Rational and integral k-regular matrices Original Research Article

In this paper we examine two possible generalisations of total unimodularity, viz., total k-modularity and k-regularity. Total k-modularity extends the permitted values for the subdeterminants of an integral matrix to the powers of k, while k-regularity sets requirements on the inverses of non-singular submatrices of a rational matrix. It is shown that the advantageous properties of totally unimodular matrices with respect to integral polyhedra can be carried over to rational k-regular matrices, namely we prove that a matrix A is k-regular if and only if the polyhedron P(A,b)={x:x⩾0, Ax⩽b} is integral for all integral vectors b the components of which have a common divisor k. Furthermore, we show that the k-regularity of an integral matrix A is equivalent to the fact that for any integral vector b all the rank-1 Chvátal-Gomory cuts for P(A,b) are dominated by mod-k cuts. We present some results on totally k-modular and k-regular matrices, as well as give non-trivial examples of 1- and 2-regular matrices. In particular, we define binet matrices, a generalisation of network matrices for bidirected graphs.