Slack matrices, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e20" altimg="si15.svg"><mml:mi>k</mml:mi></mml:math>-products, and 2-level polytopes

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