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Maximizing Quadratic Programs: Extending Grothendieck's Inequality
Abstract
This paper considers the following type of quadratic programming problem. Given an arbitrary matrix A, whose diagonal elements are zero, find x /spl isin/ {-1, 1}/sup n/ such that x/sup T/Ax is maximized. Our approximation algorithm for this problem uses the canonical semidefinite relaxation and returns a solution whose ratio to the optimum is in /spl Omega/(1/ logn). This quadratic programming problem can be seen as an extension to that of maximizing x/sup T/Ay (where y's components are also /spl plusmn/1). Grothendieck's inequality states that the ratio of the optimum value of the latter problem to the optimum of its canonical semidefinite relaxation is bounded below by a constant. The study of this type of quadratic program arose from a desire to approximate the maximum correlation in correlation clustering. Nothing substantive was known about this problem; we present an /spl Omega/ (1/logn) approximation, based on our quadratic programming algorithm. We can also guarantee that our quadratic programming algorithm returns a solution to the MAXCUT problem that has a significant advantage over a random assignment.
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Goemans, M. X., & Williamson, D. P. (1995) · Journal of the ACM · doi:10.1145/227683.227684 · Cited by 3,587
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Metadata
- Authors
- Charikar, M., & Wirth, A.
- Venue
- 45th Annual IEEE Symposium on Foundations of Computer Science
- Publisher
- IEEE
- Published
- 2004-12-23
- Pages
- 54-60
- ISSN
- 0272-5428 (ISSN-L)
- OpenAlex
- W2167681887
- Cited by
- 214
- References
- 1
- FWCI
- 8.37
- Type
- conference paper
- Language
- en
- Open access
- closed
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- Charikar, M., & Wirth, A. (2004). Maximizing Quadratic Programs: Extending Grothendieck's Inequality. 45th Annual IEEE Symposium on Foundations of Computer Science. https://doi.org/10.1109/focs.2004.39
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