Hardness of some optimization problems over correlation polyhedra

TL;DR

This paper establishes NP-hardness results for several problems related to the correlation polytope, including membership, rank, and a relaxed rank variant, highlighting computational challenges in these areas.

Contribution

It proves NP-hardness of key problems over correlation polyhedra, including a novel relaxed rank problem relevant to signal processing and statistics.

Findings

Membership problem is NP-hard.

Rank determination over the correlation cone is NP-hard.

Relaxed rank problem is NP-hard and applicable in applications.

Abstract

We prove the \textbf{NP}-hardness, using Karp reductions, of some problems related to the correlation polytope and its corresponding cone, spanned by all of the $n\times n$ rank-one matrices over $\{0,1\}$. The problems are: membership, rank of the decomposition, and a ``relaxed rank'' obtained from relaxing the zero-norm expression for the rank to an $\ell_1$ norm. While membership and rank are natural problems for any matrix cone, the relaxed rank problem occurs in some signal processing and statistical applications.