean. Resonance, 13, 2008.
8. B. Bollobas. Modern Graph Theory. Springer, 1998.
9. J. Bourgain and L. Tzafriri. Invertibility of "large" submatrices with applications to the geometry of Banach spaces and harmonic analysis. Israel J. Math, 57, 1987.
10. J. Bourgain and L. Tzafriri. On a problem of Kadison and Singer. J. Reine. Angew. Math., 420:1-43, 1991.
11. C. Boutsidis, P. Drineas, and M. Magdon-Ismail. Near optimal column-based matrix reconstruction. In FOCS, 2011.
12. C. Boutsidis, M. W. Mahoney, and P. Drineas. An improved approximation algorithm for the column subset selection problem. In SODA, 2009.
13. S. C. Boyd and W. R. Pulleyblank. Optimizing over the subtour polytope of the travelling salesman problem. Mathematical Programming, 49:163-187, 1991.
14. G. Calinescu, C. Chekuri, M. Pal, and J. Vondrak. Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing, 40, 2011.
15. B. C. Carlson. The logarithmic mean. The American Mathematical Monthly, 79(6):615-618, 1972.
16. P. G. Casazza and J. C. Tremain. The Kadison-Singer problem in mathematics and engineering. Proceedings of the National Academy of Sciences, 103(7):2032-2039, 2006.
17. C. Chekuri, J. Vondrak, and R. Zenklusen. Dependent randomized rounding for matroid polytopes and applications, 2009. arXiv:0909.4348.
18. C. Chekuri, J. Vondrak, and R. Zenklusen. Dependent randomized rounding via exchange properties of combinatorial structures. In FOCS, pages 575-584, 2010.
19. R. Y. Chen, A. Gittens, and J. A. Tropp. The masked sample covariance estimator: An analysis using matrix concentration inequalities. Information and Inference, 2012.
20. A. Deshpande and L. Rademacher. Efficient volume sampling for row/column subset selection. In FOCS, 2010.
21. H. Epstein. Remarks on Two Theorems of E. Lieb. Communications in Mathematical Physics, 31, 1973.
22. S. Fujishige. Submodular Functions and Optimization. Elsevier, second edition, 2005.
23. R. Gandhi, S. Khuller, S. Parthasarathy, and A. Srinivasan. Dependent rounding and its applications to approximation algorithms. Journal of the ACM, 53(3) :324- 360, 2006.
24. F. Gesztesy and E. Tsekanovskii. On matrix-valued Herglotz functions. Math. Nachrichten, 218:61-138, 2000.
25. L. A. Goddyn. Some open problems I like. Available at http://math.sfu.ca/goddyn/Problems/problems.html.
26. M. Goemans. Thin spanning trees. Lecture at "Graph Theory @ Georgia Tech: Conference Honoring the 50th Birthday of Robin Thomas", 2012.
27. R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge University Press, 1991.
28. R. V. Kadison and J. R. Ringrose. Fundamentals of the Theory of Operator Algebras. American Mathematical Society, 1983.
29. T. Leighton, S. Rao, and A. Srinivasan. Multicom- modity flow and circuit switching. In Proc. Hawaii International Conference on System Sciences (HICSS), 1998.
30. E. H. Lieb. Convex trace functions and the Wigner-Yanase-Dyson conjecture. Advances in Mathematics, 11:267-288, 1973.
31. M. W. Mahoney. Randomized algorithms for matrices and data. arXiv: 1104.5557.
32. Marcus, D. A. Spielman, and N. Srivastava. Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem, June 2013. arXiv: 1306.3969.
33. M. Ohya and D. Petz. Quantum Entropy and Its Use. Springer-Verlag, 1993.
34. R. I. Oliveira. Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges, 2010. arXiv:0911.0600.
35. S. Oveis Gharan and A. Saberi. Asymmetric traveling salesman problem on graphs with bounded genus. In SODA, 2011.
36. A.Panconesi and A. Srinivasan. Randomized distributed edge coloring via an extension of the Chernoff- Hoeffding bounds. SIAM Journal on Computing, 26(2):350-368, 1997.
37. P. Raghavan. Probabilistic construction of deterministic algorithms: Approximating packing integer programs. Journal of Computer and System Sciences, 37, 1988.
38. P. Raghavan and C. D. Thompson. Randomized rounding: A technique for provably good algorithms and algorithmic proofs. Combinatorica, 7:365-374, 1987.
39. B. Recht. A simpler approach to matrix completion. Journal of Machine Learning Research, 12, 2011.
40. M. Rudelson. Almost orthogonal submatrices of an orthogonal matrix. Israel J. Math., 111(1):143-155, 1999.
41. M. Rudelson and R. Vershynin. Sampling from large matrices: An approach through geometric functional analysis. Journal of the ACM, 54(4), 2007.
42. W. Rudin. Real and Complex Analysis. McGraw-Hill, third edition, May 1986.
43. D. A. Spielman and N. Srivastava. Graph sparsification by effective resistances. In STOC, 2008.
44. D. A. Spielman and N. Srivastava. An elementary proof of the restricted invertibility theorem. Israel J. Math, 190(1):83-91, Aug. 2012.
45. A. Srinivasan. Approximation algorithms via randomized rounding: A survey. In M. Karonski and H. J. Promel, editors, Lectures on Approximation and Randomized Algorithms, pages 9-71. Polish Scientific Publishers PWN, 1999.
46. A. Srinivasan. Distributions on level-sets with applications to approximation algorithms. In FOCS, 2001.
47. A. Srinivasan. Pessimistic estimators, 2003. Lecture notes for CMSC858T "Randomized Algorithms".
48. J. A. Tropp. Column subset selection, matrix factorization, and eigenvalue optimization. In SODA, 2009.
49. J. A. Tropp. User-friendly tail bounds for sums of random matrices. Foundations of Computational Mathematics, 2011.
50. J. A. Tropp. From the joint convexity of quantum relative entropy to a concavity theorem of Lieb. Proc. Amer. Math. Soc., 140(5): 1757-1760, 2012.
51. R. Vershynin. Introduction to the non-asymptotic analysis of random matrices. In Y. Eldar and G. Kutyniok, editors, Compressed Sensing: Theory and Applications, chapter 5, pages 210-268. Cambridge University Press, 2012.
52. N. Weaver. The Kadison-Singer problem in discrepancy theory. Discrete Mathematics, 278:227-239, 2004.
53. A. Wigderson and D. Xiao. Derandomizing the Ahlswede-Winter matrix-valued Chernoff bound using pessimistic estimators and applications. Theory of Computing, 4(3), 2008.
54. D. P. Williamson and D. B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, 2011.
55. P. Youssef. Restricted invertibility and the Banach- Mazur distance to the cube, 2012. arXiv:1206.0654vl.