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arXiv · 2605.02896

Hardness of some optimization problems over correlation polyhedra

Alberto Caprara·Fabio Furini·Claudio Gentile·Leo Liberti·Andrea Lodi
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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.

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Alberto Caprara, Fabio Furini, Claudio Gentile, Leo Liberti, Andrea Lodi. 2026-03-20. Hardness of some optimization problems over correlation polyhedra. https://arxiv.org/abs/2605.02896

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