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

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.