Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The positive semidefinite Grothendieck problem with rank constraint   Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of
size m x m, the positive semidefinite Grothendieck problem with
rank-n-constraint (SDP_n) is
  maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, ..., x_m
\in S^{n-1}.
  In this paper we design a polynomial time approximation algorithm for SDP_n
achieving an approximation ratio of
  \gamma(n) = \frac{2}{n}(\frac{\Gamma((n+1)/2)}{\Gamma(n/2)})^2 = 1 -
\Theta(1/n).
  We show that under the assumption of the unique games conjecture the achieved
approximation ratio is optimal: There is no polynomial time algorithm which
approximates SDP_n with a ratio greater than \gamma(n). We improve the
approximation ratio of the best known polynomial time algorithm for SDP_1 from
2/\pi to 2/(\pi\gamma(m)) = 2/\pi + \Theta(1/m), and we show a tighter
approximation ratio for SDP_n when A is the Laplacian matrix of a graph with
nonnegative edge weights.

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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
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The categories of this paper are: chem-ph
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The categories of this paper are: nlin
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The categories of this paper are: gr-qc
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The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
27
The categories of this paper are: comp-gas