A Minimax Theorem and a Dulmage–Mendelsohn Type Decomposition for a Class of Generic Partitioned Matrices

Abstract

This paper discusses an extension of the Dulmage--Mendelsohn decomposition for a certain class of matrices whose row-set and column-set are divided into couples or singletons. A genericity assumption is imposed and an admissible transformation is defined in respect of this partition structure. Extensions of the Konig--Egervary theorem and the Hall--Ore theorem are established. The latter states that the rank of such a matrix is characterized by the minimum value of a submodular function, of which the set of minimizers yields a canonical block-triangularization under the admissible transformations.