# Recognizing Cartesian products of matrices and polytopes

The 1-product of matrices S_1 ∈R^m_1 × n_1 and S_2 ∈R^m_2 × n_2 is the matrix in R^(m_1+m_2) × (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 of polytopes, which goes through the concept of slack matrix. Determining whether a given matrix is a slack matrix is an intriguing problem whose complexity is unknown, and our algorithm reduces the problem to irreducible instances. Our algorithm is based on minimizing a symmetric submodular function that expresses mutual information in information theory. We also give a polynomial time algorithm to recognize a more complicated matrix product, called the 2-product. Finally, as a corollary of our 1-product and 2-product recognition algorithms, we obtain a polynomial time algorithm to recognize slack matrices of 2-level matroid base polytopes.

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