Book contents
- Frontmatter
- Contents
- Preface
- I An Introduction to the Techniques
- II Further Uses of the Techniques
- 9 Further Uses of Greedy and Local Search Algorithms
- 10 Further Uses of Rounding Data and Dynamic Programming
- 11 Further Uses of Deterministic Rounding of Linear Programs
- 12 Further Uses of Random Sampling and Randomized Rounding of Linear Programs
- 13 Further Uses of Randomized Rounding of Semidefinite Programs
- 14 Further Uses of the Primal-Dual Method
- 15 Further Uses of Cuts and Metrics
- 16 Techniques in Proving the Hardness of Approximation
- 17 Open Problems
- Appendix A Linear Programming
- Appendix B NP-Completeness
- Bibliography
- Author Index
- Subject Index
13 - Further Uses of Randomized Rounding of Semidefinite Programs
from II - Further Uses of the Techniques
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- I An Introduction to the Techniques
- II Further Uses of the Techniques
- 9 Further Uses of Greedy and Local Search Algorithms
- 10 Further Uses of Rounding Data and Dynamic Programming
- 11 Further Uses of Deterministic Rounding of Linear Programs
- 12 Further Uses of Random Sampling and Randomized Rounding of Linear Programs
- 13 Further Uses of Randomized Rounding of Semidefinite Programs
- 14 Further Uses of the Primal-Dual Method
- 15 Further Uses of Cuts and Metrics
- 16 Techniques in Proving the Hardness of Approximation
- 17 Open Problems
- Appendix A Linear Programming
- Appendix B NP-Completeness
- Bibliography
- Author Index
- Subject Index
Summary
We introduced the use of semidefinite programming for approximation algorithms in Chapter 6. The algorithms of that chapter solve a vector programming relaxation, then choose a random hyperplane (or possibly many hyperplanes) to partition the vectors in some way. The central component of the analysis of these algorithms is Lemma 6.7, which says that the probability of two vectors being separated by a random hyperplane is proportional to the angle between them. In this chapter, we look at ways in which we can broaden both the analysis of algorithms using semidefinite programming, and the algorithms themselves.
To broaden our analytical techniques, we revisit two of the problems we discussed initially in Chapter 6. In particular, we consider the problem of approximating integer quadratic programs, which was introduced in Section 6.3, and the problem of coloring a 3-colorable graph, which was introduced in Section 6.5. In our algorithms in this chapter, we again solve vector programming relaxations of the problems, and choose a random hyperplane by drawing its components from the normal distribution. Here, however, our analysis of the algorithms will rely on several more properties of the normal distribution than we used in the previous chapter; in particular, it will be helpful for us to use bounds on the tail of the normal distribution.
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- The Design of Approximation Algorithms , pp. 333 - 354Publisher: Cambridge University PressPrint publication year: 2011