Slack matrices, k-products, and 2-level polytopes

In this paper, we study algorithmic questions concerning products of matrices and their consequences for recognition algorithms for polyhedra. The 1-product of matrices S1∈Rm1×n1,S2∈Rm2×n2 is a matrix in R(m1+m2)×(n1n2) whose columns are the concatenation of each column of S1 with each column of S2. The k-product generalizes the 1-product, by taking as input two matrices S1,S2 together with k−1 special rows of each of those matrices, and outputting a certain composition of S1,S2. Our first result is a polynomial time algorithm for the following problem: given a matrix S, is S a k-product of some matrices, up to permutation of rows and columns? Our algorithm is based on minimizing a symmetric submodular function that expresses mutual information from information theory. Our study is motivated by a close link between the 1-product of matrices and the Cartesian product of polytopes, and more generally between the k-product of matrices and the glued product of polytopes. These connections rely on the concept of a slack matrix, which gives an algebraic representation of classes of affinely equivalent polytopes. The slack matrix recognition problem is the problem of determining whether a given matrix is a slack matrix. This is an intriguing problem whose complexity is unknown. Our algorithm reduces the problem to instances which cannot be expressed as k-products of smaller matrices. In the second part of the paper, we give a combinatorial interpretation of k-products for two well-known classes of polytopes: 2-level matroid base polytopes and stable set polytopes of perfect graphs. We also show that the slack matrix recognition problem is polynomial-time solvable for such polytopes. Those two classes are special cases of 2-level polytopes, for which we conjecture that the slack matrix recognition problem is polynomial-time solvable.