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Complexity of the Positive Semidefinite Matrix Completion Problem with a Rank Constraint

Chapter
Part of the Fields Institute Communications book series (FIC, volume 69)

Abstract

We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most k. We show that this problem is \(\mathcal{N}\mathcal{P}\)-hard for any fixed integer k ≥ 2. In other words, for k ≥ 2, it is \(\mathcal{N}\mathcal{P}\)-hard to test membership in the rank constrained elliptope \(\mathcal{E}_{k}(G)\), defined by the set of all partial matrices with an all-ones diagonal and off-diagonal entries specified at the edges of G, that can be completed to a positive semidefinite matrix of rank at most k. Additionally, we show that deciding membership in the convex hull of \(\mathcal{E}_{k}(G)\) is also \(\mathcal{N}\mathcal{P}\)-hard for any fixed integer k ≥ 2.

Key words

Elliptope Correlation matrix psd matrix completion 

Subject Classifications

90C22 68Q17 05C62 

Notes

Acknowledgements

We thank A. Schrijver for useful discussions and a referee for drawing our attention to the paper by Aspnes et al. [1].

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Centrum Wiskunde & Informatica (CWI)AmsterdamThe Netherlands
  2. 2.Tilburg UniversityTilburgThe Netherlands

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