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

TL;DR

This paper proves that determining whether a partial symmetric matrix with all-ones diagonal can be completed to a positive semidefinite matrix of rank at most k is NP-hard for any fixed k ≥ 2, highlighting computational complexity challenges.

Contribution

It establishes NP-hardness results for the positive semidefinite matrix completion problem with rank constraints, extending complexity understanding of the problem.

Findings

NP-hardness for rank-constrained completion when k ≥ 2

NP-hardness of membership testing in the convex hull of the rank-constrained elliptope

Complexity results apply to partial matrices with specified off-diagonal entries at graph edges

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 $k\ge 2$.