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Marianna EisenbergNagy (M.E.Nagycwi.nl) Abstract: We consider the decision problem asking whether a partial rational symmetric matrix with an allones 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 offdiagonal 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 (Semidefinite Programming ) Category 2: Combinatorial Optimization (Other ) Citation: Download: [PDF] Entry Submitted: 03/29/2012 Modify/Update this entry  
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