Recognizing Cartesian products of matrices and polytopes

@article{Aprile2020RecognizingCP,
  title={Recognizing Cartesian products of matrices and polytopes},
  author={Manuel Aprile and Michele Conforti and Yuri Faenza and Samuel Fiorini and Tony Huynh and Marco Macchia},
  journal={ArXiv},
  year={2020},
  volume={abs/2002.02264}
}
The 1-product of matrices $S_1 \in \mathbb{R}^{m_1 \times n_1}$ and $S_2 \in \mathbb{R}^{m_2 \times n_2}$ is the matrix in $\mathbb{R}^{(m_1+m_2) \times (n_1n_2)}$ whose columns are the concatenation of each column of $S_1$ with each column of $S_2$. Our main result is a polynomial time algorithm for the following problem: given a matrix $S$, is $S$ a 1-product, up to permutation of rows and columns? Our main motivation is a close link between the 1-product of matrices and the Cartesian product… 

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