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.
|Title of host publication||Discrete Geometry and Optimization|
|Editors||K. Bezdek, A. Deza, Y. Ye|
|Place of Publication||Heidelberg|
|Number of pages||336|
|Publication status||Published - 2013|
|Name||Fields Institute Communications|