A decomposition theory for matroids. VII. Analysis of minimal violation matrices

Abstract

Let P be a matrix property that is defined for the matrices over GF(2) or GF(3), and that is maintained under submatrix taking, row and column permutations, scaling, and pivots, and when a row or column unit vector is adjoined. We propose a general matroid-based technique for investigating the minimal violation matrices of P. The problem of understanding these matrices is converted to a matroid problem involving a certain class of matroids that is closed under the taking of minors. Each matroid of the class has its elements labelled in a novel way. The use of these labels is crucial for the efficacy of the overall approach. We apply the method to the case where P is the property of regularity (a binary matrix is regular if it can be signed to become a totally unimodular real matrix). The method yields surprisingly simple and complete constructions for the following matrix classes, for which to-date no complete construction other than enumeration has been known: The minimal nonregular binary matrices, the minimal violation matrices of total unimodularity, the complement totally unimodular matrices, and the ternary matrices that are almost binary. In addition we construct a large subclass of balanced matrices none of which is totally unimodular.