We define binet matrices, which furnish a direct generalization of totally unimodular network matrices and arise from the node-edge incidence matrices of bidirected graphs in the same way as network matrices do from directed graphs. We develop the necessary theory, give binet representations for interesting sets of matrices, characterize totally unimodular binet matrices and discuss the recognition problem. We also prove that binet constraint matrices guarantee half-integral optimal solutions to linear programs.

No abstract

This paper deals with linear and integer programming problems in which the constraint matrix is a binet matrix. Binet matrices are pivoted versions of the node-edge incidence matrices of bidirected graphs. It is shown that efficient methods are available to solve such optimization problems. Linear programs can be solved with the generalized network simplex method, while integer programs are converted to a matching problem. It is also proved that an integral binet matrix has strong Chvátal rank 1. An example of binet matrices, namely matrices with at most three non-zeros per row, is given.

No abstract

a b s t r a c tWe present a polynomial time algorithm to construct a bidirected graph for any totally unimodular matrix B by finding node-edge incidence matrices Q and S such that QB = S.Seymour's famous decomposition theorem for regular matroids states that any totally unimodular (TU) matrix can be constructed through a series of composition operations called k-sums starting from network matrices and their transposes and two compact representation matrices B 1 , B 2 of a certain ten element matroid. Given that B 1 , B 2 are binet matrices we examine the k-sums of network and binet matrices. It is shown that the k-sum of a network and a binet matrix is a binet matrix, but binet matrices are not closed under this operation for k = 2, 3. A new class of matrices is introduced, the so-called tour matrices, which generalise network, binet and totally unimodular matrices. For any such matrix there exists a bidirected graph such that the columns represent a collection of closed tours in the graph. It is shown that tour matrices are closed under k-sums, as well as under pivoting and other elementary operations on their rows and columns. Given the constructive proofs of the above results regarding the k-sum operation and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any TU matrix.