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

Marianna Eisenberg-Nagy (M.E.Nagy***at***cwi.nl)
Monique Laurent (M.Laurent***at***cwi.nl)
Antonios Varvitsiotis (A.Varvitsiotis***at***cwi.nl)

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 $k\ge 2$. Equivalently, for $k\ge 2$, it is $\NP$-hard to test membership in the rank constrained elliptope $\EE_k(G)$, i.e., the set of all partial matrices with 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 $\EE_k(G)$ is also $\NP$-hard for any fixed integer $k\ge 2$.

Keywords: psd matrix completion, elliptope,

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Combinatorial Optimization (Other )

Citation:

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Entry Submitted: 03/29/2012
Entry Accepted: 03/29/2012
Entry Last Modified: 07/22/2012

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