Complexity of the positive semidefinite matrix completion problem with a rank constraint

M. Nagy, M. Laurent, A. Varvitsiotis

Research output: Chapter in Book/Report/Conference proceedingChapterScientificpeer-review

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 NP-hard for any fixed integer k ≥ 2. In other words, for k ≥ 2, it is NP-hard to test membership in the rank constrained elliptope Ek(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 Ek(G) is also NP-hard for any fixed integer k ≥ 2.
Original languageEnglish
Title of host publicationDiscrete Geometry and Optimization
EditorsK. Bezdek, A. Deza, Y. Ye
Place of PublicationHeidelberg
PublisherSpringer Verlag
Pages105-120
Number of pages336
ISBN (Print)9783319001999
Publication statusPublished - 2013

Publication series

NameFields Institute Communications
Number69

Fingerprint

Matrix Completion Problem
Positive Semidefinite Matrix
NP-complete problem
Partial Matrix
Integer
Symmetric matrix
Decision problem
Convex Hull
Partial

Cite this

Nagy, M., Laurent, M., & Varvitsiotis, A. (2013). Complexity of the positive semidefinite matrix completion problem with a rank constraint. In K. Bezdek, A. Deza, & Y. Ye (Eds.), Discrete Geometry and Optimization (pp. 105-120). (Fields Institute Communications; No. 69). Heidelberg: Springer Verlag.
Nagy, M. ; Laurent, M. ; Varvitsiotis, A. / Complexity of the positive semidefinite matrix completion problem with a rank constraint. Discrete Geometry and Optimization. editor / K. Bezdek ; A. Deza ; Y. Ye. Heidelberg : Springer Verlag, 2013. pp. 105-120 (Fields Institute Communications; 69).
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Nagy, M, Laurent, M & Varvitsiotis, A 2013, Complexity of the positive semidefinite matrix completion problem with a rank constraint. in K Bezdek, A Deza & Y Ye (eds), Discrete Geometry and Optimization. Fields Institute Communications, no. 69, Springer Verlag, Heidelberg, pp. 105-120.

Complexity of the positive semidefinite matrix completion problem with a rank constraint. / Nagy, M.; Laurent, M.; Varvitsiotis, A.

Discrete Geometry and Optimization. ed. / K. Bezdek; A. Deza; Y. Ye. Heidelberg : Springer Verlag, 2013. p. 105-120 (Fields Institute Communications; No. 69).

Research output: Chapter in Book/Report/Conference proceedingChapterScientificpeer-review

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N2 - 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 NP-hard for any fixed integer k ≥ 2. In other words, for k ≥ 2, it is NP-hard to test membership in the rank constrained elliptope Ek(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 Ek(G) is also NP-hard for any fixed integer k ≥ 2.

AB - 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 NP-hard for any fixed integer k ≥ 2. In other words, for k ≥ 2, it is NP-hard to test membership in the rank constrained elliptope Ek(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 Ek(G) is also NP-hard for any fixed integer k ≥ 2.

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T3 - Fields Institute Communications

SP - 105

EP - 120

BT - Discrete Geometry and Optimization

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Nagy M, Laurent M, Varvitsiotis A. Complexity of the positive semidefinite matrix completion problem with a rank constraint. In Bezdek K, Deza A, Ye Y, editors, Discrete Geometry and Optimization. Heidelberg: Springer Verlag. 2013. p. 105-120. (Fields Institute Communications; 69).