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The Positive Semidefinite Grothendieck Problem with Rank Constraint

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Book cover Automata, Languages and Programming (ICALP 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6198))

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Abstract

Given a positive integer n and a positive semidefinite matrix \(A = (A_{ij}) \in {\mathbb R}^{m \times m}\) the positive semidefinite Grothendieck problem with rank-n-constraint (SDP n ) is

$$ \text{maximize } \sum_{i=1}^m \sum_{j=1}^m A_{ij} \; x_i \cdot x_j, \text{ where } x_1, \ldots, x_m \in S^{n-1}. $$

In this paper we design a randomized polynomial-time approximation algorithm for SDP n achieving an approximation ratio of

$$ \gamma(n) = \frac{2}{n}\left(\frac{\Gamma((n+1)/2)}{\Gamma(n/2)}\right)^2 = 1 - \Theta(1/n). $$

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 γ(n). We improve the approximation ratio of the best known polynomial-time algorithm for SDP n from 2/π to 2/(πγ(m)) = 2/π + Θ(1/m), and we show a tighter approximation ratio for SDP n when A is the Laplacian matrix of a weighted graph with nonnegative edge weights.

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Author information

Authors and Affiliations

  1. Centrum Wiskunde & Informatica (CWI), Science Park 123, 1098, SJ, Amsterdam, The Netherlands

    Jop Briët

  2. Department of Econometrics ad OR, Tilburg University, 5000, LE, Tilburg, The Netherlands

    Fernando Mário de Oliveira Filho

  3. Delft Institute of Applied Mathematics, Technical University of Delft, P.O. Box 5031, 2600, GA, Delft, The Netherlands

    Frank Vallentin

Authors
  1. Jop Briët
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  2. Fernando Mário de Oliveira Filho
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  3. Frank Vallentin
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Editor information

Editors and Affiliations

  1. Oxford University Computing Laboratory, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Samson Abramsky

  2. Université de Bordeaux (LaBRI) & INRIA, 351, cours de la Libération, 33405, Talence Cedex, France

    Cyril Gavoille

  3. INRIA, Centre de Recherche Bordeaux – Sud-Ouest, 351 cours de la Libération, 33405, Talence Cedex, France

    Claude Kirchner

  4. Heinz Nixdorf Institute, University of Paderborn, Fürstenallee 11, 33102, Paderborn, Germany

    Friedhelm Meyer auf der Heide

  5. University of Patras and RACTI, 26500, Patras, Greece

    Paul G. Spirakis

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Briët, J., de Oliveira Filho, F.M., Vallentin, F. (2010). The Positive Semidefinite Grothendieck Problem with Rank Constraint. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds) Automata, Languages and Programming. ICALP 2010. Lecture Notes in Computer Science, vol 6198. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14165-2_4

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  • DOI: https://doi.org/10.1007/978-3-642-14165-2_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14164-5

  • Online ISBN: 978-3-642-14165-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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