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Pittsburgh, Pennsylvania, USA

Oct. 23, 2005 to Oct. 25, 2005

ISBN: 0-7695-2468-0

pp: 206-215

Sanjeev Arora , Computer Science Department, Princeton University

Eli Berger , Institute for Advanced Study, Princeton University

Guy Kindler , Institute for Advanced Study, Princeton University

Muli Safra , Institute for Advanced Study, Princeton University

Elad Hazan , Computer Science Department,Princeton University

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/SFCS.2005.57

ABSTRACT

<p>This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find \chi \varepsilon {-1, 1}^n that maximizes \chi ^{\rm T} Mx this problem recently attracted attention due to its application in various clustering settings, as well as an intriguing connection to the famous Grothendieck inequality. It is approximable to within a factor of O(log n), and known to be NP-hard to approximate within any factor better than 13/110 - \varepsilon for all \varepsilon > O. We show show that it is quasi-NP-hard to approximate to a factor better than O(Log^\gamma n) for some \gamma > 0. </p> <p>The integrality gap of the natural semide?nite relaxation for this problem is known as the Grothendieck constant the complete graph, and known to be \theta (\log n). The proof this fact was nonconstructive, and did not yield an explicit problem instance where this integrality gap is achieved. Our techniques yield an explicit instance for which the integrality gap is \Omega (\frac{{\log n}}{{\log \log n}}), essentially answering one of the open problems of Alon et al. [AMMN].</p>

INDEX TERMS

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CITATION

Sanjeev Arora,
Eli Berger,
Guy Kindler,
Muli Safra,
Elad Hazan,
"On Non-Approximability for Quadratic Programs",

*FOCS*, 2005, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science 2005, pp. 206-215, doi:10.1109/SFCS.2005.57