A generalization of totally unimodular and network matrices
In this thesis we discuss possible generalizations of totally unimodular and network matrices. Our purpose is to introduce new classes of matrices that preserve the advantageous properties of these well-known matrices. In particular, our focus is on the polyhedral consequences of totally unimodular matrices, namely we look for matrices that can ensure vertices that are scalable to an integral vector by an integer k. We argue that simply generalizing the determinantal structure of totally unimodular matrices does not suffice to achieve this goal and one has to extend the range of values the inverses of submatrices can contain. To this end, we define k-regular matrices. We show that k-regularity is a proper generalization of total unimodularity in polyhedral terms, as it guarantees the scalability of vertices. Moreover, we prove that the k-regularity of a matrix is necessary and sufficient for substituting mod-k cuts for rank-1 Chvatal-Gomory cuts. In the second part of the thesis we introduce binet matrices, an extension of network matrices to bidirected graphs. We provide an algorithm to calculate the columns of a binet matrix using the underlying graphical structure. Using this method, we prove some results about binet matrices and demonstrate that several interesting classes of matrices are binet. We show that binet matrices are 2-regular, therefore they provide half-integral vertices for a polyhedron with a binet constraint matrix and integral right hand side vector. We also prove that optimization on such a polyhedron can be carried out very efficiently, as there exists an extension of the network simplex method for binet matrices. Furthermore, the integer optimization with binet matrices is equivalent to solving a matching problem. We also describe the connection of k-regular and binet matrices to other parts of combinatorial optimization, notably to matroid theory and regular vectorspaces.