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arXiv · 1408.2270

A sub-constant improvement in approximating the positive semidefinite Grothendieck problem

Roy Frostig·Sida I. Wang
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Abstract

Semidefinite relaxations are a powerful tool for approximately solving combinatorial optimization problems such as MAX-CUT and the Grothendieck problem. By exploiting a bounded rank property of extreme points in the semidefinite cone, we make a sub-constant improvement in the approximation ratio of one such problem. Precisely, we describe a polynomial-time algorithm for the positive semidefinite Grothendieck problem -- based on rounding from the standard relaxation -- that achieves a ratio of $2/\pi + \Theta(1/{\sqrt n})$, whereas the previous best is $2/\pi + \Theta(1/n)$. We further show a corresponding integrality gap of $2/\pi+\tilde{O}(1/n^{1/3})$.

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Roy Frostig, Sida I. Wang. 2014-08-10. A sub-constant improvement in approximating the positive semidefinite Grothendieck problem. https://arxiv.org/abs/1408.2270

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