Generalisations of total unimodularity

In this paper we examine possible generalisations of total unimodularity. To this end, we introduce two concepts: total k-modularity and k-regularity. Total k-modularity extends the permitted values for the subdeterminants of a matrix to the powers of k, while k-regularity sets requirements on the inverses of non-singular submatrices. It is shown that the advantageous properties of totally unimodular matrices with respect to integral polyhedra can be carried over to k-regular matrices, namely we prove that a matrix A is k-regular if and only if the polyhedron {x : x ≥ 0, Ax ≤ b} is integral for all integral vectors b the components of which have a common divisor k. We present some results on totally k-modular and k-regular rational matrices, as well as give examples of 2-regular matrices. In particular, we define binet matrices, a generalisation of network matrices for bidirected graphs. Keywords—Total unimodularity, integral polyhedron, network matrices, integer programming